SciPost Phys. Lect. Notes 78 (2024)

Aspects of 4d supersymmetric dynamics and geometry
Shlomo S. Razamat1 , Evyatar Sabag1,2 , Orr Sela1,3 and Gabi Zafrir4,5,6
1 Department of Physics, Technion, Haifa, 32000, Israel
2 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
3 Mani L. Bhaumik ITP, Department of Physics and Astronomy, UCLA, CA 90095, USA
4 Dipartimento di Fisica, Universit‘a di Milano-Bicocca INFN, I-20126 Milano, Italy
5 YITP, Stony Brook University, Stony Brook, NY 11794-3840, USA
6 SCGP, Stony Brook University, Stony Brook, NY 11794-3840, USA

Abstract
In this set of five Lectures we present a basic toolbox to discuss the dynamics of four
dimensional supersymmetric quantum field theories. In particular we overview the program of geometrically engineering the four dimensional supersymmetric models as compactifications of six dimensional SCFTs. We discuss how strong coupling phenomena infour dimensions, such as duality and emergence of symmetry, can be naturally imbedded in the geometric constructions. The Lectures mostly review results which previously
appeared in the literature but also contain some unpublished derivations.
Copyright S. S. Razamat et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 23-05-2022
Accepted 22-01-2024
Check forupdatesPublished 12-02-2024
doi:10.21468/SciPostPhysLectNotes.78

Contents
1 Introduction

22 Lecture I: The basic toolbox
2.1 Symmetry, anomalies, and ’t Hooft anomaly matching
2.2 a-maximization and superconformal R-symmetry
2.3 Beta functions, deformations, and conformal manifolds
2.4 Supersymmetric RG-flow invariants

9
9
11
14
22

3 Lecture II: Examples of strong coupling dynamics
3.1 IR dualities
3.2 Emergence of symmetry in the IR
3.3 Conformal dualities
3.4 Exercise: Lagrangian dual of a class S theory

26
27
30
32
38

4 Lecture III: Across dimensional dualities, an example
4.1 Six dimensions
4.2 Compactification to 4d
4.3 Reduction to 5d and punctures
4.4 Four dimensions

40
42
44
46
48

1

SciPost Phys. Lect. Notes 78 (2024)

5 Lecture IV: Compactifications of the E-string
5.1 Six and five dimensions
5.2 4d theories from genus g Riemann surface with zero flux
5.3 Decomposing the surface into pairs of pants
5.4 Closing punctures and comments

56
56
58
62
70

6 Lecture V: An algorithm for deriving across dimensional dualities
6.1 The general idea
6.2 Compactification of the rank one E-string SCFT
6.3 Compactification of minimal (D, D) conformal matter theories
6.4 From tubes to trinions

81
81
84
91
96

7 Discussion and comments

103

A Superconformal algebra in 4d

113

B Leigh-Strassler argument for conformal manifolds

117

C Computing Kähler quotients with Hilbert series

118

D Class S interpretation of the exercise

119

E Basics of 6d SCFTs
E.1 The Green-Schwarz mechanism in 6d
E.2 The 6d anomaly polynomial eight-form
E.3 Example: The rank one E-string anomaly polynomial

123
124
125
127

F

127
127
128
130

Twisting, fluxes, and integrating anomaly polynomials
F.1 The topological twist and compactification
F.2 Integrating the anomaly polynomial
F.3 Examples

G Supersymmetric boundary conditions

132

References

134

1

Introduction

Quantum field theory is a framework to study the dynamics of a wide variety of quantum
systems. One of the interesting open problems in understanding the predictions of this framework is the question of strong coupling. Strong coupling problems have numerous avatars.
For example, we can start from a free field theory and turn on relevant deformations flowing
to strong coupling in the infra-red (IR). We then can ask questions about the deep IR physics
which are hard to answer directly using the UV weakly coupled starting point.
Such issues become more tractable once we introduce enough supersymmetry. In 4d,
which will be the focus of these Lectures, enough means four supercharges, i.e. minimal
N = 1 supersymmetry. With this amount of supersymmetry one can use a wide variety oftechniques, ultimately related to holomorphy of various quantities [1], to deduce non pertur-2

1

SciPost Phys. Lect. Notes 78 (2024)

batively certain conclusions about strongly coupled phases of QFTs. Typically the power ofholomorphy gives us a variety of quantities we can count: Most, if not all, of the exact results
one can obtain about supersymmetric QFTs can be related to counting problems.1

Dualities and emergence of symmetry
As an example of a strong coupling effect one can discuss is the notion of duality. A given CFT
A in the UV deformed by a relevant perturbation can flow in the IR to a theory (which canbe free, gapped, or interacting CFT) which is equivalent to a theory obtained by staring withanother CFT B and some relevant deformation. When an RG flow is involved we will referto such dualities as IR dualities. In case the deformations preserve conformality, that is they
QFTare
QFTA
B
exactly marginal,
we will refer to these effects as conformal dualities. See Figure 1. In
UVthe last 25 years, following
6d in particular [2], numerous examples of such dualities have been
conjectured
a wide variety of exact supersymmetric techniques. The canonical example
2
𝒞2B
𝒞usingA
is the IR equivalence (in a certain range of parameters) of SU(N ) SQCD with N f flavors andSU(N f −N ) SQCD with N f flavors, gauge singlet fields and a superpotential. The IR fixed point
is fully determined (or labeled) by a choice of a pair, UV CFT and the relevant deformation. The
statement of the duality is4dthat this labeling is not unique and moreover might be not unique in
an interesting way (e.g. the UV theories having different gauge groups). A canonical example
of a conformal duality is the statementthat N = 4 SYM with gauge groups G and L G (theIRG)4d reside at two cusps of the same conformal manifold (see e.g. [3] for a
QFTℳc Langlands dual ofdetailed discussion).
QFT0 = T6d

𝒞2

UV

QFT1

QFT2

QFT1

D̃

UV

QFT2

QFT3 = T4d

IR

ℳc

QFT3

D̃ − ΔD

IR
QFT2

QFT3

Figure 1: On the left we have a depiction of IR duality: Two different UV QFTs flowing
to the same conformal manifold in the IR. In this paper we will refer to theories
differing by exactly marginal deformations as equivalent. On the right we have a
depiction of the notion of UV duality: Two different QFTs in the IR obtainable as two
different deformations of a single CFT in the UV. This notions becomes very usefulonce one considers flows starting with a strongly coupled CFT leading to weakly
coupled QFTs in the IR. Examples of these include compactifications on a circle of
6d SCFTs.
Another interesting strong coupling effect is the emergence of symmetry in the IR. The
global symmetry of the IR fixed point can be larger than the one of the UV starting point andthe interesting question is whether this can be understood directly in the UV. A simple example
is SU(2) with N f flavors and symmetry SU(2N f ) describing in the IR the SU(N f − 2) SQCD
with N f flavors with only the SU(N f )2 × U(1) symmetry visible in the UV of the latter, see e.g.
1

The counting problem might be in the theory of our interest or maybe in a higher dimensional theory reducing
to it. For example the Sd partition functions in d dimensions can be related to various indices, Sd × S1 partition
functions, in d + 1 dimensions.

3

SciPost Phys. Lect. Notes 78 (2024)

CFT6d GF

6d

𝒞g , ℱ

ℳc

CFT4d

QFT

𝒞2B

𝒞2A

RG

4d

QFT4d

QFTB

UV

IR

ℳc

Figure 2: Depiction of a dualityof
QFT0 = T6d
QFT0 across= T6d dimensions. Here we give an example
UV
a duality between 6d flow and a 4d flow. The 4d theory might be UV complete
5d1(asymptotically free) or UV5dcompleted
by2 the 6d theory.

Q

UV

QFT1
QFT2
[4]. Other examples involve
emergence𝒞of supersymmetry
in the flow:QFT
e.g.
QFT2
𝒞2
1 using Intriligator2
Seiberg duality [5] the SU(2) N = 4 SYM can be obtained by starting from SU(2) × SU(2)N = 1 SQCD with the charged matter consisting of three bi-fundamentals.3
An interesting question is thus to bundle all of the scattered instances of interesting strong
coupling dynamics into some uniform framework whichIRwill give some sort of an explanationIRor an understanding of when
QFT3 = T4dℳc motivation for such
QFT3 = T4d occur. Yet anotherℳcand how these phenomenaa framework can be found by asking a sort of an inverse question: Given a strongly coupled
SCFT with given properties to find a weakly couple UV theory which flows to it. The UV theory
might exhibit less symmetry than the IR one, and in fact it often has to do so. For example,
listing all the 4d N = 2 Lagrangians leading to interacting SCFTs [9] one just does not findcandidate descriptions of many of 4d N = 2 SCFTs which can be obtained using a variety ofmore abstract techniques, e.g. coming from string theory.

QFT1

2

Dualities across dimensions
Typically one considers dualities between QFTs starting from two theories in the same number of dimensions and ending in the same number of dimensions. However, in recent years,
following the seminal work of [8], it has been realized that much can be achieved, in particular answering the questions posed above, if one discusses flows across dimensions. Let
us first then define the notion of an IR duality across dimensions. We can start from a highere = 6, and deform the theory by placing it on
dimensional CFT, in this paper we will start in D
a compact geometry of dimension D, e.g. a Riemann surface with D = 2. Studying this setup
at low energy, i.e. much lower than the scale set by the compact geometry, we will arrive ate − D dimensional QFT. An interesting question is whether there exists a D
e−D
an effective D
dimensional weakly coupled QFT flowing to the same effective theory in the IR. If such a thee deformed theory and the De − D one as being IR dual acrossory exists we will refer to the Ddimensions. See Figure 2.
See also [6] for recent discussion.
Let us mention in passing here for the experts that this duality has a class S [7,8] interpretation. One can take
two TN trinions and glue them together with N = 1 vector multiplets; flip the two maximal punctures; close thenone of the two punctures completely and the other one to a minimal puncture. This theory has the same geometricdata as the one of genus one with a single minimal puncture and N = 2 preserving flux, which is N = 4 SYM
with a decoupled hypermultiplet. The N = 2 case is then the Intriligator-Seiberg duality while for higher N one
has novel strongly coupled generalizations of this. In our Lectures we will review the notions needed to understandthe statements in this footnote.
2
3

4

ℳc

4d

ℳc

FT1

ℳc

QFT0 = T6d

𝒞2

QFT2

QFT3 = T4d

QFT1

IR

SciPost Phys. Lect. Notes
QFT 78 (2024)

QFT0 = T6d

UV

5d2

5d1

ℳc

IR

CFT4d

𝒞2

4d

UV

QFT1

QFT2

IR

ℳc

QFT3

QFT2

QFT3 = T4dℳc

Figure 3: 4d dynamics from different decompositions of the compactification geometry.
e − D dimensional QFT by a pair (CFT, deformation)
Here we thus can also label the IR D
but this might involve a higher dimensional CFT and a geometric deformation. As we will
be mainly interested in 4d physics our starting points will be in 6d (and 5d) where interesting SCFTs are all strongly coupled and do not have a simple field theoretic definition, seee.g. [10]. As the starting point is strongly coupled, such geometric constructions often leadto 4d theories with properties which are hard or impossible to engineer directly in 4d insisting on all the symmetries being manifest. This in particular led to many such theories beingreferred to as non-Lagrangian. Across dimensional duals, if existing, thus would provide fora Lagrangian definition of such SCFTs. An example of such a duality is a geometric, class
S [7, 8], construction as compactification on punctured spheres of a (2, 0) 6d SCFT of certain
N = 2 Argyres-Douglas SCFTs [11], and an alternative description of these SCFTs startingwith a N = 1 weakly coupled gauge theory in 4d [12].
As with in-dimension dualities some of the symmetries of the IR QFT might be explicit in
the UV in one description but emergent in the other. Systematically understanding such acrossdimensional dualities will give us a handle over understanding of emergence of symmetry. Infact the appearance of geometry in the construction gives us a useful knob to start and builda systematic framework to understand emergence of symmetry and duality.

4d dynamics from across dimensional dualities
The main idea behind deriving 4d dualities and emergence of symmetry phenomena from 6dstems from the following factorization property of the constructions. One considers compactifying a given 6d SCFT, T6d , on surface C such that the surface can be written as,

C = C1 ⊕ C2 ⊕ · · · ⊕ Cℓ .

(1)

Here Ci are punctured surfaces and ⊕ is the geometric operation of gluing two surfaces along
a puncture. The compactification might be parametrized by additional geometric data, suchas flux for a global symmetry supported on the surface. The operation ⊕ then associates to the
combined surface a sum of these fluxes. We will call a set of surface Ci complete if any surface C
can be constructed using these. We then first seek for across dimensional dualities associatinga pair of 4d weakly coupled CFTs and deformations (TCi , ∆i ) to (T6d , Ci ). Assuming that sucha dictionary between a complete set of surfaces Ci and 4d theories could be found one can find5IRSciPost Phys. Lect. Notes 78 (2024)

a theory dual across dimensions to (T6d , C ),(TC , ∆) = (TC1 , ∆1 ) ⊗ (TC2 , ∆2 ) ⊗ · · · ⊗ (TCℓ , ∆ℓ ) .

(2)

Here ⊗ is a field theoretic operation in 4d. This can involve gauging symmetry associatedwith punctures (in a way we will discuss in detail in the paper), adding fields, and turning on
various superpotentials couplings. The precise meaning of ⊗ will depend on the 6d SCFT andvarious other choices. For example, the complete set of Ci might not be unique; one can havedifferent types of punctures (following from UV dualities exemplified in Figure 1), etc. In thestatement (2) we have several RG flows, and thus hidden in it there is an assumption thatthese types of flows commute. Commutativity of flows is a non-trivial statement (see e.g. [13]and discussion below). However, assuming it and the statement (2) one can arrive at a large
web of consistent results which supports the validity of the assumption.
The fact that such a construction exists is highly non-trivial: However as we will discussin this review in various examples it can be worked out explicitly and by now somewhat systematically. An example of such a construction is the derivation of all compactifications of theA1 (2, 0) 6d SCFT using tri-fundamentals of SU(2) as across dimensional duals to compactifications on certain three-punctured spheres [8]. Given such a structure one can then generatevarious examples of dualities and instances of IR emergence of symmetry systematically. For
example, if a given surface can be decomposed in more than one way,
(1)

(1)

(1)

(2)

(2)

(2)

C = C1 ⊕ C2 ⊕ · · · ⊕ Cℓ = C1 ⊕ C2 ⊕ · · · ⊕ Cℓ′ ,(3)

we will obtain different field theoretic descriptions of (TC , ∆) which should be IR equivalent
by construction,
(1)

(1)

(1)

(TC , ∆) = (TC (1) , ∆1 ) ⊗ (TC (1) , ∆2 ) ⊗ · · · ⊗ (TC (1) , ∆ℓ ) ,
1

(2)

(4)

ℓ

2

(2)

(2)

= (TC (2) , ∆1 ) ⊗ (TC (2) , ∆2 ) ⊗ · · · ⊗ (TC (2) , ∆ℓ′ ) .
1

ℓ′

2

Moreover the geometry C might preserve more symmetry than some of the Ci building blocks.
Then the corresponding theories (TCi , ∆i ) would have less symmetry than (TC , ∆). However combining them together using (2) should give in the IR the expected symmetry, if theconstruction is correct. Building the necessary toolkit to discuss such correspondences anddiscussing the systematics of these constructions will be the main goal of this review.
CFT6d GF

6d

𝒞g , ℱ

QFTB

UV

QFTA

𝒞2B

6d

𝒞2A

UV

RG

4d

4d

QFT0 = T6d

UV

5d2

𝒞

2

ℳc

IR

CFT4d

QFT4d

IR

Figure 4: A depiction
QFT0 =ofT6da 6d duality. Starting from two different theories in 6d and
QFT1
QFT2
UV onedeforming them by different geometries
can flow to equivalent
theories in 4d.

QFT1

D̃

UV

QFT2

We will discuss two explicit instances of collections of across dimension dualities, one
starting with T
by pure SU(3) SYM on its tensor branch and another with
6d which is described
2
QFT
1

𝒞

QFT2

6
IR

IR

ℳc

QFT3

D̃ −

IR
QFT2

QFT

SciPost Phys. Lect. Notes 78 (2024)

T6d being the rank one E-string. These two cases turn out to be rather simple and amenable
to a variety of ad hoc techniques, which we will discuss in detail. The systematic treatment ofcompactifications starting with a large class of more general T6d can be done understanding
first compactifications on two punctured spheres. The problem of understanding reductions
on such surfaces can be directly related to understanding duality domain walls in 5d [14].
We will discuss this procedure using again E-string as an explicit example. Understanding
systematically compactifications on surfaces with more punctures can be done studying theinterplay between 6d flows and across dimension flows, which we will soon describe.

6d dualities
Of course once we allow for across dimensional dualities we can consider starting and finishingin different combinations of dimensions. There has been a lot of work for example on compactifying 6d theories down to three dimensions and constructing 3d Lagrangians for these, seee.g. [15]. However, there is another interesting phenomenon that we want to mention here.
A
B
One can start from two different 6d SCFTs, T6d
and T6d, deform by placing them on differentAB
geometries, C and C , and consider the resulting four dimensional theories. In certain casesAB
the 4d theories might be IR equivalent and we thus can refer to the pairs (T6d
, C A) and (T6d
, CB)
as being 6d dual to each other. By now there are numerous examples of such dualities, though
there is no systematic understanding of these. It is likely that to gain such an understanding one would need to exploit various string/M-theoretic constructions. For example, various
compactifications of (2, 0) theories on spheres with punctures leading to N = 2 theories in
4d turn out to be equivalent to compactifications on tori of different (1, 0) theories [16–18];
compactifications on certain punctured spheres of N5 branes probing A-type singularity is 6d
dual to compactifications on tori with flux of M5 branes probing D-type singularity [19]. We
will discuss yet another example of this phenomenon (Rank-one E-string on genus two surface
without flux is the same as minimal SU(3) 6d SCFT on a four punctured sphere).

Interplay between flows in different dimensions
6d theories do not possess interesting supersymmetric relevant or marginal deformations [20].
However, given a 6d SCFT one can trigger an interesting flow by exploring the moduli spaceof vacua, i.e. turning on vacuum expectation values for some operators. One then can wonder
how flows in six dimensions and across dimensions are interrelated. It turns out that theanswer to this question is rather interesting and understanding it gives an explicit tool toderive compactifications on surfaces with more than two punctures. The basic idea is depicted
in Figure 5. Let us consider two QFTs in 6d related by a flow triggered by a vev to some
operator O6d , QF T1 and QF T2 . Next we compactify QF T1 on some surface CA with somevalue of flux for the 6d global symmetry. Usually one can find the 4d analogue, O4d , of theoperator O6d , the vev of which connects QF T1 and QF T2 in 6d. Turning on a vev to O4d weflow then to a new QFT in 4d. A natural question is whether there is a surface CB (and somevalue of flux) such that the same 4d QFT can be reached starting with QF T2 in 6d. It turns outthat the answer is positive but CB differs from CA by the number of punctures: The difference
being determined by the value of flux for symmetries under which O6d is charged. Thus
understanding say compactifications on two punctured spheres for QF T1 we can systematicallyderive compactifications on spheres with higher number of punctures for QF T2 . We will discussin detail this procedure for a sequence of 6d SCFTs called (DN +3 , DN +3 ) minimal conformal
matter theories: (1, 0) SCFTs residing on a single M5 brane probing DN +3 singularity in Mtheory. Models with different values of N are related by RG flows reducing the value of N .
The simplest model with N = 1 turns out to be the rank one E-string. In particular this will
give us yet another derivation of the three punctured spheres for the E-string theory.
7

SciPost Phys. Lect. Notes 78 (2024)

QFT1

Vev for 𝒪6d

UV
QFT2

𝒞2A

QFT1

Vev for 𝒪4d

3d

QFT4

𝒞1 = 𝕊1

4d

QFT′1
IR

Figure 5: Commuting diagram of flows in 4d and in 6d triggered by vevs, and acrossdimensions triggered by geometry.

QFT′3

Outline of the paper
The idea of these Lectures is to allow a reader familiar with the basics of supersymmetric dynamics in 4d (e.g. at the level of first chapters of [21]) to familiarize themselves in detail,and in a self contained way, with the techniques and results used to geometrically construct
4d dynamics. We tried to either explicitly review all the needed material, or to cite a paper
discussing manifestly and in detail the needed points. Although string/M/F-theoretic considerations are very useful in understanding various aspects of our discussion, we restricted topurely field theoretic exposition for the sake of the Lectures being self contained. These Lectures are not intended to be an exhaustive review of the subject but rather a pedagogical and selfcontained exposition of a particular slice of recent developments.
The paper is structured as follows. In section 2 we overview the basic techniques to deal
non-perturbatively with N = 1 supersymmetric QFTs in 4d. This involves in particular the review of exact tools to extract information about the IR fixed point in 4d flows. In section 3 wediscuss in detail examples of IR dualities, conformal dualities, and IR emergence of symmetry.
We discuss several test cases to illustrate the applications of the techniques of section 2. Parts
of Lecture II are mainly based on results reported in [22, 23]. Next in section 4 we analyze infull detail the compactification of 6d minimal SU(3) SCFT to 4d. In particular we will discuss
the 6d SCFT itself, its reduction to 5d, and the resulting 4d theories. This is a very tractablecase (e.g. the analysis is simplified by the 6d theory not having continuous global (non-R)
symmetry). We will illustrate various understandings following from compactifications using
this example. This section is based on results of [24]. In section 5 we discuss in detail compactifications of the rank one E-string. Here we will use non generic but simple techniques to
get to the result in a non-cumbersome manner. This section is mainly based on [25] as wellas [22,26]. In section 6 we discuss a systematic approach of studying 5d duality domain walls
and commutative diagrams of RG flows. Here the discussion mainly follows the results derived
in [27]. We use these techniques to arrive at the 4d models obtained in compactifications of
the E-string theory. Finally in section 7 we discuss some generalities of geometrically engineering 4d SQFTs starting from 6d SCFTs, comment on subjects not covered in the Lectures, andmake general remarks. Several appendices include pedagogically various topics in 4d/5d/6d
supersymmetric physics as well as give more details of some of the computations in the bulkof the paper.

8

QFT′2

QFT3

ℱB

QFT3

4d

6d

𝒞2B

ℱA

QFT2

CFTUV1

CFTUV2
SciPost Phys. Lect. Notes 78 (2024)

𝒢IR
CFTIR
RG Flow
CFTUV + Δ

Δ=
6d SCFTUV + Δ

𝒢UV

6d SCFTCFTUVFigure 6: A depiction of an RG flow. We start with some CFT in the UV. This C F TU V
can be a free theory or some non trivial strongly coupled theory. We deform it by arelevant deformation ∆: It can be e.g. a relevant interaction, a vacuum expectation
value, or turning on a non trivial background. We denote the symmetry of the theory
after the deformation as GU V . In the IR the model flows to a new fixed point C F TIR .
The global symmetry of the fixed point G IR might or might not be the same as GU V .

2

Lecture I: The basic toolbox

We wish to understand the following general set of question. Starting from a CFT in the UV(C F TU V ) and deforming it by introducing a scale, we trigger an RG flow. The properties ofthe deformed QFT depend on the energy scale at which we probe the physics. We are theninterested to understand what happens when we go to extremely low energies, far below theenergy scale set by the deformation, see Figure 6 for an illustration. This question is often hardto answer, since as we lower the energy scale at which we probe the physics the description interms of the degrees of freedom of C F TU V can become strongly coupled. For example, as we
will review soon, if we start from a free theory as C F TU V and turn on a relevant deformation,a variety of interesting behaviors can emerge in the IR: The theory can flow to an interacting
strongly coupled conformal theory, it might flow to a weakly coupled gauge theory, and theflow can also be gapped with no propagating degrees of freedom remaining in the IR.

2.1

Symmetry, anomalies, and ’t Hooft anomaly matching

Without simplifying assumptions, such as supersymmetry that we will soon introduce, answering the posed questions is rather hard. Nevertheless, we do have certain tools which we canuse quite generally. Not surprisingly they have something to do with symmetry. Let us assume that after deforming C F TU V the global symmetry of the resulting QFT is GU V . Here wewill focus on zero-form continuous symmetries, though one can consider generalizations of
the discussion to higher form symmetries [28] and higher group structures [29]. Typicallythe deformation breaks explicitly some of the symmetry of C F TU V and some of the symmetry
might also be spontaneously broken: Let us assume GU V is the surviving fraction of the symmetry. This symmetry will be preserved during RG flow. Thus, we expect that the symmetryof the theory in the IR should be GU V . There are however two caveats. First, GU V might not
act faithfully in the IR. In particular in an extreme case it might not act at all, for example
when the theory develops a gap in the IR. In this case, GU V will not act manifestly in the IR.
9SciPost Phys. Lect. Notes 78 (2024)

Second, the symmetry in the IR can actually be bigger than GU V . Intuitively, some of the interactions/degrees of freedom which break a certain larger symmetry group, of which GU V isa subgroup, might be washed away/gapped in the IR. In such situations we will say that there
is an enhancement (or emergence) of the symmetry in the IR and the symmetry group G IR
might be bigger than GU V . We will denote by G IR the symmetry group of the IR fixed point if
the theory flows to an SCFT, or the symmetry of the weakly coupled gauge theory if it flows tosuch a theory.
Importantly, we can say something more about the symmetry. If a theory possesses a global
symmetry with a corresponding conserved current4 Jµ , we can turn on background gauge
fields Aµ , valued in the Lie algebra of some sub-group of the symmetry group, coupled to thisconserved current, and compute the effective action Γ [A]. As the current is conserved thegauge field comes with a gauge symmetry,
A → Ag ≡ g · A · g −1 + (d g) · g −1 ,(5)where g is an element of the symmetry (sub)group. Then we can try to promote Aµ to be
dynamical fields. However, there might be an obstruction to doing so, which goes under thename of ‘t Hooft anomaly. The obstruction comes about as the effective action of the theory
might or might not be invariant under (5),?

Γ [Ag ] = Γ [A] .

(6)

If the equality does not hold we say that the (sub)group of the symmetry has a ‘t Hooft anomaly.
In particular this means that the symmetry cannot be gauged, i.e. the gauge fields Aµ cannot
be promoted to dynamical fields. See [30] for a comprehensive introduction to the subjectof anomalies and [31] for a more recent and general discussion. The important fact about‘t Hooft anomalies is that they are quantifiable. There are various ways to understand this andlet us here mention two of them. First, the anomaly of a continuous symmetry can be captured
in D = 2n dimensions by an n + 1 point one loop amplitude involving the conserved currents.
In particular, say in D = 4, this is proportional to,aG1 ,G2 ,G3 ≡ TrR G1 G2 G3 .

(7)

Here R is the representation of the chiral fermions of the model (if we have a description of
the theory in terms of a Lagrangian) under the product of the three groups G1 × G2 × G3 , which
are the groups corresponding to the three currents. The trace is taken over this representation.
For example,
PN if G3 i all correspond to the same U(1) symmetry then the ‘t Hooft anomaly is justgiven by l=1 ql , where ql are the charges of the N Weyl fermions of the model.
Another way to think about the anomaly is as follows. The failure of the effective action to
be invariant cannot take any form and is constrained by Wess-Zumino consistency conditions to
be related to what is known as the anomaly polynomial I D+2 (where D is the number of spacetime dimensions which is assumed to be even). I D+2 is a homogeneous polynomial of degree
D+2 in characteristic classes built from gauge fields (including gravitational) corresponding to
the symmetries of the system. The coefficients of the polynomial encode the ‘t Hooft anomaliesof the theory. For example a term of the form,n123

aG1 ,G2 ,G3
6

Tr F1 ∧ F2 ∧ F3 ⊂ I6 ,(8)

captures the anomaly aG1 ,G2 ,G3 defined in (7). Here n123 is a simple numerical factor depending
on how many of the three Fi correspond to different groups. If all are the same then n123 = 1,
4
One can make this discussion more general and abstract by thinking about the conserved charge corresponding
to a zero-form symmetry as certain co-dimension one topological operators in the QFT (and higher co-dimension
when the symmetry is of a higher form) [28].

10SciPost Phys. Lect. Notes 78 (2024)

if two are the same n123 = 3 and if all are different it is equal to 6. See Appendix E for moredetails on the 6d anomaly polynomial.
‘t Hooft anomalies are useful for us since they don’t change during the RG-flow and thus
are the same in the UV and the IR: This fact goes under the name of ‘t Hooft anomaly matchingcondition. There are various ways to see this. First, we can add to the theory a number ofWeyl fermions (called spectators), such that the combined anomaly of the theory of interestand the additional fermions is zero. Then we can weakly gauge the symmetry: That is turnon infinitesimally small gauge couplings. Assuming we can always stay in a regime where thegauge coupling is small, we can flow to the IR where we decouple the spectators. As there is noobstruction to the gauging in the IR, and we decouple the same fermions, the anomaly should
not change in the process. Yet another argument for the non-renormalization is through what
is called anomaly inflow. One way to render the symmetry non-anomalous is to realize thetheory on the boundary of a D + 1 dimensional space with a background field A living in D + 1
dimensions. Then, we can add a Chern-Simons term in the bulk that cancels the anomalyof the boundary theory. This will again remove the obstruction from gauging the symmetry
which should hold in the UV as well as in the IR.
Thus, we learn that the anomaly polynomial I6 computed in the UV should be the same as
the one computed in the IR, no matter what is the IR behavior of our deformation. This is true
for the symmetries we can identify in the UV. At the IR fixed point new symmetries can emerge
and as we cannot identify them in the UV their anomalies do not have to be zero. One way tounderstand this is that we can move out of the IR fixed point by irrelevant deformations which
explicitly break the emergent symmetries and thus invalidate the ‘t Hooft anomaly matchingargument. On the other hand, if a sub-group of the symmetry group does not act faithfullyin the IR its anomaly has to be zero also in the UV due to the ‘t Hooft anomaly matchingargument.5

2.2

a-maximization and superconformal R-symmetry

Until now our discussion did not involve supersymmetry at all. Indeed, without supersymmetry we do not have at the moment useful robust tools beyond matching symmetries andanomalies to understand the physics in the IR. However with supersymmetry there are morethings that we can say. Let us start the discussion of supersymmetric theories in D = 4 withwhat the interplay between the supersymmetry and ‘t Hooft anomalies can give us.
We will assume throughout these Lectures that we are dealing with N = 1 supersymmetrictheories in 4d . Moreover we will assume that these theories possess an R-symmetry.6 A U(1)R-symmetry is a necessary ingredient of the N = 1 superconformal group, as for example it appears on the right-hand side of anti-commutation relations between supercharges Q α and theirsuperconformal counterparts Sα . See Appendix A for a summary of the N = 1 superconformalgroup in 4d. The SCFT in the UV thus has an R-symmetry which is part of the superconformalgroup. We will only discuss deformations which preserve N = 1 supersymmetry. We will also
assume that some combination of this R-symmetry and an abelian subgroup of the global symmetry group of C F TU V is not broken by the deformation, though of course as we introduce ascale the conformal symmetry is broken. In the IR, if we arrive to a conformal fixed point, weagain acquire the superconformal R-symmetry. However, the superconformal symmetry in the
UV and in the IR might not be the same symmetry. Nevertheless, under certain assumptions
that we will detail, the fact that the R-symmetry is intimately related to the superconformalgroup allows us to determine it in the IR.
5Note that this is not true for the case of spontaneous symmetry breaking. A symmetry with a non-trivial ‘t Hooft
anomaly in the UV can be spontaneously broken in the IR.
6
The R-symmetry can be broken by superpotentials or be anomalous in the UV. We thus will not deal with suchsituations.

11

SciPost Phys. Lect. Notes 78 (2024)

Any conformal theory, supersymmetric or not, in four dimensions has two important numbers associated to it: These are referred to as the a and the c conformal anomalies. The
conformal anomalies measure, among other things, the failure of the expectation value of the
trace of the stress-energy tensor to vanish when the theory is placed on a curved backgroundwith metric gµν ,ca〈T µ µ 〉 =
W2 −
E4 ,
(9)
16π2
16π2
where W is the Weyl tensor and E4 is the Euler density, both built from certain combinationsof the metric gµν and its derivatives [32].7 In superconformal theories, as the stress energy
tensor and the R-symmetry are part of the same symmetry algebra the various anomalies areinterrelated. It is possible for example to utilize these relations to arrive at the following
extremely useful statements [35],a=93
Tr R3 −Tr R ,
32
32

c=

9
5
Tr R3 −Tr R .
32
32(10)Here it is important that R is the R-symmetry in the superconformal group. Tr R3 is the ‘t Hooft
anomaly corresponding to Tr F 3 term in the anomaly polynomial we have discussed above,with F being the field strength for the R-symmetry background gauge field. Whereas Tr R is themixed R-symmetry gravity anomaly, which in the anomaly polynomial appears as a coefficientof a term involving Tr F and a certain Pontryagin class computed using the background metric.
Exploiting the interplay between supersymmetry and the R-symmetry in a conformal theory one can also arrive at the conclusion that the mixed ‘t Hooft anomalies between any global
U(1) symmetry (which is not an R-symmetry) and the superconformal R-symmetry are also
related to the mixed gravitational-U(1) anomaly [36],
Tr U(1) = 9 Tr U(1) R2 .

(11)

Moreover, the positivity of the two point function of the currents of the global U(1) symmetry
can be related to the negativity of yet another ‘t Hooft anomaly,
Tr U(1)2 R < 0 .

(12)Combining all these observations Intriligator and Wecht arrived at a very simple procedure to
determine the R-symmetry of the IR fixed point by knowing all the abelian symmetries andthe R-symmetry preserved along the RG-flow [36]. One defines the following trial a anomaly,a({α}) =93
Tr(R + αi U(1)(i) )3 −Tr(R + αi U(1)(i) ) .
32
32

(13)

Here αi are arbitrary real numbers associated to the i-th U(1) symmetry: The summation over
i is implied in the equation. Now we notice that,32 ∂a({α}) = 9 Tr(R + αi U(1)(i) )2 U(1)( j) − Tr U(1)( j) = 0 .
3 ∂ αj(14)

The last equality holds if (R+αi U(1)(i) ) is the superconformal R-symmetry following (11). On
the other hand,
16
∂
∂
βj
βk
a({α}) = Tr(R + αi U(1)(i) )β j U(1)( j) βk U(1)(k) < 0 ,27 ∂ α j ∂ αk(15)

7
In any conformal theory the a conformal anomaly is smaller at the IR fixed point relative to the UV fixed
point [33, 34].

12

SciPost Phys. Lect. Notes 78 (2024)

where βi are arbitrary real numbers and the last inequality follows from (12) assuming again
R + αi U(1)(i) ) is the superconformal R-symmetry. This thus implies that:
The superconformal R-symmetry maximizes the trial a({α}) .
The procedure of obtaining the superconformal R-symmetry discussed here is called amaximization. This statement however has an important caveat. We assume that we haveidentified all the U(1) symmetries that can possibly mix with the R-symmetry to produce thesuperconformal one. However, as we already discussed some of these symmetries can only
emerge in the IR. This possibility should always be kept in mind, as it is usually hard to rule
it out. In some cases, certain indications that this has to be the case can be derived. For
example, using certain unitarity bounds, superconformal representation theory implies that
the R-charge of any chiral operator in the theory has to be bigger or equal to 32 . The R-charge
saturates this bound only if the operator is a free chiral field. If using a-maximization leads
to certain chiral operators violating these bounds, then some of the assumptions going into
the computation have to be wrong. A natural way for this to happen is if some of the abelian
symmetries in the IR have been missed [37].
Using the relation (10) we can compute the conformal anomalies of free fields, which will
be useful for us in what follows. A free chiral superfield Q has an R-charge of 23 . This is the Rcharge of the scalar component of the superfield. The R-charge of the Weyl fermion is shifted8
by −1 and thus the conformal anomalies of the free chiral field are,

3


9 2
3 2
1
a=
−1 −
−1 =
,
32 3
32 3
48


3


9 2
5 2
1
c=
−1 −
−1 =
.
32 3
32 3
24

(16)

For the free vector superfield the R-symmetry is zero, and thus the R-symmetry of the gaugino
is +1. This implies the following anomalies,
a=

9
3
3
−
=
≡ av ,
32 32 16

c=

9
5
1
−
= ≡ cv .
32 32 8(17)

For future convenience we also define the contributions to the conformal anomalies of chiralfields of general R-charge r to be,aR (r) =93(r − 1)3 −(r − 1) ,32
32

cR (r) =95(r − 1)3 −(r − 1) .
32
32

(18)

Exercise: Show that if the free R-charge assignment is consistent with all the superpotentials and is anomaly free in a general gauge theory, it solves the a-maximization
problem.
We assume that a gauge group G is given, the number of chiral fields is dim R, and thatthe assignment of R-charge 23 to all chiral superfields is consistent with superpotentials andanomalies. We also assume that the theory has L U(1)ℓ symmetries under which the i-th chiralsuperfield has charges Qℓi . Then we define a trial superconformal R-symmetry to be,
R it r ({α}) =2+ Qℓi αℓ ,
3

(19)

8
In superspace notations this is due to the fact that the superspace coordinates have a unit charge under the
R-symmetry.

13

SciPost Phys. Lect. Notes 78 (2024)with αℓ arbitrary real parameters. Computing the trial a-anomaly,dim
XR  2322ℓ3ℓe({α}) ≡aa({α}) − a v dim G =3( + αℓQ i − 1) − ( + αℓQ i − 1)
3
3
3
i=1
=

(20)

dim
XR 

′
′′
′
2
dim R + 3αℓ αℓ′ αℓ′′ Qℓi Qℓi Qℓi − αℓ αℓ′ Qℓi Qℓi ,9i=1

we immediately see that the term linear in αℓ cancels out. This implies that taking αℓ = 0 is astationary point and it is then trivial to show that it is a local maximum. One way to see thise({α})
is to take some arbitrary direction in {α} space parametrized as αℓ = nℓ t and see how achanges with t,2e({α}) = dim R + 3
a9dim R
X


∆R3i

3

t −3

dim R
X

i=1


∆R2i

t2 ,

(21)

i=1

where we defined ∆R i = Qℓi nℓ . Obviously then t = 0 is a maximum for any choice of directionnℓ unless ∀ i ∆R i = 0. However the latter case implies that R it r ({n t}) = 23 , and thus the
R-symmetry is still the free one.
□

2.3

Beta functions, deformations, and conformal manifolds

An important question regarding CFTs is what is the collection of relevant and exactly marginal
deformations of a model leading to an inequivalent fixed point. In particular for SCFTs we will
be interested in such deformations which preserves N = 1 supersymmetry.
One type of supersymmetric deformations one can add to an SCFT is the following superpotential term,Z
W = λOd 4 x d 2 θ O + c.c. ,(22)

where O is a chiral operator and λO is a complex number. Relevant supersymmetric deformations are given by chiral operators O with scaling dimension smaller than 3. For suchdeformations λO has a positive mass dimension and thus grows with the RG-flow to the IR.
Remember that dθ has mass dimension + 12 . Note also, that for chiral operators the superconformal algebra relates the dimension to the superconformal R-charge, ∆ = 32 R; thus, theR-charge of relevant deformations O is smaller than 2.
The marginal superpotential deformations are given by chiral operators O of dimension
exactly 3, or equivalently superconformal R-charge of exactly 2, and as we will soon discussthese can be either marginally irrelevant or exactly marginal.
Given an SCFT with a subgroup G of the global symmetry group with vanishing ’t Hooft
anomaly we can couple it to dynamical gauge fields in the standard way. In particular we
introduce the superpotential term,Z
W = τGd 4 x d 2 θ Wα W α + c.c. ,(23)with τ being the complexified gauge coupling and Wα the chiral superfield with the fieldstrength Fµν as one of its components. As with other superpotential couplings τ can be either
a relevant deformation, irrelevant, or marginal. To determine which of the three possibilities
holds one needs to perform a one loop computation of the gauge beta-function: Classically14

SciPost Phys. Lect. Notes 78 (2024)

gauge-couplings are marginal in 4d. In supersymmetric theories the result of this computation
can be expressed in an elegant form [38],9βG ∝ − Tr RP G 2 ,(24)

where P refers to the fixed point under consideration. That is the beta function is proportional
to minus the ’t Hooft anomaly of the superconformal R-symmetry, denoted here by RP , and two
currents of the gauged symmetry. The proportionality coefficient is a positive number which
will not play a role for us (and can be easily fixed by computing it for simple Lagrangians). The
reason this expression can be derived is because the superconformal R-symmetry is in the same
multiplet as the stress-energy tensor. In particular, if Tr RP G 2 is positive the gauging is UV free,
if Tr RP G 2 is negative the gauging is IR free, and if Tr RP G 2 = 0 the gauging is marginal. Notethat this way of determining relevance of a gauging does not rely on a description of the theory
in terms of weakly coupled fields and thus will turn out to be useful also for strongly coupled
SCFTs. Note also that the marginality of gauging is determined by whether the superconformal
R-symmetry at the fixed point is anomalous or not: That is whether it is consistent with thegauging. This is very analogous to marginality of superpotentials: Marginal superpotentials
do not break R-symmetry.
Finally, we need to determine whether the marginal deformations we discussed are exactly
marginal, marginally irrelevant, or marginally relevant. In fact for supersymmetric theories
with supersymmetry preserving deformations the last option is not possible. The reason is
as follows [40]. When deforming a theory by a supersymmetric marginal operator, in orderfor the deformation to cease being marginal its dimension needs to change after the RG-flow.
However, such deformations are chiral and as such form shortened, protected, representationsof the superconformal symmetry group. The only way for such a deformation to cease being
marginal is for it to recombine during the flow with another shortened multiplet to form along multiplet. Studying the representation theory of the superconformal group it is possible to show that the only multiplet with which the marginal deformations can recombine areconserved currents [41], see Appendix A. The primaries of the resulting long multiplet havedimensions which must be larger than three. As such, after such a recombination the dimension of the marginal deformation, which was 3 since it was the chiral primary of the shortmultiplet, must increase and thus it is marginally irrelevant. This implies in particular that
there are no supersymmetric marginally relevant deformations. Another simple implicationof this logic is that if a deformation does not break any symmetry it is exactly marginal. See
Figure 7 for a depiction of exactly marginal, marginally irrelevant, and relevant deformations.
Also, see Figure 8 for interesting cases of RG-flows.
A careful analysis of the above logic [40] (see also [42]) leads to the following conclusion.
iiGiven an SCFT P with a space of marginal operators OP
and couplings λP, the set of exactly
marginal deformations is given by the following Kähler quotient,i{λP}/(GP )C ,(25)

where GP is the global symmetry of the SCFT P and (GP )C is its complexification. Whenthe marginal deformations involve also gauge couplings of some gauge group, we need toinclude in GP also the symmetries which exist before the gauging but are anomalous once thegauging is introduced. The coupling which transforms well under the anomalous symmetry isexp(2πiτ) with τ being the complexified gauge coupling. Note that in the exponentiation we
took τ with the positive sign.10 This procedure is very abstract and has again the advantagethat it does not rely on having a description of a theory in terms of weakly coupled fields. One
See also [39] for an important discussion.
Usually in computations of Kähler quotients both signs are allowed. However, here taking the negative signmight lead to solutions with imaginary YM coupling. Formally in perturbation theory this would give a line of
9

10

15

SciPost Phys. Lect. Notes 78 (2024)ℳ′c
μi 𝒪i𝒫

ℳc
λi ˜
𝒪 i𝒫

G𝒫

G

ga

𝒫′

𝒫2

𝒫′

νi ℛi𝒫𝒫𝒫

way to compute this quotient is to list all the (GP )C invariant independent combinations of thecouplings: Though often this is practically hard to do. We refer the reader to [48] for a detailed
analysis of such quotients in many examples as well as a discussion of how to approach the
problem in a general case.
Example 1: SU(N ) SQCD with 3N flavors(26)

Thus, according to the previous exercise it is also the superconformal one. Moreover, since thisis also the superconformal R-symmetry of the free collection of chiral fields before gauging,
the gauging is marginal. Since the superconformal R-charges at the free UV fixed point is 32 we
can turn on marginal superpotentials using chiral operators with R-charge 2 which are built
from cubic combinations of the basic fields. However, for N ̸= 3 no such operators exist. Forje (a Qe b Q c) are singlets under the gaugede = εi jk Q
N = 3 the operators B = εi jk Q i(a Q b Q kc) and Bj ki
SU(3) symmetry: These are the baryons and the anti-baryons. The analysis thus is different
for N ̸= 3 and N = 3. We start with the former.
The symmetry of the free fixed point is GP = U(6N 2 ). We choose G = SU(N ) subgroupwhich we gauge. The commutant of G in GP is GP |G = U(3N ) × U(3N ). The Kähler quotient
we need to compute is thus,{exp(2πiτSU(N ) )}/(U(3N ) × U(3N ))C .

(27)

exactly marginal non-unitary SCFTs (See for similar effects e.g. [43]). However it is not clear whether this would
make sense non perturbatively. In some cases, e.g AGT correspondence [44, 45] or 5d gauge theories [46, 47](where gauge coupling is related to real mass for instanton symmetry), going to lower half plane for τ is allowed
but it is interpreted as going to infinite coupling (and “beyond”) where the gauge theory description breaks down.
In our analysis we are dealing with weakly coupled gauge fields and thus will restrict to upper half-plane for τ.
We thank Z. Komargodski for discussions of this subtlety.

16

ga𝒪a + gb𝒪b

ga𝒪a

Figure 7: A depiction of the conformal manifold Mc and three deformations fromthe point P . The exactly marginal deformation is related to λi OePi , the marginallyirrelevant deformation is related to µi OPi , and the relevant deformation is related toνi RPi .

First, a free R-symmetry assignment to all the matter fields is non anomalous,




1 2
1 2
2
Tr RP SU(N ) = N +− 1 3N +− 1 3N = 0 .
2 3
2 3

𝒫1

gb𝒪b

SciPost Phys. Lect. Notes 78 (2024)

𝒫1

ga

𝒫2

ga𝒪 + gb𝒪
a

ga

ga

𝒫2

𝒫2

b

ga𝒪a + gb𝒪b

𝒫3

gb𝒪b

𝒫1 gb

ga𝒪a + gb𝒪b
ga𝒪a

𝒫

ga𝒪a

ga𝒪a
gb𝒪b

(𝒜)

gb

𝒫

gb𝒪b

(ℬ)

𝒫1 gb

𝒫

(𝒞)

Figure 8: Several interesting RG flow patterns. (A ) Example of a dangerously irrelevant deformation. Turning on only g b we flow back to the original CFT. Turning
on both g a and g b , during the flow g a increases while g b decreases. However, onceg a is large enough we cannot trust low orders in perturbation theory, and g b mightstart increasing again and we might flow to a new fixed point. We call g b in sucha case a dangerously irrelevant deformation. (B) An example of “dangerously relevant operator”. Turning on either g a or g b we flow to a new CFT and thus these are
relevant. However, if we first turn on g b and flow to the fixed point, g a then becomesan irrelevant deformation at the new fixed point (This is sometimes related to the socalled chiral ring stability [49,50].) (C ) Turning on either g a or g b we flow to a new
CFT and thus these are relevant. Also if we turn on the deformations sequentiallythey are still relevant at each step.
The gauge coupling is not charged under the SU(3N ) × SU(3N ) × U(1)B non-anomalous
symmetry, but transforms under the U(1)A anomalous symmetry, with exp(2πiτSU(N ) ) having charge 3N . Thus this quotient is empty and there are no exactly marginal deformations.
The gauge coupling is marginally irrelevant and the theory is free.
Next we analyze the case of N = 3 where we have additional marginal operators B ande. These transform in (84, 1)+3,+3 and (1, 84)+3,−3 under GP |G which we parametrize as
B(SU(9), SU(9))U(1)A,U(1)B , where U(1)A is the symmetry under which both quarks and antiquarks have charge +1 while U(1)B is the baryonic symmetry under which they are oppositely
charged (that is with charges ±1). The representation 84 is the three index completely antisymmetric irrep of SU(9). The gauge coupling transforms as (1, 1)9,0 . The charge of the gauge
coupling τ, or rather of exp(2πiτ), is given by the ’t Hooft anomaly Tr U(1)A G 2 . In this case
it is +1 × 21 × 18 = 9. Note that the baryonic couplings have charge −3 as all the matter fields
have charge +1. In particular this implies that we always can solve for the quotient with theanomalous U(1)A. Thus the quotient we need to compute is,{(84, 1)+3 , (1, 84)−3 }/(SU(9) × SU(9) × U(1)B )C .

(28)

One way to construct the quotient is by finding all the independent monomial holomorphic
combinations of the couplings invariant under the symmetries. In general this is a well definedgroup theoretic question which is nevertheless tricky to solve.
A way to proceed is to find a deformation which is exactly marginal. Understand whatsymmetry it breaks. Deform the theory by this deformation and repeat the process. In thegiven case a simple deformation which was found by Leigh and Strassler [51] (See Appendix
B.) in the seminal paper on conformal manifolds is,

e 1Q
e 2Q
e3 + Q
e 4Q
e 5Q
e6 + Q
e 7Q
e 8Q
e9 .
W = λ Q 1Q 2Q 3 + Q 4Q 5Q 6 + Q 7Q 8Q 9 + Q
(29)

17

SciPost Phys. Lect. Notes 78 (2024)

This deformation breaks U(1)B and each SU(9) is broken to SU(3)3 . Note that under thisbreaking,
3
X

84 → 1 + 1 + 1 + 31 32 33 +3i 3 j ,(30)

i, j=1; i̸= j
3
X

80 → 1 + 1 + 81 + 82 + 83 +3i 3 j .

i, j=1; i̸= j

P3Note that the components of the currents (the 80) in representation i, j=1; i̸= j 3i 3 j recombinewith the same components of the marginal operators (the 84). Moreover the currents of thefour U(1)s in the decomposition of SU(9)2 to SU(3)6 × U(1)4 as well as the current of U(1)B
recombine with five out of the six singlets in the decomposition of the two 84s. We are thus
left with marginal deformations in,
1 + 31 32 33 + 34 35 36 ,(31)with the singlet being the exactly marginal deformation. We thus deduce that there is one
direction on the conformal manifold on which the symmetry is SU(3)6 . Next we can turn on
the marginal deformation 31 32 33 . This will break SU(3)3 down to diagonal SU(3). Note that
under this decomposition,
31 32 33 → 1 + 10 + 8 + 8 .

(32)

In particular the two 8 recombine with two of the three SU(3) currents leaving only SU(3)symmetry and marginal operators in,
1 + 1 + 101 + 34 35 36 .

(33)

Thus we have a two dimensional conformal manifold on which the symmetry is SU(3)4 . Next
we can break in the same manner using 34 35 36 another triplet of SU(3) symmetries down
to diagonal SU(3). We will obtain a three dimensional conformal manifold with symmetry
SU(3)2 and marginal operators in,
1 + 1 + 1 + 101 + 102 .

(34)

In Lecture III we will have a geometric interpretation of this conformal manifold as corresponding
to complex structure moduli of a sphere with six marked points. We can next continue exploring
the conformal manifold by turning on operators in the 10. As from 10 one can build two
independent invariant (the fourth and the sixth symmetric powers contain singlets), each gives
two exactly marginal directions on which the relevant SU(3) is completely broken. All in all
we obtain a seven dimensional conformal manifold on which all the symmetry is broken.
Instead of starting with (29) we can start with the following,


e1 Q
e 2Q
e3 .
W = λ Q1[i Q2j Q3k] + Q(35)[i j k]Here we think of the nine quarks as forming 3×3 representation of the SU(3)×SU(3) subgroupof SU(9). The lower flavor indices in (35) are antisymmetrized as well as the gauge indicesare. This superptential is exactly marginal. This can be understood either performing the LS
analysis (all the fields have same anomalous dimensions but we have two couplings: Gauge
coupling and λ), or performing the Kähler quotient. The two terms are charged oppositely
under the U(1)B symmetry and we note that,
84 → 101 + 102 + 81 × 82 ,80 → 81 + 82 + 81 × 82 .
18

(36)

SciPost Phys. Lect. Notes 78 (2024)

Φ1

Qn

Q1

N
Q̃ n

Φ2

Q2

N
Q̃ 1

Φ3

Q3

Qn−1

N
Q̃ 2

Φn

Qn

N
Q̃ 3

Q̃ n−1Q̃ n

Figure 9: The necklace quiver of length n.
Moreover the characters of 10 and 8 are given byχ10 = χ8 − 1 + z13 + z23 +1,
z13 z23

χ8 = 1 + 1 +X

zi /z j .

(37)

i̸= j

Using these observations we can conclude that (35) breaks the symmetry toSU(3) × SU(3) × U(1)4 where the abelian symmetries are the Cartan generators of the SU(3)sbroken by (35),11
f − 1 − 1 − 80 − 80
f → 1 + 101 + χ3 ((z 2 )3 , (z 2 )3 ) + 1
f01
1 + 84 + 84
2
1

(38)

e1 − 1 − 1 − 1 − 1 .
z22 )3 ) − 81 − 8+ χ3 ((ez12 )3 , (eâ .
Here zi2 are fugacities for the Cartan of SU(3)2 and ezi2 are fugacities for the Cartan of SU(3)22
We can further break the rest of the symmetries. For example, the two zi can be broken
together by turning on the three operators in χ3 ((z12 )3 , (z22 )3 ) and the SU(3)1 can be broken
first to the Cartan and then completely by turning on operators in 101 .
□Example 2: N = 2 necklace quiver: Consider the quiver gauge theory of Figure 9. It
is well known that this theory has n exactly marginal deformations preserving N = 2
supersymmetry (show it). Show that it also possesses an additional independent exactly
marginal deformation preserving only N = 1 supersymmetry.
The matter content is conformal, i.e. the one loop beta function vanishes, and thus all thefields have free R-charges. Let us start by discussing the symmetries of the theory. We haven anomalous symmetries U(1)Ai under which the adjoint fields Φi have charge +1 and all theother fields are not charged. We have n U(1)αi symmetries under which Q i has charge +1 ande i charge −1 with all the rest of the fields not charged. Finally we have n U(1) t symmetriesQie i have charge +1, Φi and Φi+1 charge −1 while the other fields
under which fields Q i and Q
are not charged. The U(1)α and U(1) t symmetries are not anomalous. Let us consider thefollowing superpotentials,
X
e i + λ′ Φi Q i−1Q
e i−1 .
W=λi Φi Q i Q
(39)
i
i=1
−1
′
Note that the couplings λi have charges encoded in fugacities as A−1
i t i t i−1 while λi have−1
−1′charges Ai t i t i−1 . Then λi λi dressed with appropriate power of exp(2πiτi ) is a singlet foreach i. These thus give us n independent exactly marginal parameters preserving in fact N = 2
supersymmetry. Along these directions the U(1) t i symmetries are broken to a diagonal combination. Note that as all of the couplings are charged under the anomalous symmetries withnegative charge these can be taken care of by exponents of gauge couplings.
11

This is very reminiscent of the N = 1 β deformation of N = 4 SYM, see e.g. [52].

19

SciPost Phys. Lect. Notes 78 (2024)

2

ϕ

qL
4

SL

qR

2

3

SR

Φ
QL

6

4

3

QR

W = QL qL Φ − QR qR Φ
Figure 10: A quiver theory with theq SU(2) IR free gauge node in the UV. qL

qR

L

4
2
3
3
4
2
3
′
Above we turned on λ andQ
λS couplings in pairs. Now consider turning
on
only
λ
couSL
Φ
Ln
Φ
plings. Then the combination
i=1 λi (dressed by appropriate power of all exp(2πiτi )) is
QL marginalalso a singlet of all the symmetries. and thus
QL turning on only λi gives us an exactly
QR
6
deformation. Here also the U(1) t i symmetries are broken
to a diagonal combination. This6deformation is only N = 1 supersymmetric. Note that turning on only λ′ couplings
W also= QisL qexactlyL Φ − QR qR Φ
W
=
Q
q
Φ
L
L
marginal, but it is a combination of the above and the N = 2 exactly marginal couplings.
Note that (39) is not the most general
P cubic3superpotential we can write. Specifically,
for N > 2, we can also add the terms i=1 λ̂i Φi , where we use the fully symmetric cubic
invariant polynomial of SU(N ) to contract the gauge indices and form a gauge invariant. The
3 3couplings λ̂i then have the charges A−3i t i t i−1 . We can cancel the U(1)Ai charges by dressing
with appropriate powers of the gauge couplings, but we cannot cancel all the U(1) t i charges.
As such, for generic N , these operators are
However, for N = 3, we
P marginally irrelevant.
e 3 , where we use the epsilon tensor
further have the baryonic superpotential i=1 λBi Q3i + λB̃i Q
ito contract the SU(3) indices to a singlet. The λBi have the charges t i−3 α3i while λB̃i have the
B B̃charges t i−3 α−3i . We then see that the λi λi combination of couplings is uncharged underU(1)αi and together with λ̂i and the gauge couplings, can be used to form flavor symmetry
singlets. Thus, we see that for N = 3 there are additional exactly marginal deformations that
preserve only N = 1 supersymmetry.
□

Exercise: Dangerously irrelevant deformations – Consider the quiver theory of Figure
10. The circles inscribing an integer N denote SU(N ) gauge groups. The squares inscribing an integer N denote SU(N ) flavor groups. The lines denote bifundamentalchiral superfields: Fundamental representation of the group they point to and antifundamental of the group they emanate from. The theory has a superpotential denotedon the Figure. Note that the SU(2) gauge group has 14 fundamental fields and thus
naively is IR free. By sequentially analyzing the flows starting with the fixed point of
the SU(3) SQCD with six flavors and turning on first the superpotentials, show that theSU(2) gauging becomes asymptotically free.
We start with the SU(3) SQCD with six flavors. This theory is asymptotically free,
1 2− 1 × 2 × 6 = 1 > 0,
Tr R SU(3)2 = 3 +
2 3

(40)

where we used the free assignment of R-charges. Assigning R-charge 12 to all the chiral fields
20

SR

4

SL

2

3

SR

Φ
QL

4

3

QR

6

SciPost Phys. Lect. Notes 78 (2024)

W = QL qL Φ − QR qR Φ
qL

qL
4

4

2

3

SL
QL

Φ

qR
2

3

QL

6

4

3

Φ

SL
6

SR
QR

W = QL qL Φ − QR qR Φ

W = QL qL Φ

Figure 11: We analyze the flow in three steps. First, we start with the IR fixed pointof SU(3) SQCD with six flavors, add 12 chiral fields Φ and turn on a superpotential(left figure). Then we couple a second copy of SU(3) SQCD with six flavors to thetheory through a superpotential (right figure). Finally, we add four more chiral fields
φ and gauge the SU(2) symmetry as in Figure 10.
is anomaly free,
1 1− 1 × 2 × 6 = 0,
Tr R SU(3) = 3 +
2 2
2

(41)

and in fact this is the superconformal R-symmetry at the fixed point. We have one abelian U(1)symmetry which is non-anomalous under which the fundamentals and the antifundamentals
have opposite charges. We thus define a trial a-anomaly,





1
1
3
a(α) = a v × 8 + aR
+ α × 18 + aR
− α × 18 =41 − 324α2 ,
(42)
2
2
64
and the maximum is at α = 0. Next we add the 12 fields Φ and couple them with superpotential
as on the left of Figure 11. This interaction breaks some of the nonabelian symmetry. The Rcharge of the superpotential at the UV fixed point is 21 + 12 + 32 < 2 and thus it is relevant andthe theory flows in the IR to a fixed point where there is a non-anomalous R-symmetry underwhich the R-charges of q L , S L and Q L are still 21 but that of Φ is 1 (such that the R-charge ofthe superpotential is 2). We are still left with finding the superconformal R-symmetry at thatIR fixed point. The theory has two non anomalous abelian symmetries (U(1)aL , U(1) b ) so thatthe charges of the various fields are,q L : (1, 2) ,S L : (1, −1) ,Q L : (−1, 0) ,

Φ : (0, −2) .

(43)

We thus compute the trial anomaly depending on two parameters αa and α b corresponding
to the two symmetries,
1a(αaL , α b ) = a v × 8 + aR (1 − 2α b ) × 12 + aR
+ αaL − α b × 12
2




1
1
+ aR
+ αaL + 2α b × 6 + aR
− αaL × 18 .
(44)
22Here in the first line we have the contribution of the vectors and Φ and on the second line thecharged matter fields. The charges of Φ are fixed by the superpotential as stated above. We
compute the maximum of a(αaL , α b ) to be at(αaL , α b ) = (0.0045, 0.0672) ,approximately. Now we couple the second copy of the SU(3) SQCD to the theory through asuperpotential as on the right of Figure 11. The additional term in the superpotential at the
new fixed point has R-charge
1 1+ + RΦ = 1 + (1 − 2 × 0.0672) = 1.8657 < 2 ,2 221

SciPost Phys. Lect. Notes 78 (2024)

and thus it is relevant. The second copy of the SQCD adds another U(1) symmetry and wecompute the new trial a-anomaly to be,


1
L
R
L
(45)
a(αa , α b , αa ) = a v × 8 × 2 + aR (1 − 2α b ) × 12 + aR
+ αa − α b × 12
2






1
1
1
+ aR
+ αaL + 2α b × 6 + aR
− αaL × 18 + aR
+ αRa − α b × 12
2
2
2




1
1
R
R
+ aR
+ αa + 2α b × 6 + aR
− αa × 18 .
22Here we used the fact that the superpotential identifies U(1) Lb with U(1)Rb . We find the maximum of a(αaL , α b , αRa ) to be at,
αaL = αRa = 0.00135 ,α b = 0.03669 .

Finally we compute the beta function of the SU(2) gauging at the new fixed point,
1
1
1
Tr R SU(2)2 = 2 + (Rφ − 1) × 2 + (RΦ − 1) × 6 + (Rq L/R − 1) × 6 .
2
2
2

(46)

Now Rφ = 32 as these are free fields we add at this point and
Rq L/R =1+ αaL/R + 2α b = 0.5747 ,2RΦ = 1 − 2α b = 0.9266 .

We thus deduce that the anomaly in (46) is
Tr R SU(2)2 = 0.1707 > 0 ,and thus at the new fixed point the SU(2) gauging is asymptotically free although it has 14
fundamental fields.
This conclusion holds under the assumption that no emergent abelian symmetries appear
in neither step of the flow invalidating the a-maximization argument. An indication that thisassumption is wrong would be if some of the chiral operators violated the unitarity bounds
at some of the steps. However, it is easy to verify that none do. For example the field Φ has
R-charge 0.8657 > 32 after step one (I), and R-charge 0.9266 > 23 after step two (II). The
mesons and baryons have the following charges,
I: SQ = 0.9328 ,qQ = 1.1343 ,q3 = 1.9165 ,Q3 = 1.48647 ,S 3 = 1.31205 ,II: SQ = 0.9633 ,qQ = 1.0734 ,q3 = 1.7242 ,Q3 = 1.4960 ,S 3 = 1.3940 ,(47)

all of which are above the unitarity bound.
A famous example of dangerously irrelevant deformations is the case of SU(N ) SQCD
with N f fundamental flavors and a chiral field in adjoint representation Φ with superpotential
W = Tr Φn+1 . For n > 2 the superpotential is irrelevant in the UV. However, first flowing withthe gauge coupling and then turning on the superpotential it can be relevant with n > 2 for arange of choices of N and N f [53, 54].
□

2.4

Supersymmetric RG-flow invariants

We have discussed an invariant of RG flows which does not need supersymmetry, the anomaly
polynomial. The supersymmetric flows have in fact many more RG-flow invariants. These
invariants typically take a form of some version of the Witten index. Here we will discuss thesimplest example of these, the supersymmetric index [55].
22

SciPost Phys. Lect. Notes 78 (2024)

Let us briefly review the general construction of a Witten index. Consider the situationthat we have two supercharges Q and Q† such that,{Q, Q† } = δ ,(48)

with δ being a combination of bosonic charges. Now consider Vδ0 to be the subspace of thelinear space of states of the theory with |ψ〉 ∈ Vδ0 satisfying δ · |ψ〉 = δ0 |ψ〉 with δ0 > 0. Now
from (48) it follows,|ψ〉 =1{Q, Q† } |ψ〉 .
δ0

(49)

From here, since Q and Q† are nilpotent it follows that any state in Vδ0 is a linear combination
Q

Q†

0

0

of a state annihilated by Q and a state annihilated by Q† , a fact we denote as Vδ0 = Vδ ⊕ Vδ .
Q†

Q

The supercharges Q and Q† /δ0 are a one to one map and its inverse from Vδ to Vδ . From00here it follows that the index defined as the following trace over the space of states of thesystem H ,I ({u}) = TrH (−1) F e−β δnYqui i ,(50)

i=1

is independent of β as only states in H with δ vanishing contribute to it. Here qi are thecharges under the Cartan generators of the n dimensional maximal bosonic subgroup of thesymmetry group commuting with Q and Q† . Moreover, this index is invariant under continuous
deformations of the theory, and in particular is invariant under the RG flow as long as thesymmetries used to define it are preserved along the flow. Let us now discuss a concrete
example of such an index.
Consider an N = 1 SCFT. It possesses a superconformal algebra and in particular one of
the commutation relations defining the algebra reads (see Appendix A)


©
¦
3
β̇
β̇
β̇
e
e
(51)
2 Q α̇ , S = δα̇ H − R + 2 (J2 ) α̇ ,2where α̇, β̇ = ±̇ are spinorial indices. We think of the space as being S3 × R. H is the operator
whose eigenvalue is the scaling dimension ∆, and which generates translations along R. The
e areoperators Ji=1,2 are the generators of the SU(2) × SU(2) isometry of S3 . Finally, the Q’s
esupercharges and the S’s are their conformal counterparts, with R being the superconformal
e supercharges are
R-symmetry. Now in radial quantization the hermitian conjugate of the Q
α̇
†α̇
e
e
e
e
the S supercharges, S = Q . We take Q = Q −̇ , such that under the Cartan generators ( ji ) of
Ji it has charges (0, − 12 ) and has R-charge +1. The commutation relation above thus becomes,
32{Q, Q† } = H − 2 j2 − R ≡ δ .
2(52)

The operators which commute with Q and Q† are in addition to δ, ± j1 + j2 + 12 R and the Cartan
generators of any global symmetry the theory might have (which we denote by Qi ). We thencan define the superconformal index to be,
F −β δ − j1 + j2 + 12 R j1 + j2 + 21 R

I (q, p; {u}) = TrS3 (−1) e

q

p

Rank
YGF

Q

ui i .

(53)

i=1

By the general logic of the Witten index described above this index is invariant under continuous deformations of the theory preserving the superconformal algebra, that is, it is invariant
23

SciPost Phys. Lect. Notes 78 (2024)

on the conformal manifold Mc as long as we use only fugacities ui preserved by the exactly
marginal deformations.
On the other hand we also want a quantity which is an invariant of RG-flows. The above
definition of the index relies on the superconformal symmetry which is broken once the RGflow is initiated and thus we cannot use it verbatim. However, there is a simple redefinitionof the setup which allows us to define the same index without relying on superconformal
symmetry [56] (See [21] Section 7.1 for a nice summary). Instead of thinking about the index
as a counting problem we can compute it as a partition function on S3 ×S1 with supersymmetricboundary conditions along the S1 . This is a curved background and analyzing carefully thesupersymmetry algebra on it, one can find a supercharge Q which satisfies the commutation
relation (52). The charge ∆ in this setup is a suitable combination of the R-symmetry and thetranslation along the S1 . The construction needs to have an R-symmetry. The index defined inthis way is sometimes called the supersymmetric index and it is invariant under the RG-flow
as long as we use the R-symmetry which is preserved by the deformations used to start theflow. This R-symmetry does not have to be the one which becomes superconformal at the fixed
point. Nevertheless, if we choose the R-symmetry which coincides with the superconformal
symmetry in the IR, the supersymmetric index computed this way in the UV coincides with thesuperconformal index of the IR fixed point. The index as an invariant of RG-flows was first
discussed in [57, 58].
The supersymmetric index captures a lot of very interesting RG-flow invariant information
about the QFT. For example, let us understand in more detail what the index is counting.
As the index is independent of β the only states that actually contribute to it have δ = 0.
This implies that {Q, Q† } annihilates these states and thus by unitarity both Q and Q† alsoannihilate it. States which are annihilated by some supercharges form short representationsof the supersymmetry group. The number of short multiplets of a given type might change
during the RG flow. However this only can happen if a collection of short multiplets forms
a collection of long ones, or if a long multiplet decomposes into short ones. The index is aninvariant under such recombinations [55]. We can then think of the index at the fixed point asa sum over the representations of the superconformal symmetry group Rℓ and representationsof the global symmetry group G F denoted by m,Xnℓ,m χRℓ (q, p) χm ({u}) ,(54)
I (q, p; {u}) =ℓ,mwith the integers nℓ,m giving the multiplicities of these representations. These multiplicities
can change with the flow or when one moves on the conformal manifold but the index issuch that one can reorganize this sum in terms of equivalence classes of representations: Two
representations are in the same equivalence class if they contribute to the index the sameway, possibly up to a sign. This analysis was carefully performed in [41] Section 3. The
conclusion is that in each equivalence class there is a finite number of representations and one
can decompose the index in terms of net degeneracy numbers n̂ℓ,m where we now sum overthe equivalence classes denoted by [R],X
I (q, p; {u}) =n̂ℓ,m χ[R]ℓ (q, p) χm ({u}) .
(55)
ℓ,m

Although in general we can extract information from the index only about these net degeneracies, it so happens that if one thinks of the index in terms of an expansion in the q and p
fugacities, at low order in the expansion the multiplets which can contribute are very simple.
As we are making use of the superconformal symmetry group the statements below are only
true if one uses the superconformal R-symmetry to compute the index and we assume that thetheory does not contain free fields.
24

SciPost Phys. Lect. Notes 78 (2024)• The only states that can contribute at order (q p)n with n < 1 are chiral operators of
R-charge R = 2n. As n < 1 these are the relevant operators. Thus we can cleanly readoff from the index the spectrum of relevant superpotential deformations.
• The only states which can contribute at order q p are marginal operators, contributing
with positive sign, and certain fermionic components from the conserved current multiplet, which contribute with a negative sign. Thus at this order we can extract the
combination,
χ M ar g inals ({u}) − χad joint GF ({u}) .

(56)

In particular any negative contribution to the index at this order comes from conserved
currents and has to be a part of the character of the adjoint representation of the global
symmetry group. This is extremely useful to identify the symmetry of the IR fixed point.
As we discussed this symmetry might be emergent and assuming we have identified the
R-symmetry correctly the index computation can tell us reliably what that symmetry is.
We thus learn that the index provides us with a very non trivial probe about the deformations
and the symmetry of the fixed point. As an aside comment, in fact the index also encodessome other non-trivial properties such as ‘t Hooft anomalies. To extract the anomalies one cansend all of the fugacities, {q, p, ui }, to 1 and study the leading divergent behavior of the index
in this limit [13, 59–62] which turns out to neatly encode the ‘t Hooft anomalies of various
symmetries. We know that the anomalies alone do not uniquely specify a CFT. The index isa rich observable however it also does not uniquely specify a theory, as for example different
models on the same conformal manifold Mc will have the same index.12
The discussion till now was very general but the technology of computing the index israther simple. We will not derive it here and only quote the results. The details can be founde.g. in [21, 65, 66]. The technology consists of two main ingredients.
• The index of a chiral field Q with R-charge R and representation R of G F is given by,

IR,R (q, p; {u}) = exp¨∞«
X 1 (qp) R2 m χR (um ) − (qp) 2−R
2 m χ (u m )R

m=1(1 − q m )(1 − p m )

m

.

(57)

Here χR is the character of the representation R. In particular this can be neatly written
in terms of the so called elliptic Gamma functions [58],¨∞«∞
X 1 z m − (qp)m z −mY
1 − q i+1 p j+1 z −1
Γe (z; q, p) = exp=,(58)m (1 − q m )(1 − p m )1 − qi p j z
m=1i, j=0which can thus be interpreted as the index of a chiral field with R-charge zero and charge
one under the U(1)z symmetry.
• Given an index of some SCFT I1 (q, p; {u, v}) with global symmetry Gu × G v we can compute the index of the theory with the G v symmetry gauged,((q; q)(p; p))Rank Gv
I2 (q, p; {u}) =|WGv |

I Rank
YGv
i=1

d vi
I1 (q, p; {u, v}).
2πi vi I0,(adj. Gv −rank Gv ) (q, p; {v})(59)

12Theories differing solely by their higher form symmetries also might have the same index, say SU(N ) andSU(N )/ZN N = 4 SYM. However this difference can be captured by other types of partition functions, e.g. theLens index [63, 64].

25

SciPost Phys. Lect. Notes 78 (2024)

Q∞
Here WGv is the Weyl group of G v . We also define (z; y) = i=0 (1 − z y i ). The contribution 1/I0,(adj. Gv −rank Gv ) comes from the vector multiplets. It is equivalent to one over thecontribution of chiral superfield with R-charge zero in adjoint representation without
the contribution coming from the Cartan generator, χadj. Gv −rank Gv = χadj. Gv − rank G v .
The superpotentials only effect the index through the restrictions they impose on flavor symmetry and R-symmetry. This index can be computed with any R-symmetry but it becomes thesuperconformal index of the fixed point only if the superconformal R-symmetry is used.
e coupled with theExercise: Consider the simplest QFT, two chiral superfields Q and Qesuperpotential m QQ. Compute the supersymmetric index and interpret it.
First we need to determine the symmetries and the charges of the model. At the free point
the two chiral fields can be rotated by U(2) but the superpotential breaks it to U(1) underwhich the two fields have an opposite charge. We chose to normalize the charges to be ±1.
The R-symmetry of the superpotential is 2 and thus the sum of the R-charges of the two fields
is RQ + RQe = 2. We then use the first entry in the technology of computing the index above towrite the index of the system to be,

 

RQ
2−RQ
IRQ ,+1 (q, p; u) I2−RQ ,−1 (q, p; u) = Γe (q p) 2 u; q, p Γe (q p) 2 u−1 ; q, p = 1 .
(60)
The last equality is derived by direct evaluation using the definition of the elliptic Gamma
function (58). Let us try to interpret the result. The fact that the index is 1 means that only
the identity operator corresponding to the vacuum contributes to it and there are no otherprotected operators (or to be more precise the spectrum of operators is such that all protected
ones can recombine into long ones). The theory is massive and thus it is gapped and in the
IR we will not have any propagating degrees of freedom. The theory has a single state, thevacuum, and thus the index is consistent with this. Note that this is a trivial example where
the UV U(1)u symmetry does not act faithfully in the IR. For this reason it is also not important
what the superconformal R-symmetry is. In particular the trial a-anomaly is identically zero,a(α) = aR (RQ + α) + aR (2 − RQ − α) = 0 .

(61)This is a trivial example, however in the more interesting cases the computations are not much
more complicated and the physics can be extracted in a similar manner.
□
We are now ready to study interesting physical systems using the simple toolkit of nonperturbative techniques we have reviewed.

3

Lecture II: Examples of strong coupling dynamics

Let us start our discussion of the dynamics of supersymmetric field theories with several examples of interesting strongly coupled behavior. We will review the phenomenon of IR duality,
discuss the interplay between duality and emergence of symmetry in the IR, and discuss a
simple algorithm to look for different weakly coupled theories residing conjecturally on thesame conformal manifold. The purpose of this Lecture is to introduce various possible physicalscenarios and effects. In later Lectures we will discuss a way to more systematically discusssuch effects using geometric constructions.

26

SciPost Phys. Lect. Notes 78 (2024)

𝒢IR
CFTIR

CFTIR
RG Flow
CFTUV1 + Δ1

𝒢UV1

QFTIR

RG Flow
CFTUV2 + Δ2

CFTUV1𝒢UV2CFTUV2

CFTUV

Figure 12: A depiction of IR dualities. On the left we have two different UV CFTs
𝒢IR
deformed to flow to the same IR fixed point. On the right we have a UV CFT deformed
𝒢IR
SCFT(e.g. a weakly coupled asymptotically
free gauge theory) flowing in the4dIR
toIRa QFT
which close to the IR fixed
can be described by yet another weakly coupled
𝒢′UV
CFTpoint
IR
4d SCFTUV + Δ′
RG Flowtheory (e.g. an IR free gauge theory).
RG Flow

3.1

IR dualities

CFTUV + Δ

𝒢UVΔ=

6d SCFTUV + Δ

4d SCFTUV

𝒢UV6d SCFTLet us consider some of the basic examples of IR dualities
discovered
by Seiberg [2]. First, let
UV
us consider
N
=
1
SU(N
)
gauge
theories(SQCD)
withN
fundamentalchiral fields and N ff
CFTUVantifundamental ones: This is referred to as the theory having N f flavors. The matter content
is non-anomalous for any N . For N > 2 we should worry about cubic anomalies Tr SU(N )3 ,which vanishes because the matter representation is real. For N = 2 there is no difference
between fundamentals and antifundamentals but we have to have an even number of theseso that the theory will be free of a Witten anomaly [67]. For N f ≥ 3N (N f > 9 for N = 3) thetheory is IR free and thus we need to worry about UV completing the theory, as the couplings
grow when we flow back to the UV. For N f < 3N (N f ≤ 9 for N = 3) these models are
asymptotically free and thus can be thought of as deformations of Gaussian fixed points in the
UV. The beta function is such that when flowing to the IR the gauge coupling grows and we areinterested in understanding what is the effective description in the IR. We will be in particularinterested in the case of N f > N as here the dynamics turn out to be rather interesting.13 Hereis the basic statement.

e i , and no superpotential• For 23 N < N f < 3N SU(N ) SQCD with N f flavors, Q i and Q
flows to an interacting SCFT in the IR. Moreover, the SU(N f − N ) SQCD with N f flavors,qi and eqi , and N f2 gauge singlet chiral fields, Mi j , with a superpotential W = M · q · eq,flows to exactly the same fixed point. The phenomenon of two different UV theories
flowing to the same IR fixed point is called IR duality. See Figure 12 on the left for an
illustration. This range of parameters is called the conformal window.
e i , and no superpotential• For N + 1 ≤ N f ≤ 23 N SU(N ) SQCD with N f flavors, Q i and Q
flows to an IR free theory. The effective description in the IR is that of SU(N f − N ) SQCD
with N f flavors, qi and eqi , and N f2 gauge singlet chiral fields, Mi j , with a superpotential
W = M ·q·e
q. Note that the latter theory satisfies in the given range of parameters
N f > 3(N f − N ) and thus the theory is IR free. In other words the SU(N f − N ) SQCD is
UV completed here by the SU(N ) SQCD. See Figure 12 on the right for an illustration.
In addition, see Figure 13 for a quiver description of Seiberg duality.
13

For N f ≤ N the dynamics becomes trickier, e.g. it involves effects such as runaway vacua and quantum deformed moduli spaces, that we will not discuss here [2]. See [21] for an excellent review.

27

SciPost Phys. Lect. Notes 78 (2024)

M

N

Nf

Nf

Q

Nf

q

Q̃

Nf

Ñ

q̃

W = M ⋅ q ⋅ q̃2
Figure 13: A depiction of Seiberg duality with SU(N ) gauge groups.
N+2

1

N+2

1

N+2

Let us discuss some evidence
for thesestatements. First, one can check the ‘t Hooft anomalies
N+2
N+2
of the various symmetries: In the first case of the two different UV theories and in the second
N+1
N+2 coupled theories
case of the weakly
in the UVN+2
and in the IR. We
this as a simple exercise.
N +leave1
N+1
Second, one can check that the supersymmetric indices of theNrelevant
theories agree. NIn+ fact,+22
there is a mathematical proof due to Rains that they do [68]. The proof is1rather non trivial
so let us here quote a simple computation for the simplest duality, SU(2)U(1)SQCD with N f = 3
Bdual to a WZ model (See Figure 14). Following the general rules we have outlined the duality
implies the following identity of indices,(q; q)(p; p)

I

du
4πiu

1
Y
6 ±1
1
i=1 Γe ((qp) u ai ; q, p)
3 a a ; q, p) .
Γ
((qp)
=
e
i j
Γe (u±2 ; q, p)
i< j

Q6

(62)

On the left we have the index of SU(2) SQCD. The numerator comes from the six fundamental
fields which have anomaly free R-charge of 1/3. The symmetry is SU(6) and ai parametrize
Q6
the Cartan of this symmetry, i=1 ai = 1. The denominator comes from the vector superfield.
The ± in the integral is a shorthand notation for the following, f (x ±1 ) ≜ f (x) · f (x −1 ). On
the right hand side we have a WZ model with 6×5
2 = 15 chiral fields with R-charge 2/3 which
are coupled with a cubic superpotential. The SU(2) SQCD with N f = 3 flows in the IR to a WZ
model of 15 chiral fields with cubic superpotential, which flows to a free theory. The identity
above was originally obtained by S. Spiridonov [69] and dubbed elliptic Beta function as incertain degeneration limits of parameters it becomes the well known Beta function integral
identity.
Next consider a duality in the conformal window, SU(2) SQCD with N f = 4 dual to SU(2)SQCD with N f = 4 and a collection of 16 chiral fields. The index of the two dual theories is,
Q8
1
4
Y
4 ±1
1
du
i=1 Γe ((qp) u ai ; q, p)2a a(q; q)(p; p)
=
Γ
((qp)
(63)
e
i 4+ j ; q, p)
4πiu
Γe (u±2 ; q, p)i, j=1
Q4
Q8
1
1
I
−1
4 ±1
4 ±1 −1 −1dui=1 Γe ((qp) u aai ; q, p)
i=5 Γe ((qp) u a ai ; q, p)×(q; q)(p; p).
4πiu
Γe (u±2 ; q, p)
I

Here the superconformal R-charge on both sides is 21 for all the fundamental chiral superfields.
Q8
The symmetry on the left is SU(8) parametrized by ai , i=1 ai = 1. On the right the symmetry
Q
 14
1
4
in the UV is SU(4) × SU(4) × U(1). The U(1) is paramterized by a 2 ≡a. The two
i=1 i
1

1

SU(4)’s are parametrized by {a 2 ai−1 }4i=1 and {a− 2 ai−1 }8i=5 . Note that SU(4) × SU(4) × U(1)
28

N

Nf

Q

Nf

Nf

Nf

Ñ

q

Q̃

q̃

SciPost Phys. Lect. Notes 78 (2024)

W = M ⋅ q ⋅ q̃

2
N+2
1
N+2

Q

N+2

1
N+2

N+1

N+2

M

N+2

N+1
N+2

Q̃

N+2

q

1

N+1
N+2

q̃

U(1)B
Figure 14: When the number of flavors is one more than the number of colors thetheory in the IR is a WZ model with a superpotential q e
q M + M N +2 term. This phenomenon is called S-confinement. In the IR the WZ models flows to a collection offree chiral fields.
is a subgroup of SU(8) and thus if this duality is correct the symmetry has to enhance toSU(8) in the IR. This is a simple example of emergence of symmetry. As we have discussed
the supersymmetric relevant operators are invariant under RG flows and thus have to match
e j match with Mi j , Q i Q j match with qi q j , and Q
e iQejacross the duality. Indeed the operatots Q i Q
match with eqi eqj.
Exercise: Compute the index of the N f = 4 SU(2) SQCD up to order q p using the superconformal R-symmetry. What is the representation of the marginal operators under
the SU(8) global symmetry? This theory is extremely interesting. In fact it has (at least)
72 different dual descriptions [70]: (The number 72 is interesting: It is the ratio of thedimension of the Weyl group of E7 and SU(8). There is an E7 lurking behind this theory. To see it one needs to do some work [71] (See also [72]).) The two Seiberg dual
theories here are just 2 out of the 72 different duality frames. Can you find another 34?
We can use the integral expressions for the index above to compute the contribution atorder qp,
χ336 ({a}) − χ63 ({a}) .
(64)
Here 336 irrep of SU(8) appears in decomposition of Sym2 28 = 336 + 70. The 28 is therepresentation of Q ·Q and the marginal operators come from (Q ·Q)2 . We note in passing that
SU(8) is a maximal subgroup of E7 with 133 E7 = 70 + 63. The above can be written as,χSym2 28 ({a}) − χ70 ({a}) − χ63 ({a}) = χ336 ({a}) − χ63 ({a}) .

(65)

The E7 is not however a symmetry of the theory in the IR, the −χ70 comes from trace relations
and not from a conserved current. We see that E7 is lurking under the surface, and in the next
Lecture we will see some geometric importance of this.14 The positive contributions are themarginal operators and the negative are the conserved currents. Note that we can here identify
the positive and the negative contributions as these have to be characters of representationsof SU(8). Note also that there is no direction on the conformal manifold preserving the fullSU(8) symmetry as the marginal operators lack a singlet of this group.
Regarding the second part of the question: Note that to construct the dual theory weneed to split the eight fundamentals into fundamentals and anti-fundamentals, which is anarbitrary procedure for SU(2). We thus have 12 × (8!/4!2 ) = 35 different ways to do so. This
14

There is an interesting interplay between kinematic constraints, more generally chiral ring relations, and enhanced symmetries which we will not review here [73].

29

φ

φ̃

𝒳

˜
𝒳

SU(3)

Q
2

Q̃

Q

Q̃
8

SU(2)

SciPost Phys. Lect. Notes 78 (2024)

6

SU(2)

M

2

M̃

W = Mab ϵ ij Qia Qjb

W = Mab ϵ ij Qia Qjb+

Figure 15: On the left SU(2) SQCD with eight fundamental fields. On the right a
relevant deformation thereof.
immediately gives us 35 inequivalent possibilities of Seiberg duality. There are in fact another
35 + 1 dualities which were discussed in [74, 75].
□
The SU(2) SQCD with various amounts of flavors are probably the simplest supersymmetricgauge theories and already these exhibit extremely rich dynamics. We will now analyze yet
another interesting strong coupling effect that can be derived starting from SU(2) SQCD with
N f = 4 [23].

3.2Emergence of symmetry in the IR

Let us consider SU(2) SQCD with N f = 4. We split the eight chiral fields into six (Q) and two
e We also introduce gauge invariant operators M coupling through a superpotential as,(Q).
W = Mab εi j Q ai Q bj .

(66)

Note that this superpotential is relevant as at the SQCD fixed point the R-charge of the quarks
is 12 and the R-charge of the gauge singlets, which are free fields at the fixed point, is 23 . The
quiver theory is depicted in Figure 15 and charges of the various fields under the symmetriesof the model are detailed in the table below. The gauge singlet fields M and the superpotential break the symmetry of the model from SU(8) down to SU(6) × SU(2) × U(1)h . We willshow eventually that this symmetry enhances to E6 × U(1)h . We also note that SU(8) is not a
subgroup of E6 .
Field
Q
e
Q
M
e
M

SU(2)G
2
2
1
1

SU(2)
1
2
1
1

SU(6)
6
1
15
1

U(1)h

U(1) r̂

1
2
− 32

5
9
1
3
8
9
4
3

−1
3

In the table U(1) r̂ is the superconformal R-symmetry obtained by a-maximization [76]29eeand the conformal anomalies are c = 24
, a = 13
16 . Note that the operator QQ has supercon2
formal R-symmetry 3 and thus is a free field in the IR. This means that in particular we havean emergent U(1) symmetry under which this field, and only it, is charged. Emergence ofabelian symmetries might invalidate the a-maximization, however this is not the case here.
The operator does not violate the bound but rather saturates it and thus following a version of
one of the previous exercises taking into account the emergent symmetry will not violate theconclusion. We are interested however only in the interacting part of the IR SCFT and thus
we can remove the free field simply by “flipping” it [77] (See also [50].). The operation offlipping [15] an operator O is preformed by introducing a chiral field ΦO and turning on asuperpotential
W = ΦO O .
30

SU(2)

U(2)

SU(2)

φ

φ̃

SU(3)

GF = SU(2)˜𝒳

SciPost Phys. Lect. Notes 78 (2024)

2

6

SU(2)

M
W = Mab ϵ ij Qia Qjb

Q

Q̃

Q

Q̃

2

6

SU(2)

M

M̃

W = Mab ϵ ij Qia Qjb+ M̃ ϵ ijϵab Q̃ ai Q̃ bj

Figure 16: The model with E6 × U(1) global symmetry.
eQ
e is a free field the flipping is just a mass term for O, both ΦO = MeIn our case since O = Q
and O become massive and decouple in the IR. Using this superconformal R-symmetry wee (see Figure 16),
compute the index after introducing M
42

1 + 27h−1 (qp) 9 + h3 (qp) 3 + · · · + (−78 − 1)qp + . . .

(67)

The bold-face numbers are representations of E6 as we will elaborate momentarily, and h is
the fugacity for U(1)h . We remind the reader that the power of qp is half the R-charge forscalar operators and we observe that all the operators are above the unitarity bound. Let us
count some of the operators contributing to the index. The relevant operators of the modele and M which comprise the (2, 6) and (1, 15) of (SU(2), SU(6)), which gives 27 of
are QQe , a singlet of non abelian symmetries. At order q p, as we have discussed,
E6 . We also have M
assuming the theory flows to an interacting conformal fixed point, the index gets contributions
only from marginal operators minus conserved currents for global symmetries. The operatorse e ((3 + 1, 1)):
contributing at order q p are gaugino bilinear λλ ((1, 1)), QψQ ((1, 35 + 1)), Qψ
Q
These operators give the contribution,
1 − (1, 35) − 1 − (3, 1) − 1 ,which gives the conserved currents for the symmetry we see in the Lagrangian. Here ψ F is thecomplex conjugate Weyl fermion in the chiral multiplet of the scalar F . We also have operatorseQ,
e and QQM , which cancel out in the computation since the first two aree, MeQψ M M , ψ Me M
fermionic while the second two are bosonic, but are both in the same representation of thee ((2, 70)) and Qψ M Q
e ((2, 20 + 70)). Theseflavor symmetry pairwise. Finally we have Q3Q
two contribute−(2, 20)to the index, which, combined with the above, forms the character of the adjoint representationof the E6 × U(1)h symmetry. We emphasize that the fact we see −78 at order qp of the index isa proof following from representation theory of the superconformal algebra that the symmetryof the theory enhances to E6 (See the discussion around (56).), where the only assumption isthat the theory flows to an interacting fixed point. We also observe that the conformal manifold
here is a point as we do not have any positive contributions to the index. As the index at orderq p has positive contributions from marginal operators and negative from conserved currents,
cancellations can occur. However, this would imply that the symmetry of the IR fixed point iseven bigger: Adding marginal operators we have to add also currents. We cannot rule out thispossibility but we have no evidence for its existence. So, under the assumption that we haveidentified the IR symmetry correctly the conformal manifold is a point.
The enhancement of symmetry to E6 follows from Seiberg duality of SU(2) SQCD with
N f = 4. Note that we can reorganize the gauge charged matter into two groups of four chiralfields. We take four out of the six Qs and call them fundamentals and combine the othere and call those anti-fundamentals. This also decomposes the symmetry SU(6) to
two with Q
31

SciPost Phys. Lect. Notes 78 (2024)

2B
2A

6B

SU(2)

SU(2)

4B

2A

2B
4B

SU(2)

2A

Figure 17: Explanation of SU(6) × SU(2) → E6 enhancement from Seiberg duality.
SU(4) × SU(2) × U(1)h′ with a combination of U(1)h′ and U(1)h being the baryonic symmetry,see Figure 17. IR duality [2] without the gauge singlet fields will then map the baryonicsymmetry to itself while conjugating the two SU(4) symmetries and adding sixteen gauge
singlet mesonic operators. With our choice of gauge singlet fields, the flipper fields of 17 are
flipping eight of the baryons and the bifundamental gauge invariant operators form half of themesons. Thus, the duality removes half of the mesons which connect SU(2) with SU(4) andattaches the other half between the SU(4) and the other SU(2). This transformation acts only
on the symmetry leaving the quiver structurally unchanged. The action on the symmetry is as
the Weyl transformation which enhances SU(6) × SU(2) symmetry to E6 . Note that in generaldualities take two different UV theories to the same conformal manifold in the IR, but here
as the conformal manifold is a point they actually are part of the symmetry group of the IR
theory.
This is an example of a generic phenomenon of the interplay between symmetry and duality
which we will encounter in several guises in what follows. In the last Lecture we will have ageometric explanation of the enhancement of the symmetry to E6 in this example.

3.3

Conformal dualities

Next, we consider yet another interesting strong coupling phenomenon, which however does
not involve an RG-flow. We want to consider the case when a given SCFT T1 resides on a
non trivial conformal manifold. We assume that there is an interesting/useful definition ofthis particular point of the conformal manifold. The conformal manifold is then spanned byexactly marginal deformations at T1 . One thing that can happen, and often does happen,is that there is another locus of the conformal manifold, T2 , where we have a completely
different description of the theory. The theory T1 then can be thought of as an exactly marginal
deformation of T2 , and vice versa. This multitude of descriptions is called a conformal duality.
A prototypical example is N = 4 SYM with gauge group G which has N = 4 preserving one
dimensional conformal manifold parameterized by complexified gauge coupling τ. At strong
coupling a dual weakly coupled description emerges in terms of N = 4 SYM, but now witha Langlands dual gauge group L G [3] (for SU(N ) the dual is SU(N )/ZN ). In general, such aduality might relate two weakly coupled gauge theories as in the case of N = 4 SYM, a weakly
coupled gauge theory and a strongly coupled one defined in a certain way (say geometrically),
or two strongly coupled theories which have certain independent definition (see Figure 18).
32

SciPost Phys. Lect. Notes 78 (2024)

T1

T2

T1

T2

ℳc

ℳc

T1

T2

ℳc
Figure 18: Conformal dualities. Theories residing on the same conformal manifold
are usually called conformal dual to each other. A theory T1 can have its own description or can be thought of as an exactly marginal deformation of T2 , and viceversa. This tautological definition of duality becomes interesting when indeed thetwo theories Ti have a rather different independent definition. For example, one, orboth, are given by a weakly coupled gauge theory. We denote here weakly coupledpoints on the conformal manifolds as cusps as these are singular loci on the conformal manifold.
We will be interested in addressing the following question [22]. Given a conformal theory
T1 which has a conformal manifold Mc we can define certain observables of T1 which are
invariant on Mc . For example, we have already discussed that the ‘t Hooft anomalies of symmetries preserved on the conformal manifold are such an invariant, and for supersymmetrictheories also the a and c anomalies, and various indices are invariant.15 In particular the dimension of the conformal manifold dim Mc and the symmetry preserved on a generic locus of
the conformal manifold G F , are such invariants.16 Given T1 and the set of its Mc invariants
we will systematically seek for a dual conformal gauge theory T2 . Following our discussion wewill be either able to prove that such a dual does not exist, or find a conjecture for such a dual.
The conjecture will be supported by comparing robust supersymmetric properties and various
anomalies. Even given that these agree the conjecture might in principle still be wrong.
The algorithm for seeking for a dual proceeds as follows. We first write down the a andthe c anomalies of the theory T1 . These are quantities which for supersymmetric theories do
not change on the conformal manifold. We parametrize the result as,a = a v dim G + aχ dim R ,c = c v dim G + cχ dim R .

(68)

1 1
On the right hand side of each expression we have denoted by (aχ , cχ ) = ( 48
, 24 ) and
3 1(a v , c v ) = ( 16 , 8 ) the contributions to the a and c anomalies of free chiral and vector fields
respectively. The anomalies of the free chiral field are computed assigning to it R-charge 23 .
Next, the variables dim G and dim R can be thought of as the “effective number” of vector multiplets and chiral multiplets of our theory. The a and c anomalies are two independent numbers
which we can parametrize uniquely by the two independent numbers dim G and dim R.
The putative dual conformal gauge theory T2 should have the same number of vector andchiral fields as we assume it is a weakly coupled conformal deformation of a free theory. If
a and c imply that these numbers are fractional, a conformal dual gauge theory does not exist.
15
For non supersymmetric theories as there is no relation between ‘t Hooft and conformal anomalies and the canomaly in principle can change on the conformal manifold [78] (see also [79]).
16
As we have discussed the exactly marginal operators correspond to certain short multiplets, number of whichcannot change as we vary continuous parameters.

33

SciPost Phys. Lect. Notes 78 (2024)

Moreover, the number of vector fields dim G must be a sum of dimensions of non-abelian gauge
groups. Given the solution to the above problem exists, we then write down all the conformal
gauge theories with the given number of free vectors and free chiral fields with an additional
demand that the one loop beta function of all gauge couplings is zero. Importantly, the number of theories in this step is finite. If we find theories which have (one-loop) marginal gauge
couplings we need to understand whether these theories are free or possess a non trivial conformal manifold. This conformal manifold also has to have the same dimension, dim Mc , andsame symmetry on a generic locus, G F , as T1 . Here the computation is performed by computing the Kähler quotient [40] as we have discussed in the previous Lecture: The number andproperties of holomorphic combinations of couplings which are singlets under the full global
symmetry group. The final step of the algorithm is to take all the theories with matching conformal manifolds and symmetries on the general locus and do finer checks: These can include,
but are not limited to, various supersymmetric partition functions and ‘t Hooft anomalies. Any
theory passing the last step qualifies as a conjecturally conformal dual model to T1 . Importantly
if a conformal dual to T1 exists we are guaranteed to find it using this procedure.
We will now discuss in detail a simple application of this algorithm to produce a duality
between two weakly coupled conformal gauge theories [22]. As theory T1 we will take N = 1
SQCD with gauge group G2 , three fundamental fields (7) Q i , and one chiral field in the 27,
e (see Figure 19). The irrep 27 of G2 appears in Sym2 7 = 27 + 1. The one loop gauge beta
Q
function of this model vanishes,




2
2
2
Tr R (G2 ) = 4 + 1 ×−1 ×3+9×− 1 = 0,(69)
33which determines then the superconformal R-charges of all the chiral fields to be the free ones.
In the above computation 4 is the Dynkin index of the adjoint representation coming from thegauginos, 1 is the Dynkin index of the fundamental representation coming from the Q i ’s, ande
9 is the Dynkin index of 27 and coming from Q.
The non-anomalous global symmetry of the free locus of this SQCD is U(1) × SU(3). The
fundamental fields Q i are in a three dimensional representation of SU(3) (with no loss ofgenerality we can take it to be the fundamental) and we will assign these U(1) charge −3.
The field in 27 is the assigned charge +1, which renders the U(1) symmetry non anomalous,
Tr U(1) (G2 )2 = 1 × (−3) × 3 + 9 × (+1) = 0 .

(70)

Next, we analyze the marginal operators. The fundamental irrep of G2 contains a singlet in its
cubic antisymmetric power. Thus, operator γ εi jk Q i Q j Q k is gauge invariant and is a marginalone. Moreover, this operator is a singlet of the SU(3) global symmetry and has U(1) charge −9.
e An interesting group theory fact is that Sym3 27
We can also build marginal operators from Q:
has two independent invariants. These two then provide two additional marginal operatorse 3 and γ2 (Q)e 3 . Both of these operators are of course singlets ofwhich we will denote as γ1 (Q)1
2
SU(3) and have U(1) charge +3. We have one additional marginal operator which contains
e [i Q j] . Unlike the other operators this transforms under SU(3) inboth types of fields, γi j QQ
the symmetric six dimensional representation. The U(1) charge of this operator is −5. This
theory has one gauge coupling and an anomalous symmetry under which all the fields can bechosen to have the same positive charge. Thus all the marginal operators will have a positive
charge, implying that the associated coupling all have negative charge. However, the gauge
coupling carry positive charge. Thus, in computations of the conformal manifold in this case
we can always account for the anomalous symmetry by an appropriate factor involving thegauge coupling and we will not discuss this further.
Next we analyze the dimension and properties of the conformal manifold of this SQCD. To
do so we need to compute the Kähler quotient [40],{γ, γ1,2 , γi j }/(SU(3) × U(1))C .
34

SciPost Phys. Lect. Notes 78 (2024)

Q̃

Q

G2

3

Figure 19: The G2 conformal SQCD.
q

GF = SU(3)ℳc

G2 SQCD

SU(2)
SU(2)
F = SU(2)First let us take the couplings which preserve
the SU(3),{γ, γ1 , γ2 }. From these three weGcan
construct two independent singlets (γ3i γ) giving rise to two exactly marginal directions. The
φconformal manifold is thus not empty andon this twoφ̃dimensional locus we break the U(1)symmetry while preserving the SU(3) one.
We have however an additional exactly
marginal
SU(3) direction
˜𝒳
𝒳 which involves also the operator
charged under the SU(3) symmetry. For example let us consider exploring the two dimensional
manifold above. Now as only the SU(3) symmetry is preserved we need to find whether wecan build a singlet from powers of γi j . A natural singlet is a determinant of this symmetricmatrix,εi1 i2 i3 ε j1 j2 j3 γi1 j1 γi2 j2 γi3 j3 .

Since the γi j transforms under the SU(3) symmetry it is broken along this direction. However,
it is not completely broken: There is an SO(3) symmetry preserved. We can deduce this forexample by noting that if we imbed SO(3) in SU(3) such that the adjoint of the former is thefundamental of the latter the six dimensional irrep of SU(3) decomposes into a singlet anda five dimensional irrep of SO(3). The adjoint of SU(3) decomposes into a five dimensional
irrep and the adjoint of SO(3). Thus, the five dimensional fraction of the conserved current can
combine with the five dimensional part of the marginal operator to leave behind the adjoint ofSO(3) (the conserved current) and the singlet exactly marginal deformation. The only ’t Hooft
anomaly involving the SO(3) symmetry we need to compute is,214
Tr R SO(3)2 =−1 ×7×2=− .
(71)
33Here 2 is the Dynkin index of the adjoint of SO(3) of which we have 7 coming from thefundamentals of G2 . All in all we thus have a three dimensional conformal manifold on general
locus of which the G F = SO(3) symmetry is preserved.
We next ask whether we can find a conformal gauge theory T2 with exactly same properties
as above. Since T1 itself is conformal gauge theory we know that any such conformal dual has
to havedim G = dimG2 = 14 ,dim R = 27 + 3 × 7 = 48 .
The only combinations of dimensions of simple compact Lie groups which give us 14 are theG2 and SU(3) × SU(2) × SU(2). We will seek a dual with the latter choice. We need to build
a theory with SU(3) × SU(2) × SU(2) gauge group and total of 48 chiral fields forming some
representation of the gauge group. The matter content should be such that the one loop beta
function for the gauge couplings vanishes. One of the finite number of choices accomplishing
this is depicted in Figure 20. The fields X and Xe are in the 6 and 6 representations of SU(3).
We will next argue that it is plausible that the quiver theory of Figure 20 is in fact a conformal
dual to the G2 SQCD.
To support our claim we need first to compute the dimension of the conformal manifold
and the symmetry on the generic locus of this manifold. At the free locus the quiver theory has
SU(3) × SU(2)2 × U(1)2 as its non-anomalous global symmetry group. The non-anomalous
U(1)2 can be chosen as follows: We assign charges (+1, 0) to the bifundamentals (the q fields)
e are assigned charges (−1, 0); and finally the
between the two SU(2)’s; the fields φ and φ
35

GF = SU(3)SU(2)2

SU(2)2SU(3) quiver

Q̃

Q

G2

GF = SU(3)3ℳc

SciPost Phys. Lect. Notes 78 (2024)

G2 SQCD

q

𝒳

SU(2)

SU(2)

φ

φ̃

SU(3)

GF = SU(2˜𝒳

Figure 20: A quiver theory which is conjectured to be conformally dual to the G2
SQCD. The superpotential of this model includes all gauge invariant marginal terms
one can construct from the fields.
X and Xe fields have charges ( 45 , ±1) respectively. One can verify that this charge assignment
is not anomalous with respect to all three simple gauge group factors. Let us also comment
in detail on the anomalous symmetries. We have three such symmetries which we denotee has chargeby U(1)Ai . Under U(1)A1 and U(1)A2 the fields q have charge +1 while (ϕ, ϕ)(+1/ − 1, −1/ + 1) for the former/latter. Uner U(1)A3 the fields q have charge −1 while ϕ ande have charge +1. The fields in the symmetric representation are not charged under theseϕsymmetries. Note then that the gauge coupling of the SU(2) on the left of Figure 20 onlytransforms under U(1)A1 . Similarly the gauge coupling of the SU(2) on the right transforms
only under U(1)A2 , and the SU(3) gauge coupling transforms only under U(1)A3 . The gauge
couplings are charged positively under the anomalous symmetries.
Having analyzed the symmetries we discuss next the marginal operators. First one, which
we will denote by λ, is built from triangles in the quiver. These marginal operators are in(3, 2, 2)(−1,0) of (SU(3), SU(2), SU(2))U(1)2 . The second type of marginal operators, which wewill denote by λ± , corresponds to X 3 and Xe 3 , which have charges (1, 1, 1)( 12 ,±3) respectively.
5e2 which have
The third and last type of operators, which we will denote by λ′ , are X φ 2 and Xe φ±charges (1, 1, 1)(− 6 ,±1) . Under all three anomalous symmetries the λ couplings have charge
5−1 and the λ± couplings are not charged. The λ′± couplings are charge −2 under U(1)A3 , ±2
under U(1)A1 and ∓2 under U(1)A2 .
Let us first consider turning on only marginal deformations λ± and λ′± . These couplings are
singlets under the non-abelian symmetries. Under the non-anomalous symmetries the following two independent combinations of the couplings are not charged, x + = (λ′+ )12 (λ+ )(λ− )5and x − = (λ′− )12 (λ− )(λ+ )5 . Considering the anomalous symmetries the combination x + x − ischarged only under U(1)A3 negatively and thus this can be offset by an appropriate power ofthe exponent of the gauge coupling of the SU(3) gauge group. We thus have a one dimensional conformal manifold on which only SU(3) gauge coupling is non vanishing and λ± and
λ′± are turned on. These deformations break all the non-anomalous abelian symmetries aswell as U(1)A3 and an off-diagonal combinations of U(1)A2 and U(1)A1 . Along this direction
the non-abelian SU(3) × SU(2)2 symmetry is preserved.
We can build two additional exactly marginal operator using the deformation λ. This
will break all the symmetry but the diagonal combination of the two SU(2)s and SO(3) inSU(3). This comes about again as the decomposition of the representation of the marginal
operator is (3, 2, 2) → 2 × 3 + 5 + 1 and the decomposition of the conserved currents is(8, 1, 1) + (1, 3, 1) + (1, 1, 3) → 3 × 3 + 5. Two of the three 3 and the 5 recombine with36

SciPost Phys. Lect. Notes 78 (2024)

the relevant components of the marginal operators leaving behind a single singlet of SU(2).
We have broken all the abelian symmetries but the diagonal combination of the anomalous
symmetries U(1)A2 and U(1)A1 by turning on λ± and λ′± and the SU(3) gauge coupling. Under
this diagonal combination λ has a negative charge which can be offset by appropriate powers of the exponents of one of the two SU(2) gauge couplings giving two additional exactly
marginal deformation.
Thus, as we obtained for T1 , also T2 has a three dimensional conformal manifold with
G F = SO(3). Note that both duality frames have loci with enhanced symmetry which are
however different. This is not a contradiction of the duality as the two do not have to intersect.
Let us next do finer checks of the duality. First we can compute the ‘t Hooft anomaly
involving the SO(3) symmetry on the quiver side,


2
1
14
2
Tr R SO(3) =− 1 (6 × 2 × + 4 × 2) = − .
(72)
3
2
3
Here 21 is the Dynkin index of the fundamentals and 2 is the index of the adjoint. The adjoint
comes from the triplet of SU(2) bi-fundamentals (there are thus four adjoints), while thefundamentals come from the bi-fundamentals of SU(2) and SU(3) (and thus there are 6 × 2of those). Note that this precisely matches (71): The matter is very different but anomaly isexactly the same providing a rather non-trivial check of the proposed duality.
Finally, we can try to match the supersymmetric indices on the two sides of the putative
duality. Following the general technology of computing the index detailed in the previous
Lecture we have for the G2 SQCD,


2
1
1
1
2
3
3
I G2 = 1 + (h + 6 6SU(3) )(pq) + 2h + 5 6SU(3) + 9 − 8SU(3) − 1 p q + · · · ,hhhwhere we have refined the index with the full non-anomalous symmetry of the free theory. The
fugaicty h is the fugacity for the single U(1) symmetry while the bold face numbers encode
irreps of SU(3). To compute the index we need the characters of various representations ofG2 and for completeness we give them here,χ7 ({z}) = z1 + z2 +1
1
1
+ + + z1 z2 + 1 ,z1 z2 z1 z2(73)

1(χ7 ({z})2 − χ7 ({z 2 })) − χ7 ({z}) ,21
χ27 ({z}) = (χ7 ({z})2 + χ7 ({z 2 })) − 1 .
2

χad j.=14 ({z}) =On the other hand the index of the putative quiver dual is,
82

Iqui ver = 1 + (a 5 + a2 6SU(3) ) (pq) 3
(74)


12
1
1
1
1
+ a 5 (b3 + 3 ) + 2SU(2)1 2SU(2)2 3SU(3) + 6 (b + ) − 3SU(2)1 − 3SU(2)2 − 8SU(3) − 2 p q + · · · .
baba5
We again computed the index at the free locus of the quiver refining it with the full nonanomalous symmetry group. The a and b fugacities parametrize the two non-anomalous U(1)swhile the bold face numbers denote irreps of the corresponding non-abelian groups. The indices of the two theories at the free locus, (73) and (74), look rather different: This is to be
expected as at these points new non-generic symmetries emerge. We should make a comparison only refining the index with symmetries appearing on a generic locus of the conformal
manifold. In our case this is the SO(3) symmetry. This implies that we need to set to 1 the37

SciPost Phys. Lect. Notes 78 (2024)

𝒬1 :

1

−1

Φ
q

Q
6

𝒬3 :

4

SU(2)

USp(4)
1
3

𝒬2 :

Q̃

−

𝒳

1

1
2

−3

1

Figure 21: The conformal dual theory, where X is in the traceless antisymmetricrepresentation (5) and Φ is in the symmetric representation (10) of US p(4).
2

N

N-1

1

N-2

1
1
1
2
2 take,2
fugacities for all the abelian symmetries and furthermore

1
2

3SU(3) → y 2 + y −2 + 1 = 3SO(3) ,2SU(2)1 = 2SU(2)2 = y + y−1= 2SO(3) .

1
2

(75)
(76)

With this the two indices match on the nose,2

I G2 = Iqui ver = 1 + (2 + 5SO(3) )(pq) 3 + (3 − 3SO(3) ) p q + · · · .

(77)This equality had to hold if the duality claim is correct.17 We emphasize that this is not a
proof of a duality. One can perform more checks (one of which is detailed in [22]). The twodifferent gauge theories are expected to describe two different weakly coupled cusps of thesame conformal manifold.
In the following Lectures we will see additional examples of such conformal dualities.
Those examples will have a simple geometrical explanation, though for the duality presented
here no such interpretation exists at the moment.

3.4

Exercise: Lagrangian dual of a class S theory
17
Exercise: Given an SCFT in 4d with a = 15
4 and c = 4 , 26 supersymmetric relevant
deformations, and 33 dimensional conformal manifold on a generic locus of which there
are no global (non-R) symmetries, find a candidate conformal Lagrangian description.
(In A2 class S [7, 8] these are the properties of an N = 2 theory corresponding to asphere with three maximal and one minimal punctures. See Appendix D.)

The conformal anomalies are such that dim R = 63 and dim G = 13. The only choiceof gauge group which gives dim G = 13 is su(2) × usp(4). Then going over possible matter
content such that dim R = 63 and the one loop gauge beta functions vanish a candidate quiver17

We do not have a proof of this equality to all order in expansion in parameters but rather this has been checked
to some high but finite order. We need to stress here that matching the indices to any finite order in expansion
in fugacities does not constitute a proof of matching the indices precisely. There is a plethora of examples, seee.g. [80, 81], where the indices of two theories match to arbitrary high order but the theories are different.

38

SciPost Phys. Lect. Notes 78 (2024)

theory is depicted in Figure 21. Let us count the supersymmetric relevant deformations. As
the theory is free the relevant deformations are the ones given by quadratic gauge singlets,
e 2 : (1, 1, 6)0,2,0 ,Q2 : (15, 1, 1)2, 2 ,2 , Q
3Φ2 : (1, 1, 1)−2,0,0 ,q2 : (1, 3, 1)0,−1,0 ,X 2 : (1, 1, 1)0,0,−6 .

(78)Here (R1 , R2 , R3 )q1 ,q2 ,q3 are representations/charges under the global symmetry of the freetheory, (SU(6), SU(2), SU(4))U(1)1 ,U(1)2 ,U(1)3 . Note that the total number of relevant operators
is 26 as needed.
Next, we write the most general marginal superpotential,e.
W = λ1 Φ2 X + λ2Q2 X + λ3Q2 Φ + λ4 q2 Φ + λ5 q2 X + λ6QqQ(79)

We then list the representations under the global symmetry of the operators the couplings λi
couple to (the couplings are in the conjugate representations),
λ1 : (1, 1, 1)−2,0,−3 , λ2 : (15, 1, 1)2, 2 ,−1 , λ3 : (21, 1, 1)1, 2 ,2 ,3

λ4 : (1, 1, 1)−1,−1,0 ,λ5 : (1, 3, 1)0,−1,−3 ,3

λ6 : (6, 2, 4)1, 5 ,1 .

(80)

6

We also have two symmetries U(1)SU(2) and U(1)USp(4) , which are anomalous under the corresponding gauge symmetries. We define the charges under these as,
U(1)SU(2) :U(1)US p(4) :11
e = 1,[Q] = − , [q] = , [Q]3211e = −1 .
[Q] = ,[q] = , [Q]
3
2

(81)

First, assuming the symmetry is completely broken on the conformal manifold, the dimension
is given by the number of marginal operators minus the number of currents,
1 + 15 + 21 + 1 + 3 + 6 × 2 × 4 − 35 − 3 − 15 − 1 − 1 − 1 = 33 ,(82)

which is the needed dimension of the conformal manifold. Let us now compute the Kähler
quotient{λi , e2πiτSU(2) , e2πiτUSp(4) }/(SU(6) × SU(2) × SU(4) × U(1)3 × U(1)SU(2) × U(1)USp(4) )C .
(83)
First,
Λ0 ≡ λ1 λ4 λ5 (λ3 )3 ,(84)

has charge zero under all the non-anomalous abelian symmetries and under U(1)SU(2) . Under
U(1)US p(4) it has a negative charge and thus dressing this with appropriate power of e2πiτUSp(4)this charge can be offset to be zero. Moreover Λ20 is a singlet of all the non-abelian symmetries
also. This follows from the fact that Λ0 involves only couplings which are singlets of SU(4);the symmetric square of adjoint of SU(2) contains a singlet; the sixth symmetric power of 21
of SU(6) contains a singlet. Thus we establish that the Kähler quotient is not empty and thetheory is a non trivial SCFT. Along this direction the SU(2) gauge coupling is zero. What is
the symmetry preserved by the US p(4) gauge couplings and λ1,3,4,5 ? All the non-anomalous
abelian symmetries and the U(1)US p(4) are broken as the couplings are charged under all ofthem. The SU(4) symmetry is preserved. The Cartan of the SU(2) is preserved as the only
deformation charged under the SU(2) is a single adjoint. Finally, the SU(6) symmetry is broken
39

SciPost Phys. Lect. Notes 78 (2024)

to SO(6) such that 6 → 6. In particular the 21 decompose to 20′ +1 and adjoint to 15+20′ andthus 20′ of the marginal operators recombines with the same representation of the conserved
currents and we are left with a singlet of SO(6). All in all we have one dimensional conformal
manifold the marginal operators are in representations,(1, 1)0 + (6, 4)+1 + (6, 4)−1 + (15, 1)0 ,(85)

and in addition we have SU(2) gauge coupling. Here we write representations/charges interms of the preserved (SO(6), SU(4))U(1) subgroup of the global symmetry, with the U(1)being the Cartan of the SU(2) at the origin of the conformal manifold.
We have gauge coupling for the SU(2) left as well as the deformations λ2 and λ6 . We
can build an invariant under SU(4) taking the baryonic combinations (λ±)4 which are in the6adjoint of the SO(6). As λ6 has opposite charge under anomalous symmetry U(1)SU(2) to)4 (λ−)4 (dressed with a proper
the SU(2) gauge coupling we then can build an invariant (λ+66factor of gauge coupling) under all symmetries. This deformation will break the anomalous
symmetry U(1)SU(2) , the remaining Cartan of the SU(2) symmetry, as well as the two non
abelian symmetries SU(4) and SO(6). The couplings λ2 will be exactly marginal. We thus
are left with conformal manifold of dimension 33 with all the symmetry broken on a genericlocus. This completes the solution of the exercise.
□
The algorithm detailed here for the search of conformal dualities can be generalized tosearch for non-conformal dualities involving flows, as long as we can assume that the spectrumof R-charges is constraints. For examples of such generalizations see [82, 83] where a variantof the algorithm was applied to find IR duals of an E6 Minahan-Nemeschansky model [84] andof some N = 3 theories.

4

Lecture III: Across dimensional dualities, an example

In the previous Lecture we have discussed several scattered observations about IR physics ofsimple gauge theories. The question we want to ask now is whether this scattered plethora offacts has any interesting organizing principle. We will show that such an underpinning can befound by thinking about 4d QFTs geometrically. The logic is as follows.
One can consider discussing higher dimensional SCFTs. Following the notorious theoremof Nahm [85] (see also [86]) the maximal number of dimensions in which an interacting superconformal theory can reside is six dimensions. In higher dimensions a superconformal algebra(with no higher spin currents) just does not mathematically exist. There are two types of superconformal algebras in 6d, (1, 0) and (2, 0), differing by the amount of supersymmetry: The
former one having 8 real supercharges and the latter 16 real supercharges. The (2, 0) SCFTs
are conjectured to be classified by an ADE algebra. A type (2, 0) theories can be engineered forexample as the low energy effective theory residing on M5 branes in M-theory constructions.
On the other hand there is a quite huge plethora of (1, 0) SCFTs. Here also there are various
classification approaches [87,88] (See [10] for a recent review.). One important fact about the
6d SCFTs is that all of these are strongly coupled. In particular, due to dimensional analysis thegauge couplings and all superpotential interactions in 6d are irrelevant; thus, gauge theories
flow to free theories in 6d. Phrasing this more abstractly, 6d SCFTs do not possess any Lorentz
preserving supersymmetric relevant or exactly marginal deformations [20].
As we are interested here in SCFTs in 4d we can then start from some given 6d (1, 0) theory (with the (2, 0) being just a more supersymmetric special case) and place it on a compact
two dimensional space. Such compact spaces are called Riemann surfaces and are classified
40

CFTUV2 + Δ2

𝒢UV2
CFTUV2CFTUV

SciPost Phys. Lect. Notes 78 (2024)

𝒢IR

𝒢IR

4d SCFTIR

CFTIR

4d SCFTUV + Δ′

RG FlowΔ=
6d SCFTUV + Δ

𝒢′UV
4d SCFTUV

𝒢UV6d SCFTUVFigure 22: A 6d CFT can be placed on a compact geometry. At energy scales far
below the scale set by the geometry the theory should be effectively described by afour dimensional model, which would then flow to some fixed point in the IR. The
geometry can be thought of as a deformation ∆ of the 6d fixed point and triggers an
RG flow across dimensions. The resulting IR SCFT in 4d might or might not be a fixed
point of an RG flow starting from a Gaussian fixed point in 4d deformed by a relevant
deformation. If such a theory exists we have an IR duality between deformations ofa 6d theory and a 4d one.
by their genus g. We will also allow to decorate the surfaces with marked points (which will
have certain physical meaning in terms of either defects or boundary conditions). Althoughsuch a space is curved for g ̸= 1 and thus breaks supersymmetry, it is possible to perform thecompactification with a certain twist, which we will soon review, such that half of the supersymmetry is preserved. Half of eight supercharges of (1, 0) will lead to four supercharges, see
Appendix F. One way to think about the geometry is as a certain deformation of the 6d SCFT.
In the IR, far below the energy scale set by the geometry, the theory becomes effectively four
dimensional. Preserving four supercharges we will obtain an N = 1 theory in four dimensions.
This effective theory might be a strongly coupled SCFT, a theory of free chiral fields, a weakly
coupled gauge theory, or a combination of these options.
An interesting question then is whether we can find a description of this effective theory
directly in four dimensions. That is whether there is a four dimensional Lagrangian which
either flows to this theory or directly describes the conformal fixed point. Such a geometricengineering of the four dimensional models produces a huge plethora of theories. Some ofthese theories have properties such that no four dimensional model is known to produce directly, and thus sometimes these are called non-Lagrangian. If a Lagrangian description is found
one can think of the setup then as a novel type of an IR duality between a 6d theory deformed
by geometry and a 4d Lagrangian theory, see Figure 22. One way to think about such a set ofdualities is as map, a dictionary, between the set of 6d SCFTs and two dimensional geometries
into the set of four dimensional supersymmetric quantum field theories,
T 4d T 6d , C ,(86)
where C collectively labels the geometry and T 6d the choice of the 6d starting point. The whole
program can be of course imbedded in the larger structure of M-theory and there is a lot of
benefit to be extracted from this, but we will concentrate on working within the paradigmof local quantum field theory. Our goal in the following two Lectures will be to derive some41

SciPost Phys. Lect. Notes 78 (2024)

entries in this dictionary and exemplify the utility of the procedure for understanding non
trivial physics in four dimensions.
The fact that there is a geometry behind the construction leads to numerous very powerfulunderstandings about the four dimensional physics: e.g, many of the dualities and emergenceof symmetry phenomena of the type we have discussed can be explained, and in fact predicted,
from such constructions. The prototypical example of the geometric constructions is compactifications of (2, 0) theories leading to N = 2 theories in 4d. A lot has been understood aboutsuch models following the seminal work of Gaiotto [8]. Here, however we will concentrate onsome simple (1, 0) examples. Let us first consider the compactification of one of the simplest
interacting 6d SCFTs, the so called SU(3) minimal SCFT [89, 90] (or SU(3) non-higgsable
cluster [91]). We will mainly follow the discussion in [24].

4.1

Six dimensions

One way to construct an SCFT in six dimensions is as follows. We consider a gauge theory,
which as we mentioned is necessarily IR free. Without going into many details (see Appendix
E and the review [10] for more details), a generic such theory can be built from three typesof (1, 0) multiplets: A vector multiplet, a tensor multiplet, and a hyper-multiplet. We will
not need the hypermultiplets for our discussion but let us mention that the vector multiplet
contains of course a gauge field and a fermion: In terms of representations of the little group
SU(2)×SU(2) these massless states are in ( 12 , 12 ) and (0, 12 ), while the tensor multiplet contains
a tensor field (in irrep (1, 0) which is a two-form with self-dual field strength), a fermion (inirrep ( 21 , 0)), and importantly a scalar (in irrep (0, 0) of course). The scalar in the tensor
multiplet (See Appendix E for discussion of the multiplets.) can be naturally coupled to thefield strength (schematically),Zd 6 x φ F ∧ ⋆F ,(87)

and thus the gauge coupling in six dimensions can be thought of as an expectation value ofthe scalar field residing in the tensor multiplet, 〈φ〉 ∼ 1/g Y2 M . In particular one can ask what
happens if this vev is set to zero. In certain cases it is believed that the theory one obtains
is a strongly coupled CFT. Conversely, a strongly coupled SCFT might contain a moduli spaceof vacua, called the tensor branch, on which certain scalar operators acquire a vev and theeffective description in the IR is in terms of an IR free gauge theory consisting of vector multiplets, tensors, and maybe hypermultiplets. This is very reminiscent of the N = 2 dynamics infour dimensions. There an N = 2 vector multiplet contains a scalar (which in terms of N = 1
language corresponds to an adjoint scalar in an N = 1 chiral superfield in the N = 2 vector
superfield), and giving a vev to this scalar we move on the Coulomb branch of the theory on
which typically the description is in terms of abelian, IR free, gauge theories. Switching off
the vev would take us back to the SCFT point.
The effective gauge theory description on the tensor branch is constrained to be nonanomalous. As the vector multiplet contains fermions, for example, just taking (1, 0) vectors
usually leads an anomalous theory. However, the theory with a vector and a tensor in somecases can be made to be non anomalous. This comes about as naturally due to supersymmetry
the term of the form (87) is accompanied by,Zα

d 6 x B ∧ Tr F ∧ F ,(88)with B being the tensor (see Appendix E) and α being some real constant. Such a term actuallycontributes to the eight-form anomaly polynomial of the 6d theory due to the Green-Schwarz
42

SciPost Phys. Lect. Notes 78 (2024)

mechanism [92, 93] a term of the form (see Appendix E for more details)
α2 (Tr F ∧ F )2 ,(89)

and this can be used to cancel the term of a similar structure coming from the vector multiplet.
Now let us then consider a gauge theory in 6d with a simple gauge group G and a tensor
multiplet. The contribution to the eight form anomaly polynomial with four field strengths
can come in two different forms,
Tr F ∧ F ∧ F ∧ F(Tr F ∧ F )2 ,

and

(90)

with the difference being different contractions of the indices. Specifically, the second term
contracts the indices using the Cartan form squared, while the first terms uses the quartic
Casimir, which is a completely symmetric invariant polynomial of order four. For most groups,
the two terms are independent, but there are some groups that do not possess an independent
quartic Casimir, in which case the two terms become equivalent and there is only one type ofgauge anomalies. The groups for which this happens are: SU(2), SU(3), G2 , F4 , E6 , E7 andE8 .
For a theory to be non-anomalous both types of anomalies must vanish. The second typeof structure can be canceled by the introduction of a Green-Schwarz term of the form (89),provided the contribution of the vector multiplet to the anomaly polynomial comes with anegative coefficient, as in fact it does. However, the first type of structure can only be canceledby matter contributions, specifically, by introducing charged hypermultiplets. As such, if we
insist on a theory containing only tensor and vector multiplets, we must limit ourselves togauge groups that either: A) don’t have an independent quartic Casimir or b) the contributionof an adjoint fermion to this anomaly is zero. We have already listed the options realizing a)and it turns out that there is a single option realizing b), SO(8).18 Additionally we also haveto worry about the existence of a Witten anomaly [67] related to π6 (G) being non-trivial,which is the case for SU(2), SU(3) and G2 . This anomaly leads to the pure SU(2) and G2
theories being inconsistent, though miraculously, an adjoint fermion of SU(3) turns out tocontribute trivially to the anomaly so a pure SU(3) gauge theory is consistent [90]. Overall,
the pure gauge theories with a single tensor that do not suffer from anomalies and thus canbe consistent tensor branch descriptions of some SCFT, are: SU(3), SO(8), F4 , E6 , E7 andE8 [89, 90]. These theories are sometimes called minimal SCFTs in 6d.
We expect that the resulting SCFTs do not have continuous flavor symmetries as on thetensor branch we do not see any. We can evaluate the ’t Hooft anomalies of these SCFTs, using
the gauge theory description, and collect the results in an anomaly polynomial 8-form.
The ’t Hooft anomalies receive contributions from three sources: The chiral fermion in
the vector multiplet, the self-dual tensor and chiral fermion in the tensor multiplet, and theGreen-Schwarz term required to cancel the Tr(F ∧F )2 terms (including the mixed gauge-gravity
anomalies). Let us quote the contributions of the vector multiplet for gauge group G [94],−−

Tr(F 4 )Ad j−

C2 (R)Tr(F 2 )Ad j

24
Tr(F 2 )Ad j p1 (T )
48

4
− dG

−

dG C22 (R)242(7p1 (T ) − 4p2 (T ))
5760

−
.

dG C2 (R)p1 (T )48
(91)Here we use C2 (R) for the second Chern class of the SU(2) R-symmetry bundle in the doublet
representation, and p1 (T ), p2 (T ) for the first and second Pontryagin classes of the tangent
18

SO(8) has the unique property of having two different independent quartic Casimirs. As such in this casethere are actually three different anomaly structures. The existence of the two structures and the reason why thecontribution to both for an adjoint fermion vanishes is related to the special triality automorphism of SO(8), see
the discussion in [24].

43SciPost Phys. Lect. Notes 78 (2024)

bundle respectively. The constant dG stands for the dimension of the group G. Since we do
not have a quartic Casimir the Tr(F 4 )Ad j can be expressed as,Tr(F 4 )Ad j = λG Tr(F 2 )2Ad j .

(92)

We will be interested here only in the group G = SU(3) for which dG = 8 and λG = 41 . The
tensor multiplet contributes,
C22 (R)
24

C2 (R)p1 (T ) (23p1 (T ) − 116p2 (T ))
+
+
,
48
5760
2

(93)

and to cancel all gauge anomalies we need to introduce the Green-Schwarz term which contributes,

2
λG
3
1
2
Tr(F )Ad j +C2 (R) +p1 (T ) .
(94)
24
λG
4λG
Summing up all the terms, we find that the eight-form anomaly polynomial is,
I 6d =

1 9
1 3(− dG + 1)C22 (R) +(− dG + 1)C2 (R)p1 (T )
24 λG
48 λG
+

4.2

( λ15 − 7dG + 23)p12 (T ) + (4dG − 116)p2 (T )
G

5760

.

(95)

Compactification to 4d

Given the 6d SCFT which on the tensor branch is described by the SU(3) gauge theory, wewant to understand what happens when we compactify it on a closed Riemann surface. The
compactifications on lower genus surfaces, g = 0 and g = 1, do not follow the general pattern
we will want to discuss and thus we will refrain from discussing these here. Compactifying on
g > 1 surfaces we will be able to derive some very general statements.
As the Riemann surface with g > 1 is curved, supersymmetry is naively broken. To avoid
this, we twist the SU(2) R-symmetry bundle so as to cancel the curvature of the Riemannsurface for some of the supercharges which are charged under it. The supercharges transform
under the Lorentz group and under the R-symmetry so we need to turn on a certain R-symmetry
bundle so that at least some supercharges will not feel the curved background and thus willremain invariant. We can do so preserving at most 4 supercharges, corresponding to N = 1 in
4d. The twist, the non-trivial bundle for the R-symmetry, also breaks the SU(2) R-symmetry
to its U(1) Cartan sub-group, which becomes an R-symmetry in 4d. See Appendix F for moredetails on the twisting procedure.
Next we will want to deduce the various ‘t Hooft anomalies of the 4d theories. We have an
RG flow across dimensions and thus we need to deduce the six-form anomaly polynomial of
the 4d theory from the eight-form anomaly polynomial of the 6d theory. The way to do so is tointegrate the anomaly polynomial of the six dimensional SCFT on the compact Riemann surface in the presence of all the non-trivial bundles we have turned on [38] (see also Subsection
3.1 of [95]),
Z

I 4d =

I 6d [F ] ,

(96)

C

where by F we collectively denote the values of the background fields. In the current case theonly such fields are due to the non-trivial bundle for the R-symmetry due to twisting, but inmore general cases one also might have non trivial bundles for the global symmetries. This

44

SciPost Phys. Lect. Notes 78 (2024)

should lead to the anomaly polynomial of the 4d theory, which contains the ’t Hooft anomaliesof the 4d theory, at least those for symmetries descending from 6d.
In our case, we need to integrate (95) on the Riemann surface, but first we need to take thetwist into account. This is done by setting (see Appendix F) C2 (R)=−C1 (R)2 +2(1− g) t̂ C1 (R)+. . . ,where C1 (R) is the first Chern class of the U(1) Cartan of the SU(2) R-symmetry and t̂ is aunit-flux 2-form on the Riemann surface. Inserting this into (95) and integrating we find,




1 9
1
3
4d
3
I =
− dG + 1 (g − 1)C1 (R) −− dG + 1 (g − 1)C1 (R)p1 (T ) .
(97)
6 λG
24 λG
1
From the coefficient of C13 (R) we deduce the Tr R3 anomaly and from the 24
C1 (R)p1 (T ) term
we deduce the Tr R anomaly,




9
3
3
Tr R =− dG + 1 (g − 1) ,Tr R =− dG + 1 (g − 1) .
(98)
λG
λG

Finally using the relation between ’t Hooft anomalies of the R-symmetry and the conformal
anomalies (10) we deduce that these are,




3 12
1 33
a=
− dG + 1 (g − 1) , c =− dG + 1 (g − 1) .
(99)
16 λG
8 2λG
In particular for the SU(3) case at hand we get,a=123(g − 1) ,
16

c=

59
(g − 1) .
8(100)

We thus have a prediction for the existence of 4d theories labeled by g with the above
conformal anomalies. These models are also expected not to have any global symmetries.
Of course global symmetries might emerge in principle in the IR, invalidating both of thesestatements. We will assume that this does not happen and will seek the corresponding four
dimensional theories.
Let us here perform a quick computation: We assume as in the previous Lecture that these
4d theories have a conformal gauge theory description. This might or might not be true, andwe will eventually argue that it is true. With this assumption we now deduce what would be
the dimension of the gauge group in 4d and the dimension of the representation of the matterfields. We need to solve,a v dim G + aχ dim R =123(g − 1) ,16

c v dim G + cχ dim R =59(g − 1) ,8(101)

to get dim G = 32(g − 1) and dim R = 81(g − 1). We note that these numbers are integer
and thus the conjecture might be correct.19 Moreover, a natural interpretation of 32 is asfour factors of SU(3) and of 81 as three tri-fundamentals of SU(3). We will soon see how toconstruct such a gauge theory. However, first it will be worthwhile to understand what one
would expect from compactifications on punctured Riemann surfaces, and that entails going
through an intermediate step between 6d and 4d, a reduction to 5d.
19
If one repeats the same exercise for other groups for some the dimensions turn out to be integer while for2187
5
others they do not. For example for F4 (λG , dG ) = ( 108, 52) and (dim G, dim R) = ( 798
5 , 5 )(g − 1) but for E6
1(λG , dG ) = ( 32 , 78) and (dim G, dim R) = (235, 648)(g − 1).

45

SciPost Phys. Lect. Notes 78 (2024)

𝒞

5d SU(3)±9 𝒩 = 1 SYM

6d

∂𝒞

5d

𝒞

Figure 23: Compactifying the 6d SU(3) 5d
minimalon a circle with the complex
SU(2)SCFT
0 Nf = 8 𝒩 = 1 SQCD
conjugation twist is conjectured to be described in 5d by an effective theory which
is SU(3) SYM with level ±9 CS term. The UV completion of the 5d theory is the 6d
SCFT. Here C is the Riemann surface and ∂ C6dis non empty, that is we have a puncture.
Each puncture is associated with a global symmetry corresponding to the 5d gauge
symmetry, which is SU(3) in this case.

∂𝒞

4.3

Reduction to 5d and punctures

5d

Let us discuss how one can think about punctured Riemann surfaces. As the 6d theory at the
CFT point, which we compactify, is strongly coupled it is hard to analyze the punctures directlyin 6d. However, one way to think about the punctures is to elongate the region near a puncture
to a long thin cylinder and first analyze this region. This amounts to compactifying first on
an infinite cylinder of small radius and obtaining an effective 5d theory, and then cutting thecylinder with a specification of a boundary conditions in 5d. If the 5d theory is still strongly
coupled we did not achieve much. However, in certain cases, starting with certain 6d SCFTs
and compactifying on a circle to 5d with certain holonomies/twists, it is conjectured that one
obtains an effective theory in 5d which is a gauge theory.20 As in 6d these gauge theories are
IR free and there are two possibilities for their UV completion: First, it is possible that they
are relevant deformations of non trivial CFTs in 5d, and second, the UV completion might begiven by a 6d SCFT. A canonical example is the ADE (2, 0) theory which is conjectured to give
the N = 2 ADE SYM when compactified to 5d with no twists [108, 109].
If one indeed obtains a gauge theory in 5d with gauge group G 5d , one next can study supersymmetry preserving boundary conditions at the four dimensional boundary of the geometry.
The details of the choices of the boundary conditions depend on the 5d theory, however there
is a canonical set of choices, which is usually called maximal, giving a Dirichlet boundary condition to all the vector fields of the 5d theory and then whatever supersymmetry demands forthe other fields. See Appendix G for more details, as well as [26, 110] (and recent similar discussion in 3d in [111]). Importantly, since the Dirichlet boundary condition breaks the gauge
symmetry to the one which is constant along the boundary, we acquire a global symmetry
which is given by the gauge group G 5d associated with each boundary component, that is witheach puncture.
Let us specialize now this general discussion to the case at hand. It is conjectured [112]that a pure N = 1 SU(3) gauge theory with CS term at level ±9 is obtained by compactifying
the minimal SU(3) SCFT in 6d on a circle with a twist by the complex conjugation symmetry
in 6d, see Figure 23.21 Twisting here means that upon compactification of the circle (defined
There is an ongoing vigorous research into classifying such effects, see e.g. [16, 17, 96–107] for a partial list ofreferences.
21
The fact that the theories in 6d and in 5d are associated to the same group SU(3) is not a generic feature.
For example minimal SO(8) SCFT upon compactification with certain twist reduces to an SU(4)±8 gauge theory in
5d [24].
20

46

SciPost Phys. Lect. Notes 78 (2024)

by angle θ ∈ [0, 2π]) we identify the configurations at θ = 0 and θ = 2π with an action of thediscrete symmetry. Note that the 6d theory does not have any continuous global symmetries
and thus barring accidental appearance of symmetry in 5d, upon compactification we do not
expect to obtain any symmetry beyond the one associated to the KK symmetry of the circle:
The latter symmetry is identified with the topological (instantonic) symmetry of the 5d gauge
theory. Thus we expect a pure gauge theory in 5d. Matching the moduli spaces of the 6d and5d theories we obtain that if there is a gauge theory in 5d it should be SU(3): In 6d the moduli
space is three dimensional but the twist by complex conjugation reduces it to two (projectingout the dimension three Coulomb branch operator of the SU(3)), while in 5d SU(3) has a
two dimensional Coulomb branch. Finally, we need to fix the level of the SU(3) gauge group;
following various string theoretic arguments [112], for levels smaller than ±9 it is believedthat the theory is a deformation of a 5d SCFT, for level ±9 it is UV completed by the 6d SCFT,
and for higher level the theory might not have a UV completion.22
Finally, we need to specify the maximal boundary conditions for the theory at hand. As weare preserving N = 1 4d supersymmetry on the boundary, we decompose the 5d fields on theboundary in terms of the 4d multiplets: Which are N = 2 4d vector fields as the number ofsupersymmetries in 5d is eight. We choose Dirichlet for the vector fields and Neumann for theadjoint chiral (see Appendix G). This will imply that we have Dirichlet boundary conditions forchiral fermions and Neumann for anti-chiral. From this we can compute the anomaly inflow
contribution of the puncture to four dimensions. This comes from several sources: The antichiral fermions with Neumann boundary conditions and the CS term. The former are in theadjoint of the 5d gauge group SU(3), have R-charge −1, and contribute half of the contributionof four dimensional fermions.23 This gives the anomaly contributions
1
1
Tr R = −1 × × dim SU(3) = −4 ,Tr R3 = (−1)3 × × dim SU(3) = −4 ,
2
2
3
1
and
Tr R SU(3)2 = −1 × 3 × = − ,22coming from every puncture. To compute the full anomaly polynomial we also need to take
into account the computation of the integral of the 6d anomaly polynomial on the Riemannsurface which is the same as above but with g − 1 → g − 1 + s/2 where s is the number ofpunctures, and we specialize to the case of the 6d SU(3) SCFT. These give for surfaces with
punctures the anomalies,


123 
s
9
3
3
a=
g −1+
+
(−4) −(−4) s =(82 (g − 1) + 33 s) ,
16
2
32
32
32


(102)
s
9
5
1
59 
g −1+
+
(−4) −(−4) s =(118 (g − 1) + 51 s) .
c=
8
2
32
32
16
In addition the CS term contributes Tr SU(3)3 = ±9, where the sign is determined by the signof the CS term.
Next we consider how we should glue two punctures in 4d. Gluing punctures geometricallycorresponds to bringing two cylinders with boundaries together and un-doing the boundary
conditions, identifying in some way the fields in the two cylinders. Field theoretically thus weneed to gauge in 4d the symmetry associated to the punctures, SU(3) in our case. Note that
the R-symmetry is non anomalous as far as this gauging is concerned as

 

3
3
2
+ −
= 0,
(103)
Tr R SU(3) = 3 + −2
2
22

The consistency of our analysis below can be viewed as additional evidence in favor of the conjecture that levelnine theory is UV completed in 6d.
23
A way to see that the half is needed is that if we compactify a 5d fermion on a finite interval with Neumannboundary condition on both ends, we will get a fermion in 4d. Thus, each boundary will contribute half of theanomaly.

47

SciPost Phys. Lect. Notes 78 (2024)

where the first term comes from the vector fields and the second and third come from the twopunctures. Moreover, as the gauging should be non-anomalous, Tr SU(3)3 = 0, we should
glue a puncture with +9 CS term to the one with −9 CS term. Note that the anomalies (102)are of course self-consistent under this gluing: Adding anomalies with (g1 , s1 ) and (g2 , s2 ),and a vector multiplet anomalies gives anomalies with (g1 + g2 , s1 + s2 − 2). Conversely, we
could have derived the anomalies of the surface with a pair of punctures and genus g startingwith surface without them and genus g + 1 and subtracting the contribution of the vector
multiplet of SU(3).
Note that we have already discussed that the anomalies on a closed surface might be consistent with free fields, and in particular SU(3) gauge groups and tri-fundamentals. The anomalies of punctures support this observation. The Tr SU(3)3 = ±9 anomaly is consistent withhaving 9 (anti)fundamentals of SU(3), which is the number of free flavors needed for a conformal gauging of SU(3). Also Tr R SU(3)2 = − 32 = 12 × ( 23 − 1) × 9, and thus interpretable
as an anomaly of 9 (anti)fundamental free fields. We will momentarily turn to constructing
in detail the 4d theories, but let us comment here on the implications of the twist we have
performed upon compactification to 5d.
The twists in general can be turned on along any cycle on the surfaces. These particularly
include those surrounding the punctures. For the special case of a sphere with s punctures,
these are the only cycles. Furthermore, the twists around different punctures and cycles are notindependent: On a sphere the holonomy around s − 1 punctures should be the inverse of thataround the remaining puncture. In our case, since the punctures we consider must incorporatea twist, this leads to constraints on the possible theories. As the punctures have a Z2 (complex
conjugation) twist, we cannot have a sphere with odd number of such punctures. In principle
there might be punctures without a twist associated to them, however as we defined punctures
above using a 5d gauge theory description the untwisted punctures have to be of a different
nature. We will say more about this when discussing the 4d theories. On Riemann surfaces
without punctures, the twists can still be incorporated on the cycles of the Riemann surface.
When the latter is built from punctured spheres then whenever we glue two twisted punctures
we get the associated twist along the cycle spanned by the punctures.

4.4

Four dimensions

We want to find four dimensional models which will match the predictions coming from six
dimensions we discussed above. In particular, these models should have factors of SU(3)global symmetry, which we will associate to punctures, and the anomalies of these symmetries
are Tr SU(3)3 = ±9, Tr RSU(3)2 = − 32 . As we already mentioned we will conjecture that
we can build such theories from free fields and conformal gaugings. We have discovered that
constructing field theories such that they have nine free chiral fields in the fundamental ofSU(3) produces the right anomalies. The question is then how we build models from thesecollections of fields. We will next discuss some conjectures which reproduce the expectations
we have derived from six dimensions.
The main conjecture is depicted in Figure 24: We associate to the four punctured spherea quiver theory with two SU(3) gauge groups, three tri-fundamental fields, and a baryonicsuperpotential. We have a single superpotential term which is charged under the baryonicsymmetry: Thus following [40] (which we reviewed in previous sections) this interaction is
marginally irrelevant if we perturb the free theory with it. We can turn on additional baryonic superptentials rendering the theory conformal, but these will break some of the SU(3)symmetries associated to the punctures. As we have discussed in Example 1 (discussion
around (35)), SU(3) SQCD with nine flavors has a locus on its conformal manifold preserving (SU(3)1 × (U(1)2 )2 ) × SU(3)3 symmetry where the (U(1)2 )2 is the Cartan of an SU(3)248

SciPost Phys. Lect. Notes 78 (2024)

3

3
3
3

3

3

3
3

3

3

Figure 24: The four punctured sphere
description. The three3
3 and its field theoretic
3
bifundamental
fields in the middle link of the quiver have a baryonic superpotential3(which is encoded in the quiver by the fact the corresponding line is wavy). The
various quiver notations
are further
explained3 in Figure 25. We
3
3
3 have four
3 factors of˜˜˜: 𝒬1 four punctures.
+1
SU(3) global
to the−1−1+1 symmetry−1we associate+1
𝒬1 : ˜
0

+1

+1

+1

+1
0

: 𝒬2
: 𝒬3

˜
𝒬2 :
˜
𝒬3 :

0̃
˜
+1

˜
+1
˜
+1

˜
+1
0̃

3
3
such that the 9 of SU(9) flavor symmetry is (3, 3) under the two SU(3)1 × SU(3)2 symmetries.
Our four punctured sphere thus has directions on the conformal manifold along which we canpreserve two SU(3) symmetries explicitly and the two other ones are broken to the Cartan.
The baryonic symmetry is also broken. We then conjecture that somewhere on the conformal
manifold the Cartan symmetry enhances again to SU(3)s but the baryonic symmetry does notappear. We will soon give evidence for this conjecture.
Once we defined the four-punctured sphere we can construct theories corresponding to arbitrary surfaces by gauging puncture symmetries, as we have discussed. For example in Figure
26 we depict theories corresponding to genus two surface with no punctures. As we have an
explicit field theoretic description of the models one can verify explicitly that the conformal
anomalies derived from six dimensions (102) match the anomalies of the four dimensionalmodels. Note that as we conjectured above, the four punctured sphere has all the SU(3) puncture symmetries only at some strongly coupled locus of the conformal manifold, to constructgeneral theories we thus need to gauge emergent symmetries. We also stress that as we constructed a four punctured sphere but not a three punctured one, we only can construct surfaces
with even number of punctures.24
As a further check of the conjecture we can compute the supersymmetric index and determine from it the supersymmetric relevant and marginal operators. For generic surfaces theindex takes the following general form,25

I g,s ({ai }) = 1 + ((3g − 3 + s) +sX(−χ8 j (a j ) + χ10 j (a j ))) q p + · · · .

(104)

j=1

Here ai are fugacities for the SU(3) symmetries associated to the punctures. There are no
powers of q p smaller than one and thus there are no supersymmetric relevant operators. At
order q p we have the marginal deformations minus the conserved currents. We have 3g − 3 + s
terms with positive sign which are singlets of the flavor group associated to the punctures
and thus these correspond to exactly marginal deformations: We expect such deformations tocorrespond to the complex structure moduli of the compactification surface. One way to think
about these operators is to consider the reduction on a surface of the 6d stress-energy tensor.
A careful application of the Riemann-Roch theorem then determines that the stress-energy
tensor will lead to 3g − 3 exactly marginal deformations on a closed Riemann surface [113],
and the above is a natural generalization including the punctures. We have −χ8 j (a j ) factors
24
See [24] for further discussion of this issue and its relation to the fact that in order to define punctures wehave compactified the 6d SCFT to 5d with twist lines.
25
For low numbers of punctures and low genus there might be deviations from this structure.

49

SciPost Phys. Lect. Notes 78 (2024)

3

Q

2A

3

3
3

SU(2)

3

Q
3
W=

3

3
3

∑

3

λlrsϵijkϵabcQ ailQ bjrQ cks

l,r,s=1

Figure 25: Two types of tri-fundamental chiral fields we use in our constructions.
First we have the tri-fundamental with no superpotential, and then trifundamental
with one of the three SU(3) symmetries broken by a baryonic superpotential.
corresponding to the conserved currents of each puncture. Finally, for each puncture we havemarginal operators in the ten dimensional (symmetric three index) representation of SU(3)puncture symmetry. Turning these on we can break completely the puncture symmetry andacquire two additional exactly marginal directions. The spectrum of protected states that ourfour dimensional theories possess is thus also consistent with the expectations from 6d.
We have discussed what is the theory corresponding to a four punctured sphere and how
to glue two theories corresponding to two surfaces together along a puncture. The resulting
theories are labeled by the combined surface. However, one can combine the same surface byperforming gluings in different ways. For example, there are two different ways to glue thepunctures of a four-punctured sphere together to obtain a genus two surface, see Figure 26.
The fact that we conjecture that all the different ways to glue surfaces so that the topology willbe the same correspond to compactifications on the same surface implies that the different ways toglue should be equivalent. More precisely these should reside on the same conformal manifold. In
other words, the different quantum field theories corresponding to the different ways to construct
the same surface are expected to be conformally dual to each other.
Usually this fact is phrased as equivalence of different pairs-of-pants decomposition of asurface, even though here we decompose into four punctured spheres. The geometric construction thus systematically produces a collection of theories which are conformally dual toeach other. This does not explain the duality directly from the 4d point of view, but rather
by imbedding the theories in the higher dimensional setup it renders the dualities a trivial
consequence of the geometrical construction, if this is correct.
The anomalies of the different decompositions are the same by construction. However,
the indices of the different decompositions a priori might be different as these might be very
different looking theories. For the conjecture to be correct combining the four puncture spheres
to form surfaces of the same topology in different ways should give equivalent theories up to
the action of dualities. In particular, the protected spectrum of the theory has to be invariant
50

SciPost Phys. Lect. Notes 78 (2024)

under the exchange of the different factors of SU(3) symmetries associated to the punctures.
This is a highly non obvious fact. In particular if this holds for the four puncture sphere it
will hold for any surface. One can check this by explicit evaluation of the index in a series
expansion. In fact we find that a stronger statement from that which we need appears tohold true. Namely, gluing two tri-fundamentals and ignoring the baryonic symmetry, which is
broken for general surfaces on the conformal manifold, the index is invariant under exchanging
the four SU(3) symmetries. The index is given by,
Q3
1
1
I Y
2
3 1 2 −1
3 3 4dz
1
i, j,ℓ=1 Γe ((q p) ai a j zℓ )Γe ((q p) ai a j zℓ )ℓ22Q(q; q) (p; p).
(105)
6 ℓ=1 2πizℓℓ1 ̸=ℓ2 Γe (zℓ1 /zℓ2 )Q3
Q3
Here ℓ=1 zℓ = 1 with zℓ being the parameters of the gauged SU(3). Also j=1 aIj = 1 with aIparameterizing the four SU(3) flavors associated to the punctures. Although we do not havea mathematical proof, the above can be found to be invariant under permutations of the four
aI in perturbative expansion in the fugacities q and p.
Let us next make several comments on the theory corresponding to the four punctured
sphere. We can count the dimension of the conformal manifold of the theory near weak coupling. This is a special theory: The full symmetry of the theory is broken on the conformal
manifold and the dimension is 252. The large dimension of the conformal manifold is dueto the fact that in addition to the baryons, which are marginal, we have marginal operators
winding from one end to the other on the quiver.26 We have already mentioned that the superpotential we turn on preserving the puncture symmetries is built from fields having somebaryonic charge and it is marginally irrelevant at weak coupling and thus there is no conformal manifold passing through zero coupling and having the symmetries we are interested tohave, four SU(3)s and no baryonic symmetry present. However, we have conjectured thatsomewhere on the conformal manifold the required symmetries do appear. We can utilize thesupersymmetric index to check this conjecture. In particular, as we can explore the conformal
manifold preserving the Cartan generators of all the puncture symmetries we do not loose any
parameters we would want to have at the locus where all the SU(3) symmetries appear. The
index assuming the symmetry we are interested in takes the following form,!
4X
1 + q p 3 (31 , 32 , 33 , 34 ) + 1 +(−8 j + 10 j ) + · · · .
(106)
j=1

Let us count the exactly marginal deformations then assuming we have the SU(3) symmetries: we can break completely the symmetry on the conformal manifold as 10 of SU(3) has
non trivial invariants. This means that we obtain 252 exactly marginal deformations. This isconsistent with our conjectures.
We expect to have a duality acting on the line of the conformal manifold we find. Thisduality should permute different symmetry factors. How this duality acts on the couplings isa very interesting question to try and answer.
Exercise: Consider the two different ways to construct genus two surface starting witha four punctured sphere. See Figure 26. The two models should be dual to each other.
Analyze the superpotentials in the two frames, show that the theories are conformal,that all the symmetry is broken on the conformal manifold. Compute the dimensionof the conformal manifold and check that the superconformal index in the two duality
frames agrees in expansion in fugacities.
26

In the next Lecture we will recover the same model in a completely different way and there this large conformal
manifold will have a different meaning giving us an example of a 6d duality.

51

SciPost Phys. Lect. Notes 78 (2024)

3
3

3
3

3

3

3

3

Figure 26: The two duality frames corresponding to genus two surface without punctures. If the conjecture is correct the two models have to be dual to each other on
their conformal manifold. Note that the theory on the left has an interesting structure. The two gauge groups on the edges have matter content which is comprised ofthree adjoint fields, N = 4 SYM. Then, one gauges an SU(3) subgroup of the N = 4
SU(4) R-symmetry with the addition of extra fields. This theory has non generic features, in particular the conformal manifold has a non-generic structure.
This is a special case so let us analyze it in detail. We start with the “Θ” shaped quiveron the right of Figure 26. We have three tri-fundamentals of SU(3), the top Q t , the bottom
Q b , and the middle Q m . We also have three U(1) symmetries. We define U(1) t such thatQ t is charged −1 and Q m,b +1. Similarly we define two additional symmetries U(1) b andU(1)m . Note that U(1)m is anomalous under the two SU(3) gauge symmetries on the sidesof “Θ”. The U(1) t is anomalous under the bottom SU(3) and U(1) b is anomalous under top
SU(3). The fields Q m form a triplet under flavor SU(3). The marginal operators are the baryon
λ Q3m (which form a 10 of flavor SU(3)) and the product of all three fields, λ̃ Q t Q mQ b (whichforms a 3 of flavor SU(3)). The latter has charge +1 under the three anomalous symmetries.
We need to build a singlet out of the couplings under all symmetries. However, note that
there are five independent singlets we can build, {λ4 , λ6 , λ3 λ̃3 , λ̃3 λ5 , λ̃6 λ4 }. All of these havepositive charge under U(1)m and thus we cannot form a singlet under it as the exponent of thegauge couplings is also charged positively. Thus we deduce that this theory has no conformal
manifold in the vicinity of the weak coupling. However, we have conjectured that the fourpunctured sphere has a locus with four SU(3) symmetries and no baryon symmetry. We canthus gauge these symmetries in pairs to form the “Θ” shape. As we do not have any otherflavor symmetries except for the puncture ones we obtain an SCFT with a conformal manifold
dimension of which is simply given by counting marginal operators minus the currents at zero
coupling: 10 λ plus 3 λ̃ plus 4 gauge couplings minus the currents of SU(3)×U(1)3 . This gives
us a six dimensional conformal manifold with no symmetries preserved. Note that for genus
2 we would generally expect 3g − 3 = 3 deformation but here we find twice that amount.
Let us now analyze the dumbbell quiver on the right of Figure 26. The analysis is rather
different than the above. Note that we can think of this model as two copies of SU(3) N = 4
SYM glued to a tri-fundamental with additional two fundamental chiral fields, where we gauge
SU(3) subgroup of the SU(4) R-symmetry. Here the two copies of N = 4 SYM give an exactly
marginal deformation each preserving the SU(3) flavor (the N = 4 coupling). We gauge
two additional SU(3) symmetries under which the tri-fundamental in the middle and thetwo chiral fields are charged. We have three U(1) symmetries: U(1)le f t under which one
fundamental is charged +1, the trifundamental is charged +1/9, and the other fundamental
charged −1; U(1) r i ght reversing the roles of the two fundamentals; and finally U(1) M underwhich the two fundamentals have charge +1 and the trifundamental has charge −1/9. The
52

SciPost Phys. Lect. Notes 78 (2024)

3

3

3

3

1

1

Figure 27: An equivalent way to depict the dumbbell quiver. The two shaded subquivers are N = 4 SYM. As 3 × 3̄ = 8 ⊕ 1 two trifundamental fields split to (8, 3) and(1, 3) of SU(3) × SU(3). The dotted lines represent a triplet of adjoints of SU(3).
U(1)l e f t/r i ght symmetries are anomalous under the left/right SU(3) gauge symmetries respectively, while U(1) M is non anomalous. The marginal deformations are the baryon built fromthe tri-fundamental and the operator built from the product of the two fundamentals and thetri-fundamental. Under the two anomalous symmetries both operators are charged positively
and under the non-anomalous symmetry they have charges of opposite signs, −1/3 and +17/9.
Moreover some of the five singlets of SU(3) detailed above have positive and some negative
charge under the non anomalous symmetry. We thus can form a singlet under all symmetries
obtaining a conformal manifold under which all the symmetries are broken: We have ten couplings from the baryon, three from the other deformation and two gauge coupling with elevencurrents (coming from SU(3) of the tri-fundamental and the three U(1) symmetries). This
gives us 10 + 3 + 2 − 8 − 3 = 4 which together with the two deformations coming from theN = 4 pieces gives us a six dimensional conformal manifold as expected.
The two models then behave in a strikingly different manner, where while for the dumbbell quiver the conformal manifold passes through weak coupling, it does not do so for the“Θ” shaped quiver. We can understand the difference in this behavior as follows. To get thequiver associated with the genus two surface from the four punctured sphere we need to gauge
two SU(3) gauge groups that are embedded as diagonal symmetries in the SU(3)4 ⊂ SU(9)2
subgroup of the flavor symmetry of the quiver associated with the four punctured sphere. The
two distinct quivers correspond to two different ways to perform this gauging. In one we embed the two SU(3) groups as 9SU(9)1 → (3SU(3)1 , 3SU(3)2 ), 9SU(9)2 → (3SU(3)1 , 3SU(3)2 ). Gauging
SU(3)1 and SU(3)2 now leads to the “Θ” shaped quiver. Alternatively, we can embed the two
SU(3) groups as 9SU(9)1 → 3SU(3)1 × 3SU(3)1 → 1 + 8SU(3)1 and similarly for SU(9)2 and SU(3)2
groups.27 Gauging SU(3)1 and SU(3)2 now leads to the dumbbell quiver. It is possible toshow that there is a subspace of the conformal manifold passing through weak coupling, that
preserves the symmetries gauged in the second embedding. This suggests that the dumbbell
quiver indeed sits on the conformal manifold of the genus two compactifcation. However, thatis not the case for the first embedding, as shown by the analysis done here. In that case theLagrangian “Θ” shaped quiver is an IR free theory and is not really a compactification of the 6d
SCFT on a genus two surface. To preform the gluing in this case, we need to go to the locus on
the conformal manifold where only the four SU(3) groups associated with the punctures arepreserved. This occurs at strong coupling and requires breaking some of the flavor symmetrythat is to be gauged to get the “Θ” shaped quiver, and hence the lack of a 6d interpretation inthis case.
□
27
There is also a U(1) that commutes with the SU(3) group for each SU(9) group, but that would not play arole here.

53

SciPost Phys. Lect. Notes 78 (2024)

3
3

3

3

3

3

3
Figure 28: Closing a puncture of the four punctured sphere. The squares are SU(3)punctures and crosses are empty ones. On the conformal manifold at most two SU(3)symmetries are preserved. On a general locus all symmetry is broken and the model
can be interpreted as a model with ten empty punctures.
3
Closing punctures

3

3

3

We have understood what is the geometric interpretation of gauging SU(3) factors of the global
symmetry. We can also discuss other operations, such as turning on relevant deformations(which are absent here), or exactly marginal ones (which we discussed), and giving vacuum
expectation values for various operators. Let us turn our attention now to the latter operation.
We will discuss turning on vacuum expectation values (vevs) for operators in a manner that
preserves the R-symmetry: We only consider R-symmetry preserving geometric constructions
since we wish to find a geometric interpretation of the vev. This in particular implies thatthe operator we give a vev to should be charged under some global symmetry. The only such
symmetry is the puncture symmetry. We have thus a natural candidate for such an operator:
These are the marginal operators in the 10 of the puncture symmetries. Such flows will breakthe symmetry of the puncture and will leave us with punctures of different type. Since theconstruction of the punctures we have discussed involves twists, it is not possible to close
them completely: The twist has to be supported on some cycle.
As we have concrete Lagrangian QFTs corresponding to the geometric construction we cananalyze such vevs in complete detail. Parametrizing the character of the fundamental of SU(3)as 3 = b1 + b2 + b3 , with b1 b2 b3 = 1, we have (see Appendix C)
10 =

X
i̸= j

bi /b j +3X

bi3 + 1 .

(107)

i=1

In index notations we give expectation value to operator contributing with weight q p b13 setting this combination of fugacities to one, that is we give a vacuum expectation value to asingle bi-fundamental field. The effect of the flow is simple, the trifundamental associated toour puncture is removed from the theory with the symmetry, to which we gave a vacuum expectation value, and the two additional SU(3) groups are identified: The vev is to a baryonicoperator which Higgses one of the gauge SU(3) symmetries. We interpret this procedure as
closing a puncture to a different puncture which has no flavor symmetry and we will refer tothat as an empty puncture.
Let us add s′ empty punctures by starting from a theory with s′ + s punctures and closing′s of those. The anomaly is easy to compute and we obtain that empty punctures behave as

54

3

3

3

3

3

SciPost Phys. Lect. Notes 78 (2024)

3

3

3

3

Figure 29: Closing SU(3) puncture of Fig. 5 we obtain a Wess-Zumino theory with
two SU(3) and two empty punctures.
one third of the SU(3) puncture. We have,3(82(g − 1) + 33s + 11s′ ) ,32
1(118(g − 1) + 51s + 17s′ ) .
c=
16
a=

(108)

We can also compute the index and find that it is,!
sX
1 + 3g − 3 + s + s′ +(−8 j + 10 j ) q p + · · · .

(109)

j=1

The dimension of the conformal manifold preserving puncture symmetries is 3g − 3 + s + s′
which is the number of complex structure moduli, as expected. Note that the above results
also imply that triplets of empty punctures are equivalent on the conformal manifold to a fullpuncture. To break maximal punctures into empty ones we turn on the most general baryonicsuperpotentials. For example, the theory of Figure 28, the SU(3) SQCD with nine flavors, thencan be re-interpreted as a compactification on a sphere with ten empty punctures. Thus it isexpected to have 3g − 3 + s = 3 × 0 − 3 + 10 = 7 dimensional conformal manifold, as indeed
it does [40, 51].
The final picture is thus as follows. Turning on the most general exactly marginal superpotentials in the theories we have constructed corresponds to surfaces of some genus and somenumber of empty punctures. On special loci of the conformal manifold the empty punctures
can collide in groups of three and form a maximal puncture with an SU(3) symmetry associated to it.28 Let us comment in passing that this picture can be further checked by giving
vacuum expectation values to derivatives of the operators in 10 which will introduce surface
defects into the theory [119]. Such constructions are related intimately to integrable models,
and the corresponding indices should satisfy interesting sets of properties [120] which can bemathematically proven [121] providing more evidence for the conjectures.
In this Lecture we have discussed a derivation of the dictionary between a theory in 6d withno continuous global symmetry and 4d Lagrangians. Next we will discuss such a dictionary
once the 6d theory does possess continuous global symmetry, which introduces more knobs
and handles to produce interesting interplays between geometric constructions, emergence ofsymmetry and duality.
28
The fact that collisions of punctures of one type can form punctures of different type has appeared in various
contexts [25, 114–118]. For example in [116] such effects were dubbed a-typical degenerations.

55

SciPost Phys. Lect. Notes 78 (2024)

5

Lecture IV: Compactifications of the E-string

We will now repeat the analysis of the previous Lecture but in a richer setup of compactifications of the rank one E-string theory.

5.1

Six and five dimensions

Let us consider one of the simplest and most studied 6d SCFTs, the rank one E-string theory.
This model can be engineered in string theory in various ways. One of them is by taking asingle M5 brane to probe the end of the world M9 brane. Another way is by taking a single
M5 brane to probe a C2 /D4 singularity. Instead of taking a single brane, taking N branes, inthe former description one obtains the rank N E-string. In the latter description taking one M5
brane to probe C2 /DN +3 singularity one obtain what is called minimal (DN +3 , DN +3 ) conformal
matter. Thus the N = 1 case is a starting point of several sequences of theories and in fact is
one of the basic building blocks of constructing most general 6d SCFTs [87]. The rank one
theory has a rank one tensor branch on which we have a single tensor multiplet: So in thisrespect this model is even simpler than the one we discussed in the previous Lecture.
There is only a limited amount of information we will need about this model. One piece of
information is that the theory has an E8 global symmetry. Another is the anomaly polynomial,
which was computed by now using a variety of techniques [80, 94, 122]. Lastly, we will need
the conjecture that upon compactification to 5d, with a certain holonomy breaking the E8 symmetry, one obtains an effective 5d description as an N = 1 SU(2) SQCD with 8 fundamental
hypermultiplets [123]. This effective theory is UV completed by the 6d SCFT. We remind thereader that an holonomy corresponds to a non-trivial background, here for a connection to a
global symmetry, such that F is zero but A has a non-trivial integral over a closed cycle, here
the compactification circle. The value of the integral of A gives some element of the group,
and its presence leads to the breaking of the group to the subgroup commuting with the given
element. For instance, the holonomy d ia g(e iθ , e−iθ , 1, 1, . . . , 1) in SU(N ) generically breaks
the group to U(1)2 × SU(N − 2).
We will use this 6d information, as in the previous Lecture, to come up with a prediction
for the existence of a class of 4d theories: In particular we will predict their anomalies andsymmetries. The anomaly polynomial of the 6d theory is [94,122]29 (we use notations of [26]),
I 6d =

13 2
111
1
C2 (R) −C2 (R)p1 (T ) −C2 (R)C2 (E8 )248 +C 2 (E8 )24824
48
60
7200 27p1 (T )2 − 4p2 (T )1+
p1 (T )C2 (E8 )248 + 29
.
240
5760

(110)

We use the notation C2 (R) for the second Chern class in the fundamental representation of
the SU(2)R 6d R-symmetry. We also employ the notation C2 (G)R for the second Chern class ofthe global symmetry G, evaluated in the representation R, and p1 (T ), p2 (T ) for the first andsecond Pontryagin classes respectively.
The anomaly polynomial of the 4d theory compactified on a closed Riemann surface ofgenus g > 1 with no flux for subgroups of E8 is obtained by twisting the theory, defining
C2 (R) = −C1 (R)2 + 2(1 − g) t C1 (R) + . . . as before, and integrating the 6d anomaly polynomial
on the surface. The result is [26],a=75(g − 1) ,
16

c=

43
(g − 1) ,8

All the rest of the anomalies vanish.
29

See Appendix E.3.

56

Tr R (E8 )2 = −(g − 1) .

(111)

𝒞

6d

∂𝒞

SciPost Phys. Lect. Notes 78 (2024)

5d𝒞

5d SU(2)0 Nf = 8 𝒩 = 1 SQCD

6d

∂𝒞

5d
Figure 30: Compactifying the 6d rank one E-string on a circle with a certain holonomy breaking the E8 symmetry to SO(16) is conjectured to be described in 5d by aneffective theory which is SU(2) SQCD with eight flavors. The UV completion of the
5d theory is the 6d SCFT. Here C is the Riemann surface and ∂ C is non empty, we
have a puncture.
Now, we can compactify while also turning on fluxes for various abelian subgroups of E8 .
For a flux in a single U(1) ⊂ E8 this entails substituting C1 (U(1)) =R −z t + C1 (U(1) F ) which
naturally generalizes to many U(1)s. Here z is the value of the flux, C C1 (U(1)) = −z, and byU(1) F we denote the first Chern class of the corresponding symmetry in 4d . This leads to avariety of theories in 4d labeled by fluxes to the various abelian symmetries. We will not quotethe most general results here. Let us just mention that the most general compactification will
be labeled by the flux for the Cartan generators of the E8 symmetry. To specify this flux weshould choose some basis of U(1) symmetries. It is convenient to do so by considering theSO(16) maximal subgroup of E8 , 248 → 120 + 128. The choice of such an SO(16) subgroupwill be later discussed in more details. Then we parametrize the flux by stating its values forthe natural SO(2) subgroups of SO(16) and encode it in a vector, an octet, of fluxes we denote
by F .
Turning on fluxes will break the E8 symmetry to the subgroup commuting with the choiceof flux. In particular we will have abelian symmetries in 4d. As usual in such cases the superconformal symmetry of the 4d fixed point will be some mixture of the Cartan of the 6d
R-symmetry, preserved by twisting, and all the abelian symmetries in 4d.
Finally let us discuss the punctures. We specify the maximal boundary conditions giving
the 5d SU(2) gauge fields Dirichlet boundary conditions. Here, unlike in the pure SU(3) caseof the previous section, we also have matter fields. In terms of N = 1 4d supersymmetry these
are an octet of hypermultiplets, that is 16 chiral fields. We give eight of the chiral fermions inthis multiplet Neumann boundary conditions and the remaining ones get Dirichlet boundary
condition. These fermions are not transforming under the SU(2)R R-symmetry coming from six
dimensions: This becomes the SU(2)R R-symmetry of 4d N = 2 under which the fermions in
the hypermultiplet do not transform. Thus the inflow contribution to the anomalies involving
the R-symmetry only comes from the vector multiplet and gives,
Tr R = −1 ×1
3× dim SU(2) = − ,22
andTr R3 = (−1)3 ×

Tr R SU(2)2 = −1 × 2 ×1= −1 .
2

1
3
× dim SU(2) = − ,
2
2
(112)

The matter fields do contribute to the anomalies of the subgroups of E8 global symmetry. The
octet of hypermultiplets transforms as a fundamental of the U(8) subgroup of SO(16) ⊂ E8 .
Thus the anomalies of a given U(1) will be determined by the inflow of each one of the fermions
which obtained Neumann boundary condition. Say we choose a U(1) ⊂ E8 such that the eight57

SciPost Phys. Lect. Notes 78 (2024)

fermions with Neumann boundary condition have charges qa=1···8 , Then, the contribution to
the anomaly of this U(1) from the inflow is,
Tr U(1) =

8
X
1
×2×
qa ,
2
a=1

Tr U(1)3 =

8
X
1
×2×
(qa )3 ,2
a=1

Tr U(1) SU(2)2 =

8
1 1 X
× ×
qa .
2 2 a=1

Here the source of 12 is explained in footnote 23, the 2 comes from the fact that the fermionsare doublets of the puncture SU(2), and the additional factor of 12 in the last anomaly comesfrom Tr SU(2)2 in the fundamental representation. Let us mention here that we have a choiceof which eight out of the sixteen chirals gets a Neumann boundary condition. All these choices
are in principle equivalent up to the action of the Weyl group of SO(16). However if we havedifferent choices for different punctures on the same surface, the relative difference becomes
physically significant. These different choices when defining the types of punctures will bereferred to as different colors of punctures.
The octet of the fundamental chiral fields with Neumann boundary conditions is expected
to give rise to an octet of chiral operators in the fundamental representation of the SU(2)puncture symmetry in 4d. These operators will have R-charge +1 under the R-symmetry inherited from 6d. We will refer to such operators as “moment maps”. The name is motivatedby analogy with compactifications of the (2, 0) theory. If one repeats this procedure in thecase of compactifications of the (2, 0) theory, say AN −1 type, the model in 5d is SU(N ) N = 2
SYM, and in 5d N = 1 language the matter content has an adjoint hypermultiplet. We again
give Neumann boundary conditions to a chiral half and Dirichlet to the other one. In 4d thesymmetry associated to the puncture is SU(N ) (the gauge symmetry of the 5d theory) andthe chiral in the adjoint representation with the Neumann boundary condition becomes themoment map operator in the N = 2 flavor current multiplet in 4d. In the case at hand there
is only N = 1 supersymmetry in 4d, but the appearance of the “moment map” operators has
the same origin, and hence the name.
Let us use the above to construct the anomalies with punctures,
 



75 
s
9
3
3
3
3
a=
g −1+
+
−
−
−
s=
(25 (g − 1) + 11 s) ,
16
2
32
2
32
2
16





(113)
43 
s
9
3
5
3
1
c=
g −1+
+
−
−
−
s = (43 (g − 1) + 20 s) .
8
2
32
2
32
2
8
Note that in general since the punctures break the symmetry and introduce a U(1) ⊂ U(8),and fluxes also introduce U(1)s, the above are not the superconformal anomalies but rather
the ones computed with the 6d R-charge, and these can be viewed just as encoding the ‘t Hooft
anomalies of the 6d R-symmetry using (10),
Tr R3 = −13 (1 − g) + 5 s ,Tr R = 11 (1 − g) − 7 s .

(114)

With this concrete information about the 6d and the 5d theories we are ready to search forthe 4d theories corresponding to the compactifications.

5.2

4d theories from genus g Riemann surface with zero flux

Let us now conjecture that the 4d theories obtained by compactifying the rank one E-string on
a closed genus g > 1 surface with no flux can be described by a conformal Lagrangian. From
the anomalies (111) we obtain that,a=75(g − 1) = 16 a v + 81 aχ (g − 1) ,
16

c=

58


43
(g − 1) = 16 c v + 81 cχ (g − 1) . (115)8

SciPost Phys. Lect. Notes 78 (2024)

9

3

3

9

10

3
𝒳

Figure 31: The conformal dual of rank one E-string compactified on genus two surface with no flux. We can turn on any exactly marginal superpotential.

3
3

3

3

99
3

3

3

3

9

9

9

3
3

Figure 32: The conformal dual of rank one E-string compactified on genus g surface
with no flux. The number of gauge nodes is 2g − 2. Here we have the example ofg = 4. The Figure is taken from [25].
The number of vectors is thus dim G = 16 (g − 1) and the dimension of the representation is
dim R = 81 (g − 1). These numbers are integers and thus there is a chance that the conjecture
is correct.
The above value for dim G for genus two surface can be obtained by taking either two
SU(3) groups or an US p(4) and two SU(2) groups: We will find a candidate dual with theformer option. See Figure 31. Note that this is the same model we obtained in Section 4.4 tocorrespond to a four punctured sphere compactification of minimal SU(3) SCFT, and here we
give it a different interpretation as genus g = 2 compactification of rank one E-string with noflux.30
We already have discussed properties of this model in previous sections, so let us here justremind that this theory has a 252 dimensional conformal manifold: A fact that we will soon
relate to various geometric objects, such as complex structure moduli of the corresponding
Riemann surface and the dimension of the space of flat connections on that surface.
For genus g > 2 we will seek a simple generalization of the above, and the simplest guess isto have 2g −2 SU(3) gauge groups which will amount to dim G = 16(g −1). The quiver theory
is depicted in Figure 32. The number of fields is as expected, dim R = 81 (g −1). The marginal
operators can be built from baryons and gauge invariant triangles on the quiver. The Kähler
quotient computing the dimension of the conformal manifold is not empty: On a generic locusof the conformal manifold all the non-R global symmetry is broken and the dimension of the30

Such equivalences of compactifications are common but deep understanding of them is lacking at the moment.
For example, the rank N E-string on a torus with no flux is the E8 Minahan-Nemeschansky theory [124] which canbe interpreted as an A6N −1 (2, 0) theory compactified on a sphere with three punctures of certain types [17].
Spheres with punctures of 2 M5 branes probing a Zk singularity [125] can be mapped to tori compactifications
with flux of minimal (DN +3 , DN +3 ) conformal matter with the number of punctures and k on one side maps tocertain combinations of flux and N on the other side [19, 126].

59

SciPost Phys. Lect. Notes 78 (2024)

conformal manifold is
dim Mc = 251(g − 1) = 3g − 3 + 248(g − 1) .
This general counting fails for the case of the genus two quiver and has to be done more
carefully: We obtain an additional accidental marginal deformation as above, see [22].
The above expression for the dimension of the conformal manifold is what one would
expect from 6d. First, we have the 3g−3 deformations we would associate to complex structure
moduli. Second, we have 248(g − 1) additional deformations which have a natural meaning.
Note that 248 is the dimension of E8 . These deformations then can be associated with flat
connections for the E8 global symmetries one can turn on upon compactification on the surface.
These flat connections are parameters of the compactification, which in this case turn out tobe exactly marginal. A different way to phrase this is that we can consider the proper “KK”
reduction of the conserved current of the E8 symmetry in six dimensions, and performing the
Riemann-Roch analysis one gets the 248(g − 1) exactly marginal deformations [113].31
Thus we conjecture that the quiver of Figure 32 describes the compactification of rank one
E-string on genus g surface with zero flux. In particular we expect the symmetry to enhance to E8
somewhere on the conformal manifold.32
Although on generic locus of the conformal manifold all the symmetry is broken, turning on non-generic superpotentials we can preserve some symmetry on sub-manifolds of theconformal manifold. For example, turning on cubic superpotentials corresponding to various
triangles in the quiver identifies the various SU(9) symmetries. To have a non vanishing Kähler quotient we also turn on baryonic superpotentials for the bifundamental fields. This breaks
all the symmetries but a diagonal combination of all the SU(9)s. The Kähler quotient is not
empty as the baryons and the triangle superpotentials have opposite charges under the baryonic U(1). The baryons charged under the SU(2)s of the free point are in 4 and thus have asinglet in the fourth symmetric power. Finally, the triangle superpotentials are in a bifundamental of two SU(9) groups and thus a di-baryon of these is an invariant. We note that underthese deformations only, the marginal operators minus the conserved currents take the form,
3g + 3 + 84 + 84 + 80 (g − 1) ,(116)
where g − 1 84 and 84 come from baryons and g − 1 80 coming from operators winding
between SU(9) groups. We note that SU(9) is a maximal subgroup of E8 such that
248 E8 → 84SU(9) + 84SU(9) + 80SU(9) .
All the representations of SU(9) appearing in the index are expected to combine into E8 representations under this branching rule. This can be verified at least in expansion of the index
in the fugacities. We thus obtain as expected (g − 1)248 E8 exactly marginal operators. Furthermore,
2
1
Tr R SU(9)2 = × ( − 1) × 3 × (2g − 2) = −(g − 1) ,2
3
In the g = 2 case we have one additional deformation which is not explained by the general geometric logic:
We do not know what the origin of it might be, but one possible explanation is that the 6d SCFT has certain surface
operators which we can wrap on the Riemann surface and obtain local operators in 4d. For low genus these
operators might have low charges and for example be exactly marginal. It will be very interesting to understandthese issues in detail.
32
There are other constructions of these models in 4d [26, 118]. The other constructions are not completely
Lagrangian as they rely on gauging certain emergent symmetries in the IR. Nevertheless they can be used tocompute protected quantities such as indices, and one can check that the indices in all the descriptions agree. This
serves as a consistency check of the various conjectures.
31

60

SciPost Phys. Lect. Notes 78 (2024)

9

3

3

9

10

3

3

𝒳

𝒳

3

3

3

3

9

9

9

9

10

Figure 33: A conformal duality which we want to interpret as two different pair ofpants decompositions of a genus two surface. The fields denoted by X and X are in
two index symmetric representation of SU(3).
which matches the result expected from the strongly coupled theory where the SU(9) is embedded inside E8 (111) (Note that the imbedding index of SU(9) in E8 is 1). Note that actually
if the theory has an E8 point on the conformal manifold, or rather a 3g − 3 dimensional locus
of such, with (g − 1)248 E8 marginal operators, we cannot go out of it preserving SU(9), butrather only the Cartan subgroup: This is what the Kähler quotient tells us, we can constructinvariants out of powers of the adjoint which preserve the Cartan but break the non-abelian
structure. Thus, all we can say is that the SU(9) preserving sub-manifold of the conformal
manifold passing through the weak coupling point might intersect the Cartan preserving submanifold passing through the E8 point.
Finally, one can wonder whether the quiver theory we have found is the only conformal
gauge theory corresponding to the given compactification. The answer is actually that there
are more, and all of these are conformal dual to each other. Let us give here the example ofgenus two in some detail. We can match the anomalies with the two quivers in Figure 33:
The one on the left we have already discussed, but also the one on the right fits the bill. The
conformal anomalies work simply because it is again a conformal theory with two SU(3) gauge
groups and the number of fields is 2 × 3 × 10 + 2 × 6 + 3 × 3 = 81. The gauge nodes are indeed
conformal as the beta functions vanish,
Tr SU(3)3 = (−1) × 10 + (+1) × 3 + (+7) = 0 ,




1
2
5
2
Tr R SU(3)2 = 3 + ×− 1 (10 + 3) + ×− 1 = 0.
2
3
2
3

(117)

We used the fact that the cubic Casimir of the two index symmetric representation of SU(N )is 4 + N and the Dynkin index is N 2+2 . We can turn on baryonic superpotentials, the cubic
symmetric power of the fields X and X and the deformation running across the quiver. We
leave the analysis of the Kähler quotient as an exercise. We just mention that the symmetry
can be broken completely on the conformal manifold. The number of marginal operators at
the free point is 120 × 2 + 1 + 2 + 55 × 2 + 10 × 10 = 453. The non-anomalous symmetry at
the free point is SU(10) × SU(10) × U(1)3 , dimension of which is 99 + 99 + 3 = 201. All in
all the dimension of the conformal manifold is then 452 − 201 = 252 as expected. We thus
conjecture that the two quivers are dual to each other.33 Note that as in the G2 SQCD dual toa quiver theory we discussed in Lecture II, the two weakly coupled cusps are very different:
They have different symmetry and different number of marginal operators. However going on
the conformal manifold of both there is a chance that we can get from one to the other, andthat is what the conjecture of the duality is about.
33

This duality can be related to a sequence of Seiberg dualities [127].

61

SciPost Phys. Lect. Notes 78 (2024)

In fact there is a natural interpretaion of the two theories being two different pair of pants
decompositions of the genus two surface, as depicted in Figure 33. The point is that we canthink of a line ending on the same gauge node with same orientation as a two index symmetricplus an antifundamental, 3 × 3 → 6 ⊕ 3 (this is in contrast to a bi-fundamental ending on thesame node which is an adjoint and a singlet). This duality thus suggests that we should be able
to “cut” the bifundamental lines in the quiver and obtain theories corresponding to pairs-ofpants: Three puncture spheres. This is what we will do next, and by doing so we will discover
how to compactify the rank one E-string on an arbitrary surface with an arbitrary amount of
flux.

5.3

Decomposing the surface into pairs of pants

Let us start with the quiver in Figure 31 and try to decompose it into a gluing of two threepunctured spheres. We have discussed that a puncture of rank one E-string compactifications
should correspond to an SU(2) symmetry. Thus we should rewrite the quiver as an SU(2)3
gluing of two SCFTs. It is straightforward to do so using the S-confining Seiberg duality of
Figure 14. We detail this rewriting in Figure 34. The quiver contains triangular blocks withsuperpotential terms of the form, h · e
h · Q i + Q3i . Such triangular blocks can be exchanged
using the S-confining duality by an SU(2) SQCD with N f = 3. We have three such blocks
corresponding to the three bi-fundamental fields Q i , and thus can perform this procedure foreach one of them. In the end we obtain the last quiver in the chain of dualities of Figure 34
having a superpotential which is a combination of quartic and sextic terms.
Next we conjecture that each one of the two SU(3) N f = 6 SQCD blocks, which are glued
together by gauging the SU(2)3 subgroup of their global symmetry and turning on the superpotential, corresponds to a three punctured sphere compactification with a certain amount offlux to be determined soon. Let us then discuss these SQCD models in detail.
The three punctured spheres
We analyze next the N = 1 SU(3) N f = 6 SQCD interpreted as a compactification of therank one E-string on three punctured sphere. The model is depicted in Figure 35. The global
symmetry of the model is SU(6) × SU(6) × U(1). However, following our disucssion in theprevious section we should think about this symmetry in terms of specific sub-groups. In particular, taking one of the SU(6) factors (say the one rotating the fundamentals) we decompose
it into SU(2)3 × U(1)2 maximal subgroup. We then associate the three SU(2) factors with thethree maximal punctures. The three remaining U(1)s (two coming from the decompositionof SU(6) and the baryonic one) together with the remaining SU(6) will form a maximal subgroup of the six dimensional E8 symmetry. This statement is motivated by the discussion in
the previous section and we will test it more soon, but here let us mention that the embeddingof this subgroup in E8 is as follows,
E8 → E7 × U(1) → (SU(6) × SU(3)) × U(1) → (SU(6) × U(1)2 ) × U(1) .

(118)

The superconformal R-symmetry of all the chiral fields is 12 . However this is not the
R-symmetry which we obtained decomposing the genus two surface into trinions. The Rsymmetry we will use is obtained by mixing the baryonic U(1) symmetry with the superconformal one. In particular, we assign an R-charge 1/3 to the fundamentals and 2/3 to theanti-fundamental chiral fields. As this is the R-symmetry we obtained decomposing higher
genus surfaces with zero flux it coincides with the Cartan generator of the su(2) R-symmetryof the six dimensional E-string theory.
If we are to interpret the SU(2) symmetries identified above with puncture symmetries
we also need to find the associated octet of moment map operators. As we have discussed in
62

SciPost Phys. Lect. Notes 78 (2024)
2
3

3

1
3

2

s1
2
3

2
3
2
3

hi

9

s̃ 1
h̃i

2
3

2
3

2
3

1
3

s3
1
3

3

2

s2
1
s
3 1

2
3

2
2

s̃ 2 1
3

1
3

3

s̃ 1

s2
1
s
3 1

2
3

1
3

s̃ 3

2
3

hi

1
3

3

8

Qi

h̃i

3

1
3

hi

3

3

Q1,2

2
2

Q1
s̃ 2
1
3

1
3

3

s̃ 1
2
3

hi

h̃i

7

h̃i

6
Figure 34: Cutting the edges of the quiver step by step employing the S-confining
Seiberg duality of Figure 14. In the first step, going from the first to the second quiver,the field h9 maps to s12 , eh9 to es12 , and Q 1 to s1e
s1 . The superpotential of the second
quiver involves terms of the form (s1es1 )2Q i +(s1es1 )Q2i +s12e
s12 (Q 1 +Q 2 )+s1e
s1 hi hi . In thesecond step we exchange yet another triangle in the quiver with an SU(2) N f = 3
sector, and similarly in the last step. The final superpotential has quartic terms of
the form (si h j )(es i h j ) and sextic terms of the form (si s2j )(es i (sej )2 ). The numbers in red
denote a choice of R-symmetry. As this is the superconformal R-symmetry for the
first quiver and it follows from duality for the rest, it is the superconformal choice
here for all the quivers.
Subsection 5.1, we expect the moment map operators to be in the fundamental representationof only one SU(2) puncture symmetry and have R-charge +1. Indeed we have a variety ofoperators with these properties: For each puncture SU(2) we have six mesons and two baryons
with these properties. In more detail, the charges of the moment map operators (encoded interms of their fugacities) are given by,
M x = 2 x ⊗ 6u4 /v 2 w2 ⊕ 1u6 v 12 ⊕ 1u6 w12 ,
M y = 2 y ⊗ 6 v 4 /u2 w2 ⊕ 1 v 6 u12 ⊕ 1 v 6 w12 ,(119)Mz = 2z ⊗ 6w4 /u2 v 2 ⊕ 1w6 u12 ⊕ 1w6 v 12 .
We denoted the three octets of moment maps by M x, y,z with x, y, and z denoting the Cartan
generators of the corresponding SU(2) puncture symmetries. Further, 6 denotes the characterof the fundamental representation of the SU(6) symmetry. The sextets of the SU(6) are built
from the mesons and the singlets from baryons. The baryon which has charges 1u6 v 12 is built
from two quarks charged under SU(2) y and one quark charged under SU(2) x . Importantly, as
can be seen from (119) the three punctures of the trinion have moment map operators charged
differently under the various symmetries. These punctures thus are of different types: Such
63

SciPost Phys. Lect. Notes 78 (2024)

F

6
1
u 2v 2 w 2

2
3

u6

2
x

1
3

v6

3
1
3

w

6

1
3

2
y

2
z

Figure 35: The SU(3) SQCD in the middle of the conformal window reinterpreted as a
4d theory dual across dimensions to the rank one E-string placed on a three punctured
sphere with a particular value of flux. The assignment of R-charges (denoted in red)follows from the decomposition of higher genus surfaces into trinions. This also canbe identified with the Cartan of the six dimensional R-symmetry. The Figure is takenfrom [25].
types of punctures are usually called colors and, as we have discussed in Subsection 5.1, canbe attributed to different choices in defining the boundary conditions in 5d.
Finally, let us compute the various ‘t Hooft anomalies of the model involving the Rsymmetry,




2
1
Tr R = 8 +− 1 × 18 +− 1 × 18 = −10 ,
3
3
3

3

1
2
3
− 1 × 18 +− 1 × 18 = 2 ,(120)
Tr R = 8 +
3
3


1
1
Tr R SU(2)2x, y,z = ×− 1 × 3 = −1 .
23These anomalies match the predictions for a three punctured sphere in (112) and (114). As
the anomalies above are insensitive to the choice of flux we can match them without specifying
the flux associated to the three punctured sphere we discuss.

Gluing back to closed surfaces: S-gluing
We have obtained a 4d theory dual across dimensions to the E-string compactified on a threepunctured sphere with some value of flux, value of which we need still to determine. Let us
now discuss how to combine these trinions to obtain theories corresponding to more generalsurfaces, and in particular closed surfaces. We know what needs to be done as we obtained
three punctured spheres by decomposing a closed surface, but it is worth to understand thegluing more systematically. The gluing we discuss first is called an S-gluing following thenomenclature of [26].34
34

S-gluing can be defined for compactifications of any 6d SCFTs, as long as we have a notion of the maximal
puncture: see e.g. [117, 125, 128, 129].

64

SciPost Phys. Lect. Notes 78 (2024)

M̃ D = M′N,

𝒞

M̃′D = MN

W = MN ⋅ M′N
6d

SU(2) (M′ , M̃′ )(MN, M̃ D)
N
D𝒞′∂𝒞 = ∂𝒞′
5d

e D)
Figure 36: Explanation of S-gluing in 5d. We have the 16 fields on the left (MN , Me ′ ). We gave eight of these Dirichlet (label D) and eight Neuand on the right, (MN′ , M
Dmann (label N ) boundary conditions. Now we are undoing the boundary conditions
by gauging the SU(2) symmetry and identifying M D′ with MN and M D with MN′ , andthus having the full set of dynamical fields at the four dimensional boundary given
by MN and MN′ .
The basic idea of gluing is to undo the 5d boundary conditions. This amounts to turning
the background vector fields coupled to the flavor symmetry at the boundary to be dynamical.
This corresponds to the gauging of the puncture SU(2) symmetry here. However we also
have the matter fields of the 5d theory, some of which we lost due to Dirichlet boundary
conditions and thus need to reintroduce. In S-gluing we identify the operators of one puncture
as the ones corresponding to the fields obtaining the Dirichlet boundary condition in the otherpuncture, and vice versa. The fields obtaining Dirichlet and Neumann boundary conditions
have opposite charges and that is exactly what the superpotential above does for us. See Figure
36 for detailed illustration.
The S-gluing procedure for the E-string is thus performed as follows. Given two maximal
punctures one gauges the diagonal combination of the two puncture SU(2) symmetries andcouples the moment maps of the two punctures (the octets of operators M and M ′ ) with asuperpotential,′

W =M ·M ≡8
X

Mi Mi′ .

(121)

i=1

Our field theoretic construction some of the moment maps are mesonic (quadratic in fields)
and some are baryonic (cubic in fields). Thus the superpotential above has quartic and sextic
interactions in fields. As the moment maps have R-charge one the superpotential is consistent
with our choice of R-symmetry. Moreover, it is easy to verify that this R-symmetry is also notanomalous.
If we take two identical theories and S-glue them to each other because of the superpotential the charges of the moment maps of the two theories are identified with conjugation. In
particular that means that if we associate some flux to one of the theories we need top associatea negated flux to the other one. The flux of the associated to the surface after S-gluing is zero.
Here there is an important assumption that the flux associated to the combined surface is thesum of fluxes associated to the ingredients: This is an assumption motivated by consistency of
the arguments following from it.

65

SciPost Phys. Lect. Notes 78 (2024)

2
1
3

u −6

w −6

2
1
3
1
3

2

v −6

3

1
3

2

u6

1
3

u 2v 2 w 2
2
3

y

v6

2

1
3

1

2
3

6

3

u 2v 2 w 2w −6(uw) 7312(uw)− 7

12

2
3

7

2
u6

2
3

3

1
3
1
3

w6

u −6

2

2
3

w6

2
1
3
1
3

2

Figure 37: On the left we have two trinions S-glued to each other. There are various
superptential terms consistent with the charges denoted on the Figure. On the right
we utilize the S-confining Seiberg duality of SU(2) SQCD with three flavors to simplify the quiver theory. The Figure is taken from [25].

Exercise: Analyze the dynamics of gluing the two three punctured spheres (trinions)
into a genus two surface.
As the gluing procedure contains quartic and sextic superpotential terms one shoul wonder
whether these are relevant interactions or not. One can study this problem step by step, as
we have done in Lecture II is by turning on only a single interaction at each step so that thisinteractions is relevant. We will not do this in detail here and leave the analysis as an exercise.
The solution can be found in [25]. Let us just mention that the sequence of interactions
is: Gauging the puncture symmetry first, quartic superpoential second, and finally the sextic
superpotential. See Figure 37 for the illustration of S-gluing. We then can continue gluing
the rest of punctures in pairs to obtain a genus two surface. One needs to be careful to glue
together punctures so that no global stmmetry is broken by the interactions: This implies
S-gluing together punctures so that the corresponding moment maps have opposite charges
under global symmetries. Gluing two different surfaces together this can be always done as
the procedure simply amounts to identifying the symmetries of the two surfaces in certain way.
However when we glue punctures on the same surface we have no room anymore for arbitrary
identifications: We need to glue punctures of the same “color” together. Field theoretically
one can flue different punctures together, however in this case some of the symmetries will be
broken. The geometric reason for this can be traced to issues with discrete fluxes and twists [26,130]: We will not discuss such issues here. The procedure can be generalized to constructing
general genus g surfaces as depicted in Figure 38.
□

Gluing back to closed surfaces: Φ-gluing
The gluing we have discussed identifies the charges of the moment map operators of the two
glued punctures with conjugation. Let us now discuss a different gluing, the Φ-gluing in thenomenclature of [26], which in particular identifies the symmetries of the moment maps of66

SciPost Phys. Lect. Notes 78 (2024)

Figure 38: S-gluing trinions into a genus g surface with no flux (g = 4 in this example). The Figure is taken from [25].
the two punctures without conjugation. In particular when Φ-gluing two theories of the samekind together the corresponding fluxes are summed and not subtracted.
The Φ-gluing includes gauging the puncture SU(2) symmetry and identifying the momentmap operators of the two punctures using the superpotential,
W=

8
X


Mi − Mi′ Φi ,

(122)

i=1

where Φi is an octet of fields in the fundamental representation of the gauged symmetry which
are added to the model. The 5d explanation of this procedure is as follows. In this gluing
the missing fields, the ones which obtained Dirichlet boundary conditions, are the octet offundamental chiral fields Φ while the fields obtaining the Neumann boundary condition are
the moment maps of the two punctures which are identified, see Figure 39. Note that Φand the moment maps have opposite charges as should be the case for chiral halves of thehypermultiplet in 5d. The Φ-gluing is analogous to the N = 2 gluing in the compactificationsof (2, 0) theory [8]. This field there can be viewed as the N = 1 chiral adjoint superfield in
the N = 2 vector multiplet.
Let us make here two comments. First, from the discussion above it is clear that there is a
simple and useful way to relate the two types of gluings, and in fact generalize them. First, let
us consider taking a theory with a puncture and corresponding moment maps M and deformthe theory with the superpotential,
W = ΦM · M ,(123)

where Φ M is an octet of chiral fields in the fundamental representation of the puncture SU(2)symmetry. This superpotential removes M from the chiral ring of the theory and the procedure
is usually called “flipping M” (with the field Φ M being the flip field) [15]. Now notice that thefield Φ M has opposite charges to M and becomes the moment map of the new theory.35 In
particular the procedure of say S-gluing two punctures can be thought of as first flipping oneof the punctures and then Φ-gluing them,
W = ΦM · M + Φ · M ′ − ΦM
→ W = M · M′ ,
(124)
35

Such flips of punctures, or “signs” of punctures, are very common in various geometric constructions. See forexample [12, 131].

67

SciPost Phys. Lect. Notes 78 (2024)

M̃ D = M̃′D = Φ,𝒞

M′N = MN

W = (MN − M′N ) ⋅ Φ
6dSU(2) (M′ , M̃′ )(MN, M̃ D)
N
D𝒞′∂𝒞 = ∂𝒞′
5d

e D)
Figure 39: Explanation of Φ-gluing in 5d. We have the 16 fields on the left (MN , M′ e′and on the right, (MN , M D ). We gave eight of these Dirichlet (label D) and eight
Neumann (label N ) boundary conditions. Now we are undoing the boundary conditions by gauging the SU(2) symmetry and identifying M D′ with M D and Φ and MN
with MN′ , and thus having the full set of dynamical fields at the four dimensionalboundary given by Φ and MN = MN′ .
where in the last step we have integrated out the massive fields Φ and Φ M . In a similar way
the Φ-gluing can be thought of as first flipping one of the punctures and then S-gluing.
Finally, the procedure can be generalized when we Φ-glue some of the components of the
moment maps and S-glue the remaining ones. That is we glue by gauging the puncture SU(2)symmetry and turn on the superpotential,
W=

k
X
i=1

′
Mi Mσ(i)
+

8
X



′
Φi · Mi − Mσ(i)
,

(125)

i=k+1

where k is some number between zero and eight and σ ∈ S8 is a permutation of the eight
moment maps. This procedure is subject to some global obstructions having to do with consistency of various fluxes of the glued theories. In field theory these obstructions will appear
concretely as the demand that gaugings should be free of Witten or chiral anomalies. We will
not discuss these subtleties here.
Let us note that if the vector of fluxes of the two glued theories is F and F ′ , the components′of the vector of fluxes of the glued theories are Fi + (−1)ind(i) Fσ(i)where ind(i) is zero if thecomponent i is Φ-glued and 1 if it is S-glued.
Exercise: Analyze the dynamics of the Φ-gluing of two three punctured spheres into afour punctured sphere.
The dynamics of the Φ-gluing is more intricate than that of the S-gluing. For example gauging the puncture SU(2) symmetry we have fourteen fundamental chiral fields which naively
renders it IR free. However, one again can seek for a particular sequence in which to turn on all
the interactions so that at each step the interactions will be relevant. In fact we have alreadyanalyzed most of this dynamics in an exercise in Lecture I (see Figures 10 and 11), and we will
leave the rest as an exercise. See [25] for more details. Let us stress that in principle it might
happen that the interactions turn out to be irrelevant: This by itself does not invalidate theidentification of the across dimension duals. What that would imply is that the field theoretic
description in 4d is an effective one and we should think of the 6d construction as of its UVcompletion.
□

68

SciPost Phys. Lect. Notes 78 (2024)

1

1 v6w12

2
1
3
1
3

2

1

1
v 6u 12

u6

w6

u6

3

v

1
2
u v 2w 2

6

y

1
3
2

u w
v4
2
3

2

2

1
3

v

1

6

2
3

6

3

1
u 2v 2 w 2

w6

2
1
3
1
3

2

Figure 40: The field theoretic description of two trinions Φ-glued to each other. The
fields Φi flipping the moment maps are denoted by red lines: The dotted lines flip
the baryonic operators with charges v 6 u12 and v 6 w12 . The notation is that the dotted fields are in the singlet representation of the symmetry they point to and in thefundamental representation of the symmetry they emanate from. There are two setsof flipped baryon operators coming from each one of the two glued trinions with thesuperpotential given in (125). Note the way the quiver is drawn it might appear
that the dotted fields couple asymmetrically to the two trinions, however because of(125) the coupling is symmetric. The Figure is taken from [25].
Using the Φ-gluing we can identify what is the flux we should associate to the theory we
conjectured is obtained by a compactification on the three punctured sphere. We can Φ-glue
the punctures of two three punctured spheres in pairs and obtain a genus two surface. Doingso we obtain that the superconformal R-symmetry of the theory is,
Rsc = R +1 p
( 5 − 1) (qu + q v + qw ) ,
54

(126)

where qu,v,w denote charges under U(1)u,v,w . This is done by performing a-maximization andassuming as usual that there are no accidental emergent U(1) symmetries in the IR. Here the
a-maximization gives a non-trivial result, as opposed to S-gluing, since the charges of various
abelian symmetries in the field theory do not appear symmetrically to conjugation. Using thisR-symmetry we can compute the conformal anomalies,pp
5 5 473 5 27
a=
+
,
c=
+
.
(127)
4
16
2
8
One can wonder whether there is a compactification of the E-string theory on a genus two
surface with some value of flux giving these conformal anomalies after a-maximization. In
fact there is: We can take the anomaly polynomial of the E-string theory and integrate it overgenus two surface with one unit of flux in a U(1) which breaks the E8 global symmetry of the
6d SCFT to E7 × U(1). We will not perform the detailed analysis here as it requires a significant
build up, but it can be read off easily from the 6d results in [26] (See Appendix F.3.). It is thennatural to associate half a unit of this flux to each three punctured sphere we have obtained.
Finally we should detail what is the combination of the abelian symmetry explicitly visible
in the Lagrangian of our SQCD model which corresponds to the U(1) with the flux. A natural
69

SciPost Phys. Lect. Notes 78 (2024)

such combination is a diagonal U(1) of U(1)u , U(1) v , and U(1)w . This is also supported by
the structure of mixing of the three symmetries with the R-symmetry in (126). Then two othercombinations of the abelian symmetries together with the SU(6) should build the E7 . In fact
E7 has an SU(3)×SU(6) maximal sub-group and the precise imbedding of various symmetries
is,E8 → E7 × SU(2)u6 v 6 w6 → SU(6) × SU(3) u8 , v8 × SU(2)u6 v 6 w6 ,(128)

w4 v 4 w4 u 4with the U(1)s in (118) being the Cartan generators of SU(3) and SU(2). The flux we havederived preserves the E7 × U(1) subgroup of E8 , however the three punctured sphere only
manifests a subgroup of this. The reason is that we have punctures and these come alsowith a choice of a subgroup of E8 : Choice of U(1) × SU(8) under which the moment maps
are charged. This choices comes about, as we discussed, since we need to specify boundary
conditions of the puncture in 5d. The choice of the boundary conditions breaks further thesymmetry of the surface. However, if we consider surfaces without punctures, the symmetry of
the theory should be the one consistent with the choice of flux: In our genus two case we thus
would expect an E7 × U(1). This symmetry is not manifest in the Lagrangian but we expect theconformal manifold of the IR theory to have a locus with this symmetry. One way to test this is
to compute the supersymmetric index from the explicit field theoretic description. Computing
the index of a genus two surface using Φ gluing for all punctures we obtain,



1
1
I g=2,FU(1)×E =1 = 1 + 3 + 1 + 133 E7 + 3 12 12 12 + 2 6 6 6 56 E7 q p + · · · . (129)
7
u v w
u v w
This expression is written using the 6d R-symmetry. Because the superconformal R-symmetry
is different, see (126), the operators with negative uvw charge are actually relevant and contributions with zero charge are marginal. We observe that all the contributions combine into E7
representations in the index. This is a rather non-trivial further evidence that our identificationof the three punctured sphere and the flux is correct. Let us note in passing that in addition tohaving representations of E7 the multiplicities of the various operators (e.g. u12 v112 w12 appearing
thrice above) can be also predicted from six dimensions [113] and for general genus the field
theoretic computations agree with such predictions.
Finally, a theory corresponding to a four punctured sphere is expected to have a duality
group acting on its conformal manifold exchanging punctures of the same type. These are thedifferent pairs of pants decompositions we already discussed in the previous Lecture. One way
to check that the duality holds is to consider the index of the four punctures sphere and verify
that it is invariant under exchanging fugacities parametrizing the symmetries of punctures ofthe same type. This exercise can be performed for both S- and Φ−gluings, and indeed at leastin expansion of fugacities (as was also done in the previous Lecture) the result is consistent
with such dualitis. We leave this as an exercise here and the reader can also consult [25] formore details.

5.4

Closing punctures and comments

As we have discussed in Lecture III there is yet another field theoretic operation which has anatural geometric meaning: Giving vevs to operators charged under puncture symmetries andclosing the punctures. Before addressing this issue here let us expand a bit on the definition of
the flux. To specify the flux of the system in a more general way than we have done so far weneed to specify the flux of all the Cartan generators of E8 . There are many choices of this octetof U(1) symmetries. Given a puncture a natural choice is in terms of the eight U(1)s rotatingeach one of the components of the moment map. Thus the flux is specified by a vector F which
has eight components. A natural choice of these eight U(1) symmetries is to take the Cartans
70

SciPost Phys. Lect. Notes 78 (2024)of U(8) ⊂ SO(16) ⊂ E8 . The choices of these U(1)s depend on the choices of the punctures.
Different choices are related by an appropriate linear transformations of the U(1)s.
Given the octet of fluxes we can understand what is the sub-group of E8 that it preserves.
The preserved symmetry is generated by the roots of E8 which are orthogonal to the vector of flux. See e.g. Appendix A of [132] for a detailed discussion. In our parametrization we write the flux in terms of the SO(16) maximal subgroup of E8 with the roots of theSO(16) subgroup being 21 (±1, ±1, 0, 0, 0, 0, 0, 0) and permutations thereof. To find the symmetry we also need to complete the SO(16) to E8 by adding the spinor weights, which are1
4 (±1, ±1, ±1, ±1, ±1, ±1, ±1, ±1): Here we need to make a choice and either allow only evennumber of minus signs (corresponding to a spinor) or only odd number (corresponding to aco-spinor). For the constructions we discuss here all odd turns out to be the relevant choice.
We will soon discuss this more and here proceed with making this choice. We then check
which of the roots of SO(16) and which spinorial weights are orthogonal to the given vectorof fluxes: These roots and weights will build the root system of the preserved symmetry.
Let us give a relevant example. Consider the vector of fluxes,

F = (−1, −1, 0, 0, 0, 0, 0, 0) .

(130)

The roots of SO(16) ± 21 (1, −1, 0, 0, 0, 0, 0, 0) and (0, 0, · · · ) (with ellipses standing for any possible choice) are orthogonal to F . These roots span an SU(2) × SO(12) subgroup of SO(16).
The spinorial weights which are orthogonal to F are of the form ± 14 (1, −1, · · · ) with odd number of minuses. These form the representation (2, 32) of (SU(2), SO(12)). Such spinorial
weights, along with the Cartan generators, build the root system of E7 × U(1). Thus the flux
vector (130) preserves E7 × U(1) subgroup of the E8 symmetry of the 6d theory.
Exercise: Show that the flux F = (−1, −1, 2, 0, 0, 0, 0, 0) preserves E6 ×SU(2)×U(1) andF = ( 12 , 12 , 12 , 21 , 21 , 12 , 12 , − 12 ) preserves E7 × U(1) . The latter vector of fluxes is equivalent
by Weyl transformation to (130).
The roots of SO(16) orthogonal to F = (−1, −1, 2, 0, 0, 0, 0, 0) are 12 (0, 0, 0, ±1, 0, · · · , ±1, · · · )which span SO(10), as well ± 21 (1, −1, 0, · · · ) which span SU(2). The co-spinor weights orthogonal to F are ± 41 (1, 1, −1, ±1, · · · ) where in the brackets we have an even number of negative
choices of ± signs. These weights form a 16 and a 16 of SO(10) charged oppositely under a
U(1)a symmetry. The adjoint of the preserved symmetry thus is in the following representationof (SU(2), SO(10))U(1)a ,U(1) b ,(1, 1)0,0 + (1, 1)0,0 + (3, 1)0,0 + (1, 45)0,0 + (1, 16)+1,0 + (1, 16)−1,0 ,(131)

where we added all the Cartan generator. The (1, 1)0,0 + (1, 45)0,0 + (1, 16)+1,0 + (1, 16)−1,0
builds the 78 representation of E6 and thus the symmetry is E6 × SU(2) × U(1).
Let us look at F = ( 12 , 12 , 21 , 12 , 21 , 12 , 21 , − 12 ). The roots orthogonal to it are ± 12 (1, −1, 0 · · · , 0)and all permutations excluding the last element as well as ± 21 (1, 0 · · · , 1) and all permutations
excluding the last element. This spans the roots of SU(8). The spinorial weights orthogonal to the flux are ± 41 (±1, ±1, · · · , 1) with three negative choices of ± signs. This builds the
2×7!/(4!3!) = 70 dimensional representation of SU(8). The representations 63+70 compose
the adjoint representation of E7 . The symmetry preserved by the flux is thus E7 × U(1).
□In fact we claim that the vector of fluxes associated to the three punctured sphere is (130).
To make this statement meaningful we need to make a choice of puncture which will determine
the basis of U(1) symmetries. However, since the three punctures appear rather symmetricallythe statement above is true for any choice of one out of the three punctures of the trinion. The
71

SciPost Phys. Lect. Notes 78 (2024)

flux is −1 for U(1) symmetries coming from the baryonic components of the moment maps
and the 0s to the mesonic ones. As we saw above this flux preserves E7 × U(1) symmetry,
consistently with our previous discussion.
Let us perform a consistency check of this claim. We assume that the flux is (130)when computed using the M x moment maps of (119) and show that the same also holds
for M y (and by symmetry also for Mz ). To show this we translate the fluxes into theSU(6) × U(1)u × U(1) v × U(1)w basis,
6Fu + 12Fw = −1 ,6Fu + 12F v = −1 ,4Fu − 2Fw − 2F v = 0 ,

(132)

1
which gives Fu = F v = Fw = − 18
. By the symmetry of this result it is immediate that (130) is
also the flux computed with respect to the other two punctures.
With this understanding of the fluxes let us discuss closing a maximal puncture. A generalidea already discussed in the previous Lecture is that given a puncture we can “close” it by turning on a vacuum expectation value to one of the components of the moment map operator. We
might also need to introduce certain fields and superpotential terms to match the anomalies
with the expectations from the theory associated to the surface without the puncture. Intuitively, the puncture is removed as the vacuum expectation value (vev) breaks the symmetry
associated to the puncture.36 In addition the flux associated with the surface we obtain after
giving the vev will be shifted. The value by which we shift the vector of fluxes depends onthe charges of the moment map component which obtains the vev. For the E-string the precise
procedure of closing a puncture was derived [26]. We will not repeat the computation here
but rather state the result.
Let us assume that the charges of the octet of moment map operators are ui x ±1 where theui are combinations of fugacities for the Cartan of E8 and x is the Cartan of the puncture SU(2)symmetry. Let us then label the components of the moment map operator charged ui x ±1 as
Mi± . We can give the vev to any component and for concreteness we will choose to turn on avev to M1+ . Following the derivation in [26] we will need also to introduce chiral superfields,
which we will denote by Fi , and couple them with the following superpotential,

W = M1− F1 +8
X

Mi+ Fi .

(133)

i=2

A natural choice for the basis of U(1) symmetries is in terms of the moment maps of thepuncture being closed. In this basis the flux of the theory in the IR is shifted,∆F = (2, 0, 0, 0, 0, 0, 0, 0) .
This vector of shifts is determined by (in fact proportional to) the vector of charges of theoperator which received the vev.
Let us apply this procedure choosing different components of the moment maps. We have
two different types of components, baryonic and mesonic, and we will discuss them separately.
First, we close a puncture giving a vev to a baryonic component of the moment map. Following
our rules above the flux of the resulting theory is,

F + ∆FB = (1, −1, 0, 0, 0, 0, 0, 0) .
This in fact is equivalent to the flux in (130) by action of the Weyl group of E8 and thus
preserves the same symmetry. However, if we turn on a vev to one of the mesonic compenentsof the moment map and follow the same rules we obtain,

F + ∆F M = (−1, −1, 2, 0, 0, 0, 0, 0) .
36
More precisely it breaks one combination of U(1) symmetries, the Cartan generator of the puncture symmetries
and symmetries coming from six dimensions. We can always choose to parametrize the broken symmetry by theone associated with the puncture.

72

SciPost Phys. Lect. Notes 78 (2024)

As we have seen this flux preserves an E6 × SU(2) × U(1) subgroup of E8 .
Given the above procedure, since we have field theoretic constructions of the trinions, we
can explicitly study the procedure of closing punctures. For example, a baryonic vev will Higgs
completely the SU(3) gauge symmetry of the trinion. We will be left with a WZ model in the
IR. On the other hand, the mesonic vev Higgses the SU(3) gauge symmetry of the trinion only
to SU(2). See Figure 41 for an illustration. We will soon discuss this case in detail in an
exercise.
Before proceeding with the detailed discussion of closing punctures let us make an additional comment. Given a puncture with an octet of moment maps Mi we can introduce
for example a single chiral field Φ and flip one of the components, say by ontroducing thesuperpotential,
W = Φ · M1 .
(134)
This procedure will remove M1 from the chiral ring and replace it with Φ. Importantly the
charges of Φ, which is the new moment map component, are opposite to M1 . This flipping of
the moment map component does not alter the flux associated to the surface: It just changes
the color of puncture. Here let us comment again on the choice of spinor versus co-spinor
to compute the flux. Flipping even number of components of the moment map indeed isconsistent with our prescription to determine the flux as it is part of the Weyl symmetry ofSO(16). However, flipping odd number of components does naively change the flux. This canbe remedied by declaring that if we flip odd number of components we compute the flux using
the spinor and not the co-spinor. Notice that the SU(2) punctures of the trinion theory have
odd number of fundamentals and thus have a Witten anomaly [67]. Flipping even number ofmoment maps will keep the anomaly while flipping odd number will remove it. Thus we canjust say that if we compute the flux relative to puncture symmetry with Witten anomaly we
use the co-spinor and if the puncture has no Witten anomaly we compute the flux with thespinor.
Let us now analyze what happens if we close punctures with a flip. Say we flip a mesoniccomponent. Then the vector of fluxes is unchanged as F of (130) has zeros at the location of
the mesonic components. Turning on the vev, the flux is shifted to,F + ∆F M = (−1, −1, 2, 0, 0, 0, 0, 0) ,which is the same E6 × SU(2) × U(1) preserving flux as we obtained without flipping. Notehowever that the vev here is turned on to the flip field which is a singlet under the gauge
symmetry. Thus, we do not have any Higgsing but rather generate various mass terms forgauge non-invariant fields. We will then obtain again an SU(3) SQCD albeit with smaller
number of flavors, namely N f = 5 SU(3) SQCD. This theory should be then related to the one
obtained without the flipping (as the flux and the surface are the same). The latter is SU(2)SQCD with N f = 5, see Figure 41. The two theories are indeed related in a non-trivial way
due to a Seiberg duality.
Exercise: Closing a puncture by giving a vev to a mesonic moment map, withand without flip, derive the theories with fluxes F = (−1, −1, +2, 0, 0, 0, 0, 0) andF = (−1, −1, −2, 0, 0, 0, 0, 0). Applying Seiberg duality to the theory with the flip compare the two models.
We will perform the analysis of the flows triggered by vevs using the supersymmetric index.
The basic observation [119] is as follows. Given an operator O in a 4d theory that contributes
(O )
(O )
(O ) 1 (O )(O )(O ) 1 (O ) Q
Qi
to the index with weight X ≡ q− j1 + j2 + 2 R p j1 + j2 + 2 Ruand O can acquire ai ivev, the index will have a pole when we tune the various fugacities so that X = 1. The residue

73

SciPost Phys. Lect. Notes 78 (2024)

5

1
1 c1 w3v3u3
uvw 7

wvu 7

c1

c1

1
2

1
2

2
Z

1

1

2

0
c1w 4
u 2v 2

2

2
x

1

c1u 4
1
v 2w 2

1

1

v 11
c1
uw
1
2

0

1
2

w4
u 2v 2Z

1

5

1

v 5u 5 12 0
c
w 1
7 7
1 u w
c1− 2 5v

1

11
c1− 2 3 3 3
wvu

2

0

1

u4
w 2v 2v 5w 5 12
c
u 1

2
x

1

Figure 41: On the left we have the model obtained by closing a puncture with amesonic vev with no flip. On the right we have the model obtained by closing apuncture with a mesonic vev with a flip. Both theories correspond to a tube with flux
breaking E8 to E6 ×SU(2)× U(1). The crosses are flip fields flipping the baryons built
from the bi-fundamentals. The Figure is taken from [25].
of the pole, removing the Goldstones for the symmetry broken by the vev, is the index of thetheory obtained in the IR. Let us apply this for the theories at hand.
The supersymmetric index of the trinion is,I
d t1 d t2(q; q)2 (p; p)2
1(t r inion)I(x, y, z) =
(135)
Q3
6
2πi t 1 2πi t 2
Γe (t i /t j )
i̸= j

×

3
Y

6


Y

1
1
1
1
Γe ((qp) 6 u6 t i x ±1 )Γe ((qp) 6 v 6 t i y ±1 )Γe (qp) 6 w6 t i z ±1
Γe (qp) 3 (uvw)−2 c j t i−1 .

i=1

j=1

Q6
Here ci parametrize the SU(6) symmetry group ( j=1 c j = 1). Let us close the puncture
symmetry of which is parameterized by y. First we give a vev to the moment map (denote it41
as M3+ ) with weight X + = (qp) 2 wv2 u2 c1 y +1 . Note that the integrand of (135) does not havea pole when we tune X + to be 1. However, the integration contour is pinched leading to adivergence. Let us look at the following poles of the integrand in t i ,1

Outside: t i = (qp)− 6 v −6 y ±1 ,Inside:1t i = (qp) 3 (uvw)−2 c1 .

Outside/Inside stand for poles outside/inside the unit circle integration contour. Here weassume first that the fugacities for flavor symmetries are on the unit circle and |q|, |p| < 1.
This determines which poles are inside the unit circle contour for t i integration variables and1 2 2
which are outside. Setting now X + = 1 we can take y = (qp)− 2 wv 4u a1−1 . Then we have,1

Outside: t i = (qp) 3 (uvw)−2 c1 ,2

t i = (qp)− 3 (uw)2 v −10 c1−1 ,1Intside: t i = (qp) 3 (uvw)−2 c1 .
11Note then that the poles in say t 1 at t 1 = (qp)− 6 v −6 y −1 and t 1 = (qp) 3 (uvw)−2 c1 collide fromboth sides of the t 1 contour pinching it and leading to a divergence. We can consider poles int 2 or t 3 which will be related to the above by the action of Weyl symmetry giving an equivalent1
result. Let us then plug the value X + = 1 and t 1 = (qp) 3 (uvw)−2 c1 in the integrand of (135)74

SciPost Phys. Lect. Notes 78 (2024)

to compute the residue,I(q; q)(p; p)dt
1(q; q)(p; p) × ResX + →1 I
(x, y, z) →(136)
2
2πi t Γe (t ±2 )

 
 

1
1
−1
−1
×Γe u7 vwc1 2 t ±1 x ±1 Γe uvw7 c1 2 t ±1 z ±1 Γe (qp) 2 (uw)−1 v 11 c12 t ±1




Y

6
6


8
4
Y
1
1
1
v
u
±1
−3
2
±1
2
Γe qpΓe (qp) 2 (uvw) c j c1 t×cΓe (c j /c1 ) Γe (qp) 2 2 2 c1 x
u4 w4 1 j=2
v w
j=2


4
1 w
±1
×Γe (qp) 2 2 2 c1 z.
v u
(t r inion)

11

Here we have defined t = t 2 (qp) 6 (uvw)−1 c12 . The multiplication of the residue by (q; q)(p; p)
is to remove the Goldstones. Now, notice that the contribution of the last line in the square
brackets of (136) is from moment maps M3− and the mesonic components of M j+ ( j = 4 · · · 8)
which are removed by adding additional terms to the superpotential (133) when closing the+puncture. We also add flip fields for the two baryonic moment maps M1,2
and thus the index
of the theory we obtain after closing a puncture with the mesonic vev is,
I
 


1
(q; q)(p; p)
dt
1− 21 ±1 ±1(tube)
7
7 − 2 ±1 ±1(x, z) =I1
Γe u vwc1 t x
Γe uvw c1 t z
2
2πi t Γe (t ±2 )
6

Y


1
1
1
1
×Γe (qp) 2 (uw)−1 v 11 c12 t ±1
Γe (qp) 2 (uvw)−3 c j c12 t ±1
(137)
j=2

 


 
 
4
1 w
1
1
u4
1
±1
±1
×Γe qp 4 4 14 c1 Γe qp 4 4 14 c1 Γe (qp) 2 2 2 c1 x
Γe (qp) 2 2 2 c1 z
.
u v w
w v u
v w
v u
This is an SU(2) gauge theory with N f = 5 with additional gauge singlet fields and a superpotential. The quiver is depicted in Figure 41. The flux of this model in terms of the moment
map symmetries of the puncture we close is F = (−1, −1, 2, 0, 0, 0, 0, 0).
e ± with the superpotential
Next we can first flip the moment maps M3± by adding a field M
3e∓,
W = M3± · M
3e + . This operation simply amounts to turningand then close the puncture by giving a vev to M
3−
on a mass term to the quarks building M3 . The resulting pole in the index comes solely from1 2 2
e + which is X̃ + = (qp) 2 w 4u c −1 y +1 . The index of the theory in the IR isthe contribution of M13v
then just the one obtained by taking (135), setting X̃ + = 1 and adding the flip fields determinedby (133),(q; q)2 (p; p)2(tube)
I2(x, z) =
6

I

3
Y
1
d t1 d t21Γe ((qp) 6 u6 t i x ±1 )Q32πi t 1 2πi t 2
i̸= j Γe (t i /t j ) i=1


6

Y


1
1
v 10
3 (uvw)−2 c t −1
6 w6 t z ±1
×Γe (qp)
t
c
Γ
(qp)
Γ
(qp)
i
1
e
i
e
j
i
u2 w2
j=2

 
Y


6
2
2
w
u
u4 w4 −1 −1−1−1×Γe (qp) 10 10 c1 Γe (qp) 10 10 c1
Γe (qp) 8 c j c1
.
v u
v w
v
j=2


− 13

(138)

This is an SU(3) gauge theory with N f = 5 with additional gauge singlet fields and a superpotential. The flux of this model in terms of the moment map symmetries of the puncture we
75

SciPost Phys. Lect. Notes 78 (2024)

close is F = (−1, −1, −2, 0, 0, 0, 0, 0). Let us perform Seiberg duality for this theory. The dual
theory of SU(3) with N f = 5 is an SU(2) with N f = 5 with the mesons flipped. The baryonic symmetry is identified between the two duality frames while the mesonic symmetries areconjugated. The resulting index is,
 

I
 5 5
7 7
1
1 u w
(q; q)(p; p)1
dtv w 12 ±1 ±1−(tube)
I2(x, z) =Γe (qp) 2 5 t ±1 c1 2
Γe
c t x
2
2πi t Γe (t ±2 )
u 1
v
 5 5
Y


6


1
v u 21 ±1 ±1
w2 −1− 21 ±1
−3
−1
× Γe
c t z
Γe (qp) 2 (uvw) c j c1 t
Γe (qp) 10 10 c1
w 1
v u
j=2
(139)
u2
× Γe (qp) 10 10 c1−1v w
(tube)

Y61

Γe (qp) 2

j=2

4
1 w
u4
±1
2
(qp)
Γ
c
x
c j z ±1 .
ej
w2 v 2
u2 v 2

(tube)

are very similar they are not exactly the same. They haveand I2
Note that although I1
exactly the same types of punctures as both are obtained by closing a puncture of a different
theory in different ways. However let us compute the flux in terms of the symmetries of say
the x puncture. In the basis of the y puncture we started from, as we mentioned above, theflux is F = (−1, −1, 2, 0, 0, 0, 0, 0). Let us compute this flux in terms of the basis {u, v, w, ci },6F v + 12Fu = −1 , 6F v + 12Fw = −1 ,

(140)

1
4 F v − 2 F u − 2 F w + F c 1 = 2 , 4F v − 2 F u − 2 F w − F c 1 = 0 .
5This gives us
§
ª
1
1 5
{Fu , F v , Fw , Fc1 } = − , 0, − ,
.
12
12 3
From here the flux in terms of the moment maps of the x puncture (119) is,
1 3 3 1 1 1 1 1
F1 = − , − , , − , − , − , − , −.
2 2 2 2 2 2 2 2

(141)

(142)

Let us repeat now the exercise with flux F = (−1, −1, −2, 0, 0, 0, 0, 0),
6F v + 12Fu = −1 ,6F v + 12Fw = −1 ,

(143)

1
4 F v − 2F u − 2F w − F c 1 = 0 .
5

4F v − 2Fu − 2Fw + Fc1 = −2 ,This gives us


§
ª
1
1
1
5
F u , F v , F w , F c1 = − , − , − , −.
36 9 36 3

From here the flux in terms of the moment maps of the x puncture (119) is,
3 1 3 1 1 1 1 1
F2 = − , − , − , , , , ,.
2 2 2 2 2 2 2 2

(144)

(145)

Note that although the punctures are the same the fluxes F1 and F2 are different. Both fluxes
preserve E6 × SU(2) × U(1) and are related by Weyl transformation exchanging the two baryonic symmetries and flipping the signs of the mesonic symmetries.37 However there is no
The fact that E6 × SU(2) × U(1) is preserved follows from (− 23 , − 12 , − 23 , 12 , 12 , 12 , 21 , 12 ) being obtained from(−1, −1, −2, 0, 0, 0, 0, 0) by change of basis, while for the latter we established the symmetry already. It is though instructive to rederive this fact. Taking the roots of SO(16) the former flux preserves explicitly SU(2)×SU(6)×U(1)2 .
The co-spinorial weights enhance one of the U(1)s to SU(2) while also giving (2, 20) of (SU(2), SU(6)) enhancing
it to E6 .
37

76

SciPost Phys. Lect. Notes 78 (2024)
2
0

2

2

2

2

1

0

6

2

2

6

1
1
0

2

0

2

1

2
2

0

0

0

0

2

1

8

1

2

0
0

2

Figure 42: On the left we have the theory resulting in Φ-gluing together either two
copies of first or the second tube. The quiver can be thought of as drawn on a spherewith the two half-squares identifies as the same SU(2) symmetry on a pole of thatsphere. This theory is associated to the E6 × SU(2) preserving flux. On the left we Φglued first and second tube to obtain a theory preserving E7 × U(1) flux. The explicit
quiver drawn is obtained by using Seiberg duality relating SU(2) with three flavors
to a WZ model.
redefinition of symmetries which will make both punctures and fluxes of the two theories thesame. Φ-gluing two copies of each one of the tube theories into a torus, and thus getting rid of
the punctures, we will obtain two completely equivalent models preserving E6 × SU(2) × U(1)symmetry (related by simple redefinitions of symmetries), as is expected. Summing the fluxes,
F1 + F2 = (−2, −2, 0, 0, 0, 0, 0, 0) we obtain E7 ×U(1) preserving vector and thus Φ-gluing tube
one to tube two we will obtain a different theory preserving this symmetry. See Figure 42 andwe will discuss these models in more detail in the next Lecture.
Let us see explicitly how the gluing of the two different tubes works. The index of the
Φ-glued theory is,I I(q; q)(p; p) 2
dx
1
dz
1
(tube)(tube)I E7 t orus =I(x, z)I2(x, z)±222πi x Γe (x )2πiz Γe (z ±2 ) 1



6
2 2

 
Y
1
1
1 v w
× Γe (qp) 2 u−6 v −12 x ±1 Γe (qp) 2 u−6 w−12 x ±1
Γe (qp) 2 4 ci−1 x ±1
(146)
u
i=1



6
2 2

Y
1
1 v u
1
−6
−12
±1
−6
−12
±1
−1
±1
Γe (qp) 2 4 ci z× Γe ((qp) 2 w v z )Γe (qp) 2 w u z.
w
i=1
The contributions from the second and third lines are for the Φ fields introduced in gluing thetwo punctures. Let us first perform the x integral. Collecting all the fields and using (60)identity (index of a couple of fields which can form a mass term is equal to 1) we getI(q; q)(p; p)
dx
Ix =
(147)
2
2πi x



 

5 5 1
1
1
−1
Γe u7 vwc1 2 t ±1 x ±1 Γe ( v uw c12 t̃ ±1 x ±1 )Γe (qp) 2 u−6 v −12 x ±1 Γe (qp) 2 u−6 w−12 x ±1.
×Γe (x ±2 )77

SciPost Phys. Lect. Notes 78 (2024)

Here we distinguish the fugacities of the SU(2) gauge groups of the tube theories by denoting
one as t and another as t̃. This is an SU(2) SQCD with N f = 3 and thus is dual using Seiberg
duality to a WZ model of the gauge invariant mesons and baryons (62),


 
 
5
1
1 uw − 1
1 uv
1 w− 12 ±1
6 6 6 ±1 ±1
±1
±1
2
2
I x = Γe (u v w t t̃ )Γe (qp) 2 11 c1 t
Γe (qp) 2 11 c1 t
Γe (qp) 2 7 7 c1 t̃
v
w
u v


 10 10 
5
1


1
v
v w
×Γe (qp) 2 7 7 c12 t̃ ±1 Γe u14 v 2 w2 c1−1 Γe
c1 Γe qpu−12 v −12 w−12 .
2
u w
w
(148)
Performing the z integration we obtain the same expression but with w and u exchanged.
Combining everything together we obtain the following index,I I(q; q)(p; p) 2
1
1
dt
d t̃
I E7 t orus =±22
2πi t Γe (t )2πi t̃ Γe ( t̃ ±2 )
 

2 
1 vw − 1
1 wu − 1× Γe (u6 v 6 w6 t ±1 t̃ ±1 )Γe qpu−12 v −12 w−12
Γe (qp) 2 11 c1 2 t ±1 Γe (qp) 2 11 c1 2 t ±1
v
u


6




5
Y
1
1
1 uv
1
1
v
− 21 ±1
−3
±1
±1
2
2
×Γe (qp) 2 11 c1 t
Γe (qp) 2 (uvw) c j c1 t
Γe (qp) 2 7 7 c1 t̃
w
u w
j=2

 
Y
6
5
1
1
1 w
1
u5 21 ±1
− 12 ±12 ±122×Γe (qp) 7 7 c1 t̃Γe (qp) 7 7 c1 t̃Γe ((qp) 2 (uvw)−3 c −1cj
1 t̃ ) .
w v
u v
j=2
The quiver description of this theory is depicted on the right of Figure 42. On the first line
we have the bifundamentals of the two SU(2) gauge groups and the flip fields (denoted bycrosses on the figure), the second line give an octet of fundamentals under one SU(2) and thelast line an octet under the other SU(2) group. Computing the index of this theory one canobserve that the protected states form representations of E7 ×U(1) with the U(1) parametrizedby u6 v 6 w6 . For example, taking antisymmetric squares of the octets and building invariants
we can obtain operators in 28 + 28 which build the fundamental representation 56 of E7 . Let
us now consider simplifying this theory by deforming it. Consider giving a vev to the flip fields
denoted by crosses in the quiver. These are not charged under the SU(8) and thus we do notbreak the E7 structure. These vevs give mass to the bifundamental fields and we remain withtwo copies of SU(2) N f = 8 SQCD coupled together through a superpotential W = M1 × M2
with Mi being the 28 mesons/baryons of each copy. This theory was discussed in detail in [71]and dubbed the the E7 surprise as it was shown to exhibit E7 symmetry. The discussion here
shows this surprise in a geometric context.38
Let us now consider taking one of the E6 × SU(2) × U(1) tubes and Φ-gluing the twopunctures to each other. The two punctures are of different types and thus we will break apart of the symmetry doing so.39 In Φ-gluing the moment map symmetries of the punctures
are identified through the superpotential. The moment map symmetries of our two puncturesdiffer by exchanging u and w and thus these two symmetries are broken to a diagonal one.
The broken u/w parametrizes the SU(2) part of the E6 × SU(2) × U(1) symmetry. To seethat let us discuss how to derive the character of the adjoint of the preserved symmetry from38

The more precise statement is that on some locus of the IR conformal manifold of this theory the SU(8)symmetry enhances to E7 . In fact it was argued in [72] that on that locus (point) the symmetry is actually E7 ×U(1).
The U(1) is accidental in our discussion.
39
This has an interesting interpretation in 6d. The flux associated with this tube is not properly quantized, andso to be consistent must be accompanied by certain non-trivial connections in the non-abelian flavor symmetry,here in the SU(2) part. These lead to the breaking of this symmetry once the tube is closed to a torus, see [26, app.
C].

78

SciPost Phys. Lect. Notes 78 (2024)

Φ1

h −1

6

1

ϕi

h −2

1

2

0

h −3 1

Qi

0

Φ3

6

Φ2
vev for ϕiΦ12

M = Φ22

2

h3

Φ3

2

2

Figure 43: On the left the theory obtained by gluing two punctures of theE6 × SU(2) × U(1) tube together. On the right we give vev to the flip fields anduse Seiberg duality of SU(2) with N f = 3 to obtain a simple theory with E6 symmetry. The cross is a field flipping the mesons/baryons in the 15 of SU(6) built fromthe bifundamental field.
the vector of fluxes. Let us take the weight of the moment maps of the puncture in termsof which we compute the flux to be {ai }8i=1 . These can be thought of as fugacities for the
Cartan of SO(16) subgroup of E8 . For concreteness let us take the flux (145). The components
a2−1 +a4 +a5 +a6 +a7 +a8 is the character of the preserved U(6) = U(1)a ×SU(6) while a1−1 +a3−1
is the character of preserved U(2) = U(1) b × SU(2). The E6 is built from SU(2) × SU(6). One
combination of the U(1)a/b enhances to SU(2). Looking at the co-spinorial weights preserved
Q8
Q8
1
1
by the flux, (a1 a3 /(a2−1 i=4 ai )) 2 parametrizes the U(1) while z = (a1 a3 (a2−1 i=4 ai )) 2 thepreserved SU(2). Plugging in the values of the ai for the moment maps of the x puncture we
obtain that z = (w/u)12 while all the other characters are independent of u/w. The resulting
theory is depicted in Figure 43. The quiver has explicitly SU(6) × SU(2) × U(1)h symmetry
with the SU(6) × SU(2) expected to enhance to E6 . The superpotential of this theory is,
W = Q 1Q 2 Φ23 + Φ1 Φ2Q 1 + Φ1 Φ2Q 2 + Q21 φ1 + Q22 φ2 .

(149)

We can derive the superconformal R-symmetry by parametrizing the trial R-symmettry as
R + h qh to compute,a(h) =27h (1 − 9 h2 ) ,
2

(150)

1
which is minimized for h = 3p
. We note that all the operators are above the unitarity bound3
evaluating their R-charges using this R-symmetry. Let us write the supersymmetric index. We
will use the more convenient rational value of h = 2/9 to write the expression, which is a good
approximation to the superconformal one:
1

5

2

I E6 t orus = 1 + 3h−6 (qp) 3 + 27 E6 h−4 (qp) 9 + (6h−12 + h6 )(qp) 3
7
9

1
3

(151)
8
9

+ 27 E6 h−2 (qp) + 3h−6 (q + p)(qp) + 3 × 27 E6 h−10 (qp) + 10h−18 (qp) + · · · .
Here we have defined,
27 E6 = 6SU(6) × 2SU(2) + 15SU(6) .
79

(152)

SciPost Phys. Lect. Notes 78 (2024)

First we see that the states form representations of E6 . Second, we notice that at order q p using
the superconformal R-symmetry we have 0.40 This implies in fact that the number of exactly
marginal deformations is 7. This follows as we have explicitly preserved SU(6)×SU(2)× U(1)symmetry which would contribute corresponding currents with negative signs at this order (seediscussion around (56)). To obtain zero we need to have marginal operator in the adjoint of
this group. Such single marginal operator will contribute exactly marginal operators numberof which is the rank of the group (computing the relevant Kähler quotient this follows immediately). Moreover assuming that somewhere on that conformal manifold the symmetry
enhances to E6 × U(1) we will have to have a marginal operator in the adjoint of this groupwhich again would give a 7 dimensional conformal manifold: Consistently with the above.
Let us now try to simplify the theory of Figure 43 by deforming it but preserving the E6
symmetry. One way to do so is to give a vev to the flip fields φi . These fields are only charged
under U(1)h and thus the deformation should preserve the E6 symmetry. Doing so the bifundamental fields Q i acquire a mass and decouple in the IR. Note also that on a general point of
the conformal manifold we would also expect the fields Φ3 to acquire a vev. One way to see
this is by noting that giving a vev to φi we are demanding that h−6 qp = 1. However the weight1of the field Φ3 is h−3 (qp) 2 and thus it will be also weighed as +1 in this limit. The fact that thisvev can be generated also follows from the exactly marginal superpotential term Q 1Q 2 Φ23 that
we have here: This superpotential breaks no symmetry. Let us assume that however the vev
for Φ3 is not generated (say when the above quartic superpotential is not turned on). Then,as Q i acquire mass the SU(2) gauge group on the left has only six fundamentals in the IR andthus is dual to a WZ model of the gauge invariants M = Φ22 . These fields couple to the rest
through W = Φ21 M + M 3 which preserves the SU(6) symmetry (and also the E6 ). The resulting
quiver is depicted on the right of Figure 43. Note that this is the theory we analyzed in Section
3.2 as an example of emergence of E6 symmetry. The two models differ by the superpotential
term M 3 which does not break the E6 symmetry as it is a singlet of SU(6) and SU(2): Moreover we expect this superpotential term to be irrelevant and vanish in the deep IR.41 Yet again
thus we have derived geometrically an instance of emergence of IR symmetry in 4d. Finally
if we do generate a vev for Φ3 this will just lead to a WZ model of fields Φ1 (bifundamentalof SU(2) × SU(6)) and M (15 of SU(6)) with superpotential W = Φ21 M + M 3 , which again
preserves E6 . The superpotential is marginally irrelevant in the IR but if we compactify to 3d
it would be relevant leading to an SCFT with E6 global symmetry [133, 134].
□
Finally, let us comment that the fact that we can interpret the SU(3) SQCD in the middleof the conformal window as a trinion of the E-string theory gives us a geometric way to understand its Seiberg (self)duality [2]. See [25] for details. We will see in the next section that
one can also find a geometric meaning for SU(N + 2) SQCD in the middle of the conformal
window: The case of N = 1 is the E-string theory discussed here, while higher N cases correspond to generalizations to compactifications of the minimal (DN +3 , DN +3 ) conformal matter.
The bottom line is that the Seiberg duality of SU(N + 2) SQCD in the middle of conformal
window can be directly related to the Weyl group of SO(4N + 12) [25].
With h = 2/9 we have at that order 10 states but these are charged under U(1)h and thus will move away once
we stick to the superconformal R-symmetry.
41
This can be argued for by first turning on the gauge coupling, then the M Φ21 superpotential and then M 3superpotential.
40

80

SciPost Phys. Lect. Notes 78 (2024)

6

Lecture V: An algorithm for deriving across dimensional dualities

In the previous Lectures, we discussed how one can try to conjecture various N = 1 four
dimensional field theories that result from compactifications of 6d SCFTs. The method used
there was to determine the expected ’t Hooft anomalies of the model, from integrating theanomaly polynomial of the 6d SCFT on the compactification surface, and then search for a 4dmodel having the same anomalies. To actually make progress usually one also needs some
additional assumptions, like that the model is conformal with a conformal manifold passing
through weak coupling. An interesting aspect in this approach is that we do not attempt totackle the reduction directly. Rather our aim here is to build a 4d model that flows to or sitson the same conformal manifold as the 4d theory expected from the 6d reduction, without therelation between the two being immediately apparent.
This method was used to great effect in the previous Lectures. Nevertheless, it has several
shortcomings. First, as we mentioned, it usually requires additional assumptions regarding
the nature of the 4d N = 1 theory42 that may be wrong. Also the reliance on anomalies, whilevery convenient, is restricted to spacetimes of even dimensions. As such it is appealing to also
look for other methods by which 4d N = 1 theories, resulting from the compactification of 6d
SCFTs, can be determined.
Here we shall present a different method that can be used to conjecture such 4d N = 1
theories. Unlike the previous method, here we shall actually try to follow the reduction process
to determine the 4d theories. However, the reduction itself is in general quite complicatedand difficult to follow. Nevertheless, here we can use an observation we made in the previous
Lecture. Specifically, we noted that the resulting 4d theories are usually sensitive only tovery rough properties of the surface, e.g. its genus and total flux. Other properties, e.g. theexplicit metric of the surface or field strength of the flux, usually affect the result at worst
through marginal operators, and in many cases only through irrelevant ones. We can exploit
this for our purpose, and choose these to take a very special form for which we can follow the
reduction.
We shall next discuss the method, which consists of two parts. In the first one we shalldiscuss a general method to conjecture the 4d theories resulting from the compactificaion of
6d SCFTs on flat surfaces, that is tori and tubes, with flux. This method follows our previous
discussion and relies on following the reduction process in a specific limit where it is easy toanalyze. After that, we shall introduce a method that allows us to exploit this to also understand the compactificaion of 6d SCFTs on more generic surfaces, notably spheres with morethan two punctures.

6.1

The general idea

Consider the compactification of a 6d SCFT on a tube, which is just a sphere with two punctures. To set the stage, we take the coordinates of our 6d spacetime to be x 1 , x 2 , . . . , x 6 , withx 6 being the circle direction and x 5 the interval direction of the tube. We shall also take thebounaries of the interval to be at x 5 = ±a. We want to determine the reduction of the theory
to four dimensions. We can analyze this by first reducing along the circle spanned by x 6 , toget a 5d theory on the 4d spacetime spanned by x 1 , x 2 , x 3 , x 4 and the interval spanned by x 5 .
As we noted in the previous sections, when compactified to 5d, 6d SCFTs can flow to 5d gauge
42

The specific nature of the assumption is usually the R-charges of the chiral fields under the superconformal
R-symmetry, from which the a and c central charges of the theory can be determined in terms of the number ofvector and chiral fields. The simplest assumption is that all chiral fields have R-charge of 23 , that is the theory is
conformal at weak coupling. It is also possible to explore other choices where there are chiral fields with morethan one value of R-charge, see for instance [82, 83].

81

SciPost Phys. Lect. Notes 78 (2024)

theories if a proper holonomy is introduced. Here we shall assume that such a holonomy isincorporated. We shall later on see that it plays an important role in the introduction of flux.
We can then reduce along the circle to get the 5d gauge theory on the 4d spacetime times aninterval. At the boundaries of the interval we need to put boundary conditions, which canbe expressed as boundary conditions on the 5d bulk fields as these approach the boundary,as done in Lectures III and IV. Throughout this Lecture we shall take the boundary conditions
to be of the same type as those used in the previous Lecture, that is the ones associated withmaximal punctures. As the theory now is just an IR free 5d gauge theory, we can then reducealong the interval and get the 4d theory, which is just built from the 5d matter that is consistent
with the boundary conditions.
So far we have discussed the case without flux. However, to deal with interesting cases
requires the introduction of flux. It turns out that we can make progress by representing theflux as a variable holonomy. Specifically, we pointed out that we usually need to turn on anholonomy on the circle so that the 6d SCFT reduces to a 5d gauge theory. However, this holonomy is not unique, where in general we can have many different holonomies leading to a 5dgauge theory. For instance, say the holonomy is in a non-abelian flavor symmetry. In that case,it breaks the flavor symmetry to a U(1) group, where the holonomy resides, plus its commutant in the non-abelian group. We can now act on this holonomy with Weyl transformationsof the broken group that are not part of the commutant. This should map the holonomy to an
equivalent one, but which does not commute with the original holonomy. As such the space ofholonomies leading to 5d gauge theories is in general quite large. Additionally, it is possiblethat a single 6d SCFT can reduce to multiple different 5d gauge theories, depending on theholonomies chosen, and indeed we shall present examples of such behavior later on.
We can consider introducing a variable holonomy. That is we introduce a backgroundvector field coupled to the flavor symmetry and give it the particular profile,A = M2 + (M1 − M2 )θ (x 5 ) δ(x 6 )d x 6 ,with M1 and M2 being some matrices in the adjoint representation of the flavor symmetry of
the 6d SCFT. This suggests that we have holonomy of Tr(M1 ) around the circle direction x 6
for x 5 > 0, but a holonomy of Tr(M2 ) for x 5 < 0. As such the holonomy essentially jumps at
x 5 = 0 between the two values. Finally we note that the presence of the variable holonomy
implies the presence of flux on the surface as F = (M1 − M2 )δ(x 5 )δ(x 6 )d x 5 ∧ d x 6 .43 The idea
now is to take our flux on the surface to be generated by such a variable holonomy, with M1and M2 chosen so that the holonomy at both x 5 > 0 and x 5 < 0 is such that the 6d SCFT flows
to a 5d gauge theory.
The advantage of this approach is that we can now make progress on analyzing the reduction to 4d by first reducing to 5d. This is as our system in 5d now reduces to an IR freegauge theory living on the region x 5 > 0, and another IR free gauge theory living in the region
x 5 < 0. These two gauge theories are separated by a domain wall at x 5 = 0. The reduction
to 4d is now straightforward for the regions in the bulk, and we just expect to get the partof the 5d IR free gauge theory matter that is consistent with the boundary conditions. Theseshould be supplemented by the fields living on the domain wall, and as such the main problem
is reduced to understanding domain walls between 5d gauge theories that are UV completed by
6d SCFTs.
Let’s consider the domain walls in a bit more detail. As we mentioned we expect there to
be some 4d QFT living on the domain wall. We will assume that we can derive a description ofthis QFT in terms of an explicit gauge theory construction. Additionally there should also be
43
We can smear the δ-functions here to make the F continuous. Moreover, later on we will also relate certain
constructions with fluxes to punctured surfaces.

82

SciPost Phys. Lect. Notes 78 (2024)

A6 =x5 = − a
A6 = −

x5 = 01ϕ
2πr6dx5 = a

5d

1
ϕ
2πr6d

𝒞

Figure 44: Illustration of a flux domain wall. the horizontal axis is the 5th dimension,x 5 and the vertical axis denotes the value of the holonomy on the circle aroundx 6 . The holonomy interpolates between two asymptotic values which give rise toeffective gauge theory description in 5d. F ≠ 0∂⟨ϕ⟩ ≠ 0

boundary conditions on the bulk 5d fields as these approach the domain wall. One operator
that we expect to generically exist is a chiral operator in the bifundamental
representation ofF≠0
F =the0 two sides of the domain wall.
the 5d gauge symmetries on
This
is
as
the domain wall∂⟨ϕ⟩ ≠ 0
should extrapolate between thetwosymmetries.
Specifically,
say
our
5d
gauge
theory has∂⟨ϕ⟩ = 0
matter hypermultiplets which transform under the flavor symmetry. We should have such
hypermultiplets at both sides of the domain walls. However, there should be only one flavor
symmetry, so there should be some mechanism that identify the symmetries acting on the twosides of the domain wall. The presence of the bifundamenal operator gives such a mechanism.
Specifically, let us denote such a chiral operator by B, and break the hypermultiplets to twochiral fields in conjugate representation which we denote as X + , Y+ for one side of the domain
wall and X − , Y− for the other. Then for a hypermultiplet in the fundamental representation of
the gauge group, the superpotential X + BY− does the job. For higher representations, we need
the superpotential X + B n Y− .44 The components X − , Y+ can also be coupled to B with similar
superpotentials.
As such, we see that we generically expect to have a bifundamental chiral operator on thedomain wall. However, we may have many more fields and such a chiral might actually be
a composite. In fact we expect that most domain walls will have rather complicated matter
living on them. To illustrate this, consider the case where we have two domain walls. Thatis we take a variable holonomy such that it has one value for say −a < x 5 < −b, another for−b < x 5 < b and another for b < x 5 < a. We then have two domain walls, located at x 5 = ±b.
Now we can consider the limit where b → 0. In this limit the two domain wall collapse to asingle domain wall directly extrapolating between the holonomy at the two edges. We expect
the fields living on this domain wall to be the same as the fields living on the two domain walls
from which it is made, as well as the fields living on the bulk between the two. We expect thelatter to contribute the gauge fields of the 5d theory, and as such we expect the single domain
wall to contain gauge fields.
To make progress, it is convenient to make the following assumption. Specifically, we shallassume that there exist a domain wall such that the fields living on it can be described solely interms of chiral fields interacting through superpotential terms. Of course this assumption may
be wrong, and such a domain wall may not exist. However, making this assumption allows us
44

There are some subtle issues here for SO groups with spinor matter that we shall ignore here.

83

SciPost Phys. Lect. Notes 78 (2024)

to make progress in analyzing the reduction and as such we shall make it. This assumption
will later be checked by a detailed study of the anomalies of the resulting model. Once wedetermined one domain wall, we can determine more complicated ones by gluing together thisbasic domain wall. Finally, we need to consider the flux associated with this tube. In principal
the flux should be related to the difference in the holonomies at the two sides of the domain
wall. However, in this approach we don’t actually control the holonomies, rather the domain
wall is selected such that the fields living on it are particularly simple. As such we in general
don’t apriori know what the flux is, and will generally determine it by matching anomalies andsymmetries.
Next, we shall illustrate this method with various examples.

6.2

Compactification of the rank one E-string SCFT

As our first example, we consider the case of rank one E-string SCFT that we discussed in theprevious Lecture, but now from the domain wall point of view. First, recall that the rank 1
E-string is a 6d SCFT with E8 global symmetry. It can be compactified to 5d with a suitableholonomy such that it flows to the SU(2) gauge theory with eight fundamental hypermultiplets. Here we shall rely on this fact to study the dimensional reduction of this 6d SCFT on
tubes with flux.
We proceed as outlined previously. Specifically, we first reduce on a circle to 5d. As previously explained, we expect to be able to represent the flux as a variable holonomy, which wetake to be such that the theory flows to an SU(2) gauge theory with eight fundamental hypermultiplets on the subspace x 5 > 0, a different SU(2) gauge theory with eight fundamental
hypermultiplets on the subspace x 5 < 0 and a domain wall at x 5 = 0. The theory away fromthe domain wall is thus an IR free 5d gauge theory and its reduction can be easily analyzed.
Here we also need to consider the boundary conditions at the two punctures. Recall thatthese give Dirichlet boundary conditions to the 4d N = 1 vector multiplet component of the
5d N = 1 vector multiplet at the boundary. Therefore, as before, we expect the SU(2) gauge
symmetry to become non-dynamical at the boundary leading to an SU(2) global symmetrythat we associate with the punctures. Additionally we have the eight fundamental hypermultiplets. Close to the boundary we can decompose them to two 4d N = 1 chiral fields in
opposite representations which we shall denote as X i , Yi , with i = 1, 2, . . . , 8 denoting thechosen hypermultiplet. We then need to give Dirichlet boundary conditions to one of themand Neumann to the other for each i. The exact choice doesn’t matter too much, as we cantransform between them by flipping the fields as explained previously, but for presentational
purposes, it is convenient to take the boundary conditions to be such that the fields Yi receive
the Dirichlet boundary conditions for x 5 > 0 while the fields X i receive the Dirichlet boundary
conditions for x 5 < 0.
Finally we need to address the fields living on the domain wall. For the case at hand thesetypes of domain walls were studied in [135], and we can in principle just use the results foundthere. However, here we shall take a slightly different approach. Specifically, as we explained
previously, we shall assume that the fields living on the domain wall can all be expressed interms of free chiral fields. These should contain at least a chiral field in the bifundamental
representation of the two SU(2) groups at the sides of the domain wall, interacting with thehypermultiplets as we outlined previously. As we shall soon see when we study the anomaliesof this model, this is not enough to match the anomalies and we must add one more chiralfield, which turns out to be a singlet flipping the quadratic SU(2)×SU(2) invariant made fromthe bifundamental. For now, we shall add it to the matter content, but will return to this issue
later.
The resulting 4d theory we obtain is shown in Figure 45. Here the two global SU(2)symmetry groups are the ones associated with the punctures and the SU(8) is the subgroup of84

SciPost Phys. Lect. Notes 78 (2024)

1 t

X+

2
y

8
2 t

4

0 1/t

B

2 t

t 1

F
2

t

Y−
2

2

W = X+ B Y− + F B

1

2

the 6d global symmetry that rotates the fields X i and Yi . The SU(2)×SU(8) bifundamentals on
the two sides comes from the part of the bulk hypermultiplets receiving Neumann boundary
conditions. Finally, the SU(2) × SU(2) bifundamental and singlet field come from the domain
wall. The fields interact trough various cubic superpotentials.
The theory has several global symmetries. First it has the SU(2) × SU(2) non-abelian
symmetries coming from the punctures, and an SU(8) which is a subgroup of the 6d global
symmetry. Additionally, we can define two U(1) global symmetries consistent with the superpotential, one of which, denoted as U(1) t is also associated with the 6d global symmetry. It isdefined such that the fields X + and Y− have the same charge under it. There is another U(1)global symmetry, under which the fields X + and Y− have opposite charges, but it is in generalanomalous once we glue the tubes to closed surfaces.45 Finally, there is a U(1) R-symmetry,
which is convenient to define as the one coming from the Cartan of the 6d SU(2) R-symmetry.
Since the fields X + and Y− came from components of the 5d hypermultiplets, they should haveR-charge 1 under it. The superpotential then forces the bifundamental, B, to have R-charge 0
and the flip field to have R-charge 2.
Exercise: Consider the anomalies of the tube and check that these match the 6d expectation. Note that we have not determined the flux associated with this tube yet.
However, are there anomalies that are independent of the flux? If so do these matchthe 6d expectations?
As we mentioned there are two sources of contributions to the 4d anomalies from the 6dones. One is from the integration of the anomaly polynomial on the surface and the otheris from the punctures. The latter are independent of the flux so we are led to consider theformer. The only non-trivial contribution can come from the flux, and as such must come fromthe integration of the C2 (E8 ) term. As such there are several anomalies that will not receive
contribution from it, and so are independent of the flux. These are the Tr(U(1)R ), Tr(U(1)3R )Closing the two punctures and obtaining the theory corresponding to a sphere with flux, this U(1) symmetry
can be identified [136] with the Cartan generator of the SU(2) isometry of the sphere which becomes a global
symmetry in 4d (see [95, 137]).

85

0

1/t 2

t
8

z

Figure 45: The basic tube theory for the compactifications of the rank one E-string.
The SU(2) symmetries correspond to the two punctures; the two fundamental octets
come from the bulk fields in the two domains separated by the wall; and the bifundamental field with the flip field come from degrees of freedom residing on the domain
wall.

45

4

2 t4

1

1/t 2

0

SciPost Phys. Lect. Notes 78 (2024)

and Tr(U(1)R F 2 ) for F some flavor symmetry. The contribution to all these anomalies will
come only from the punctures for which we can evaluate:
3 3
Tr(U(1)R ) = Tr(U(1)3R ) = − − = −3 ,2 2
Tr(U(1)R U(1)2t ) = Tr(U(1)R SU(8)2 ) = 0 .

(153)
(154)

Now let’s consider comparing them with the anomalies we observe in the tube. For theones only involving the R-symmetry we have that:
Tr(U(1)R ) = Tr(U(1)3R ) = −4 + 1 = −3 ,(155)

where the first term is the contribution of the bifundamental B and the second term is thecontribution of the flip field.
For the one involving the flavor symmetry we have that:
Tr(U(1)R U(1)2t ) = −22 + 4 = 0 , Tr(U(1)R SU(8)2 ) = 0 ,(156)

where again the first term in the first equation is the contribution of the bifundamental B andthe second one is the contribution of the flip field.
We see that the anomalies indeed match. However, note that for this matching to work
it is important that we have the flip field. This is one way to understand why we must add
it, as otherwise the anomalies won’t match. Furthermore, this necessitates that it has U(1)R
R-charge of 2 and U(1) t charge of 4. The requirement that the charges have this value thenalso determines the superpotential.
□
Gluing two tubes to form a torus
Having formed a conjecture for the 4d theory associated with the compactification of the 6d
E-string SCFT on a tube with flux, we next want to test this conjecture and in the process
determine the flux. To do this it is convenient to work with closed surfaces with no punctures,
and as such we shall take two such tubes and glue them together. The gluing process is done as
we explained in the previous Lectures. Specifically, here the moment map operators associatedwith the punctures are the SU(2)×SU(8) bifundamentals. We first note that these are charged
differently under the 6d global symmetry for the two punctures so the punctures then havedifferent colors. What we shall do is take two tubes and glue the punctures of same colorof each tube together. As we are gluing punctures with same color, the gluing we need toperform is Φ gluing, so we gauge the SU(2) global symmetry of the two punctures with an
N = 1 SU(2) vector multiplet with eight fundamental chiral fields, the fields Φ. These arethen coupled to the moment map operators associated with the two glued punctures via aquadratic superpotential. The latter just becomes a mass term, leading to the fields Φ andsome of the moment map operators being integrated out. We end up with the theory shownin Figure 46.
We can next study the resulting theory. We first note that the U(1) R-symmetry giving
free R-charge to all the chiral fields is anomaly free, suggesting that the theory might be a
conformal theory at weak coupling. As such, we could have in principle arrived at this theory
using our previous strategy. We can then ask whether it is actually conformal or not. We firstnote the two flip fields should decouple as there cannot be any quotient for the symmetries
acting only on them. Using the same methods as in the previous Lectures, one can show thatthe remaining theory has a non-trivial quotient so we get a 4d SCFT and two decoupled freefields.

86

SciPost Phys. Lect. Notes 78 (2024)

1 t

X+

2
y

8
2 t

4

t 1

F

0 1/t 2

B

2 t

4

t

Y−
2

2

1

W = X+ B Y− + F B

1/t 2

t
8

z

2 t4

2

0

2

1

1/t 2

0

Figure 46: The theory corresponding to gluing two basic tubes into a torus. Notethat this is the same model we have obtained on the right side of Figure 42 using a
different method.
The first thing we want to do is check whether this theory has the right properties to indeed
be the dimensional reduction of the 6d E-string SCFT on a torus with flux, and if so to determine
the flux. To do so we first evaluate the superconformal index. As we are mainly interested
in matching with expected properties from 6d, we shall calculate the index including the flip
fields and refining only with respect to symmetries existing in 6d, that is we shall ignore thefact that the flip fields should decouple. The result we find is,


1
2
1
1
4
8
2
3
3
1 + 2t (pq) + (pq)
+ 3t + t (28 + 28) + 2t 4 (pq) 3 (p + q)
(157)
4
t
+ 2pq(2t 12 + t 6 (28 + 28)) + . . .
One thing we note in this index is that it forms characters of E7 . Specifically, we have that56 E7 → 28 + 28. As such it is tempting to assign to it a flux of value 1 preserving the U(1) × E7
±10
subgroup of E8 . Indeed we have the branching rules: 248 E8 → 10E + 1±2
E7 + 56 E7 + 133 E7 .
7This suggests that the 6d global symmetry is related to the symmetry we see in 4d by:
248 E8 → (t 4 + 1 + t14 )1 + (t 2 + t12 )(28 + 28) + 63 + 70.
The claim can be further supported by looking at the ’t Hooft anomalies of the model.
Specifically, we can calculate the ’t Hooft anomalies of the model finding:
Tr(U(1)R ) = Tr(U(1)3R ) = Tr(U(1)R U(1)2t ) = 0 ,Tr(U(1) t ) = 24 ,Tr(U(1)3t ) = 96 ,Tr(U(1)2R U(1) t ) = −8 ,

(158)

2

Tr(U(1) t SU(8) ) = 2 ,with the rest vanishing trivially. These anomalies indeed match the anomalies expected fromthe compactification of the rank one E-string on a torus with unit flux preserving the U(1) × E7
subgroup of E8 (which can be read off from Appendix F.3). Here we use the expected embedding of the 4d symmetries in the 6d global symmetry suggested by the index, and the following
embedding index: ISU(8)→E7 = 1.
All this is consistent with the flux being of value 1 preserving the U(1) × E7 subgroup of
E8 . Since we got this theory from gluing together two copies of the tube we introduced, weare lead to associate with the tube a flux of 12 preserving the U(1) × E7 subgroup of E8 . We
87

SciPost Phys. Lect. Notes 78 (2024)

8
1

2
y

1

2
0

1

2

2
0

2
Z

Figure 47: Combining two basic domain walls. This domain wall corresponds to flux
which is twice the one for the basic tube.
can write this flux in our flux basis as 21 (1, 1, 1, 1, 1, 1, 1, 1).46
Combining domain walls
We have seen how our method can be used to conjecture 4d theories associated with thecompactification of 6d SCFTs on tubes by relating the problem to the behavior of domain
walls in 5d and assuming there exist a particularly simple type of the latter. Next, we want tobuild on that and also get models corresponding to the compactification on tubes with othervalues of flux. We can do so by combining two domain walls to form more complicated domain
walls.
Specifically, consider the configuration where we now have two domain walls, that is wetake our holonomy to have the value M1 on one side of the interval, M2 in the middle of theinterval, and M3 on the other side. As we explained previously, we can also view this as one
domain wall extrapolating directly between M1 and M3 , though it is easier to analyze this if we
instead view it as multiple domain walls.47 In particular, as we have determined one domain
wall, we can ask how can we combine that one domain wall to form more general ones.
The most straightforward way to do so is to simply chain them together. That is we take thetwo domain walls to be identical. If we do this then we can repeat our previous analysis andarrive at the theory shown in Figure 47. Here as before, the SU(2) × SU(8) bifundamentals
at the two edges come from the component of the bulk hypermultiplets that receive Neumannboundary conditions. The SU(2) × SU(2) bifundamental and its flip field come from the domain walls, where now we have two of them. In the middle however, we now have a gauge
SU(2) group, as the N = 1 vector receives Neumann boundary conditions at the domain wall.
This suggests that the adjoint chiral receives Dirichlet boundary conditions there, explaining
its absence. Finally, we again have the SU(2) × SU(8) bifundamental that comes from thecomponent of the bulk hypermultiplets that receive Neumann boundary conditions, now atthe domain wall. The end theory is just the theory we get if we glue two tubes to themselves.
This of course is as expected as the resulting surfaces would indeed be itself a tube, but now
with two domain walls. As such the flux associated with this tube is just twice the flux of theoriginal tube, which in our basis is (1, 1, 1, 1, 1, 1, 1, 1).
The more interesting case is when we glue the tubes with a Weyl symmetry twist. We notedthat the flux associated to the tube should be such that it breaks E8 into U(1) × E7 . We have46
Here we are using the basis of the roots of E8 were its SO(16) spinor roots are taken to have an even numberof minus signs. This is as the puncture SU(2) symmetry of the tube theories here does not have Witten anomaly.
47
If M1 = M3 the combined domain wall becomes trivial.

88SciPost Phys. Lect. Notes 78 (2024)

1 ty

2
1

4

0 1/t

ty

−1

4

1

2

t y −1

4

1

−1

t̃ ỹ

2

2

1

1

ty

t̃ ỹ

(a)

2
0 1/ t̃

4

1

t̃ ỹ
2
1

t̃ ỹ−1(b)

Figure 48: Combining two basic domain walls with non trivial identification of symmetries between the two copies.
many different choices on how to embed this flux inside E8 . For instance, consider one suchchoice. Then we can get an equivalent, yet different choice by acting on the flux with part of
the Weyl group of E8 that is broken by the flux. Since these are equivalent fluxes, the tube
theories that we have presented will be the same for both of them, and so also if we glue eachone of them to another copy of itself. However, if we glue together two tubes preserving a
different E7 inside E8 , then the resulting tube will be different. In particular, it will no longer
preserve an E7 subgroup. We shall next illustrate how this is done with an example.
We first consider breaking the SU(8) global symmetry to U(1) y ×SU(4)×SU(4) as follows:
8 → y41 + 1y 42 . To see why this is interesting it is convenient to also break E8 into thesesymmetries, where we have:





1
1
1
1
4
4
2
2
248 E8 → t + y + 2 + 4 + 4 1 + t + 2
y + 2 (61 + 62 )
(159)
y
t
t
y




1
1
+ t 2 + 2 (41 42 + 41 42 ) + y 2 + 2 (41 42 + 41 42 ) + 61 62 + 151 + 152 .
ty
The interesting property that we shall soon make use of, is that the adjoint of E8 is symmetric
under the exchange t ↔ y, 41 ↔ 41 . This is part of the Weyl group of E8 . To illustrate
this, consider the SO(16) subgroup of E8 . This decomposition induces also a decomposition ofySO(16) → U(1) t ×U(1) y ×SU(4)×SU(4) such that 16 → t y42 + yt 41 + t1y 42 + t 41 . This implies
that this transformation reduces to the Weyl transformation acting as charge conjugation onfour out of eight SO(2) independent subgroups of SO(16).
Let’s consider two copies of the tube we presented, which we have included in Figure 48.
Here we have used different fugacities for the symmetries of the two theories to stress thefact that we have not yet identified the symmetries between the two theories. The simplest
identification is to take: ỹ → y, t̃ → t, 4R1 → 41L and 4R2 → 42L . This makes the two tubes
identical and we can glue them to get the tube we previously mentioned.
However, in light of what we have seen from the decomposition of the adjoint of E8 , weL

can also make the identification: ỹ → t, t̃ → y, 4R2 → 42L and 4R1 → 41 . This will still map theE8 symmetry of the underlying 6d SCFT to itself, but with some action of the Weyl symmetry
89

SciPost Phys. Lect. Notes 78 (2024)

4
1 ty

1 ty

2

0 1/t
1 t y −1

2

2

0 1/y
1

ty

4

1

2

2

t −1 yFigure 49: The two domain walls of Figure 48 combined together with half of theboundary conditions being Neumann and half Dirichlet.
of E8 . The two tubes are then not completely identical, as can be seen from the fact that thematter representation under the symmetries are slightly different. Specifically, we note thatwhile the SU(2) × SU(8) bifundamental on the lower right of Figure 48 (a) and lower left of
Figure 48 (b) have the same charges, the ones on the upper right of Figure 48 (a) and upper
left of Figure 48 (b) have opposite charges.
We can next glue the two tubes together. Here the gluing is done with Φ gluing for thefour SU(2) × SU(8) bifundamental chiral fields with the same charges, and with S gluing
for the four SU(2) × SU(8) bifundamental chiral fields with opposite charges. This leads tothe tube shown in Figure 49. We can also understand this tube as follows. Recall that theSU(2)×SU(8) bifundamental chiral fields come from the bulk eight hypermultiplets consistent
with the boundary conditions. As such, if they have the same charges, then that suggests thatthe same component of the hyper receives Neumann boundary conditions on both ends, while
if they have opposite charges then different components receive boundary conditions at thetwo ends. As the boundary conditions at the edges should be the same as on the domain wall,
this suggests that fields with the same charges will receive Neumann boundary conditions on
the two domain walls, and so survive the 4d reduction, while for those with opposite charges
every component should receive Dirichlet boundary conditions at least on one of the domain
walls.
We next want to determine the flux associated with the new tube in Figure 49. This should
be given by the sum of the fluxes of both tubes, when evaluated in the same basis. For the firsttube we can take the flux to be 21 (1, 1, 1, 1, 1, 1, 1, 1). For the second one the flux is the same,
but in a different basis related to the one of the other tube by the Weyl transformation. As thelatter is of order two we can undo it by performing it again. Recall that we determined that
the Weyl action we consider here should act on the eight SO(2) subgroup of SO(16) by charge
conjugating four and leaving the rest invariant. This suggests that in the basis of the first tube,
the flux of the second tube is 21 (1, 1, 1, 1, −1, −1, −1, −1). We then conclude that the flux ofthe tube in Figure 49 is (1, 1, 1, 1, 0, 0, 0, 0), which is a half flux preserving the U(1) × SO(14)subgroup of E8 .

90

SciPost Phys. Lect. Notes 78 (2024)

These results can be checked by gluing two copies of this tube and studying its anomalies
and superconformal index. This method can also be generalized to build many more domain
walls, and as such 4d theories associated with compactifications on tubes, for many othervalues of flux. We refer the reader to [26] for more information on both subjects.

6.3

Compactification of minimal (D, D) conformal matter theories

Having illustrated the basic idea by studying the compactifications of the E-string SCFT, we
next wish to further elaborate on interesting phenomena and considerations that can appear
in this construction. For this we consider another example, now involving the compactificationof the 6d SCFT known as the minimal (DN +3 , DN +3 ) conformal matter on a torus with global
symmetry fluxes. For brevity, we shall usually refer to this SCFT as minimal D type conformal
matter.
The minimal D type conformal matter
The D type conformal matter theories is a family of 6d SCFTs. This family can be engineeredin string theory as the theories living on n M5-branes probing a C2 /DN +3 singularity [10,138].
The specific case of n = 1 is known as the minimal case and is the one we will consider here.
We note that for N = 1, this SCFT is just the rank one E-string theory that we have discussed
in detail. As such, we can think of this family as a generalization of the previous case.48 One
advantage of this generalizations is that as we still have only one M5-brane, the tensor branchof these theories is still one dimensional.
These theories have a convenient field theory realization that we shall next study. Specifically, starting from the SCFT point we can deform it by giving a vacuum expectation value
to the scalar in the tensor multiplet, that is by going to the tensor branch. This initiates an
RG flow that in many cases ends with a 6d IR free gauge theory, whose coupling constantis identified with the scalar vev, see Appendix E. For the case at hand, this 6d tensor branchtheory turns out to be a US p(2N − 2) gauge theory with 2N + 6 fundamental hypermultiplets.
As such we can also think of this 6d SCFT as a UV completion of this 6d gauge theory. The
gauge theory as an SO(4N + 12) global symmetry algebra rotating the 2N + 6 fundamental
hypermultiplets, which turns out to also be the global symmetry algebra of the 6d SCFT.49
We can next consider the compactification of this theory to 5d. From our previous discussion, we shall be mainly interested in possible 5d gauge theory descriptions that can emerge
from such a reduction with a suitable holonomy. One intriguing property that this family of
6d SCFTs has is that there are actually three different possible 5d gauge theories one can obtain [96, 97]. The first has gauge group US p(2N ) and 2N + 6 fundamental hypermultiplets,
the second has gauge group SU(N +1), no Chern-Simons term and 2N +6 fundamental hypermultiplets, and the last has gauge group SU(2)N , bifundamental hypermultiplets connecting
the groups into a linear quiver, and four fundamental hypermultiplets for each of the SU(2)groups at the ends of the quiver. Note that these all degenerate to SU(2) + 8F for N = 1.
48
There are other possible generalizations. For instance, we can keep N = 1, and take n to be arbitrary, leading
to non-minimal D4 type conformal matter theories. Alternatively, we can use the constructions of the E-string as
the theory living on one M5-brane in the presence of an M9-plane, and generalize to the case of N M5-branes.
This class of theories is known as the rank N E-string theories. We note that all theories in the three families are
distinct, save for the rank one E-string.
49
As the 6d SCFT and 6d gauge theory are related by flow, the usual caveats regarding the relation between theirsymmetries apply. Interestingly, in this case, while the two share the same global symmetry algebra, they don’t
share the same global symmetry group. Specifically, the gauge theory has an SO(4N + 12)/Z2 global symmetrygroup. However, the 6d SCFT has a S pin(4N + 12)/Z2 global symmetry group, see the discussion in [19]. This
comes about as the 6d SCFT has a Higgs branch operator in a chiral spinor representation of SO(4N + 12), whichbecomes massive once we go on the tensor branch. Finally, we note that for N = 1 this Higgs branch operator
actually becomes a conserved current multiplet, leading to the enhancement of SO(16) → E8 .

91

SciPost Phys. Lect. Notes 78 (2024)

This phenomena has profound implications on the study of 4d compactification. First, wenoted that punctures can be associated with boundary conditions of the 5d gauge theory, butwhat happens when there are multiple such theories? In that case it appears that we canassociate a family of punctures to each 5d gauge theory description. Specifically, we can use
the boundary conditions to define a maximal puncture associated with each 5d gauge theory
description. Each puncture will have the gauge symmetry of the 5d theory has its associatedglobal symmetry, and carry its own type of moment map operators. We can then choose topartially close the puncture by giving a vev to the moment map operators. This creates a familyof punctures starting from each maximal puncture.50
Naturally, we can also use the three possible descriptions to construct different boundary
conditions. As before we can try to build domain wall theories for domain walls between thesame description, as we did for the E-string case, for each of the three gauge theory descriptions. Furthermore, we can consider domain walls where there are entirely different gauge
theory descriptions on each side. This leads to six different possibilities. We can then try toconjecture the resulting 4d theories by assuming that the theory living on such domain walls ismade of chiral fields, and check whether or not such a conjecture can reproduce the properties
expected from theories corresponding to the compactifications of these 6d SCFTs. This analysis has been carried out for this family of theories, and out of the six possibilities only threeyield sensible results:51 SU(N + 1) − SU(N + 1), SU(N + 1) − US p(2N ) and SU(2)N − SU(2)N .
Next we shall further illustrate the use of the domain wall method by performing this analysis
for the SU(N + 1) − SU(N + 1) case. For the analysis of the SU(N + 1) − US p(2N ) case, and formore details on the analysis of the SU(N + 1) − SU(N + 1) case, we refer the reader to [19],while for the analysis of the SU(2)N − SU(2)N case, we refer the reader to [126].
The SU − SU tube
Following the previously outlined procedure, we can attempt to conjecture a 4d model associated with the compactification of the minimal D type conformal matter theory on a torus
with punctures. Here we shall concentrate on the cases built from the SU − SU domain wall,
so we take the punctures at the two ends to be SU(N + 1) maximal punctures. Following ourprevious discussion, we then make the ansatz for the theory shown in Figure 50. Here we have
the SU(N + 1) global symmetries associated with the two maximal punctures at the two endsof the tube. These come from the 5d gauge symmetry in the bulk that becomes a global symmetry as the vector receives Dirichlet boundary conditions. Additionally, we have the 2N + 6
fundamental hypermultiplets. Half of the chiral fields in them receives Neumann boundary
conditions, while the other half receives Dirichlet boundary conditions. As the representation
for generic N are complex, the representation under the SU(N + 1) puncture global symmetryof the surviving hypermultiplets depends on which component receives the Neumann boundary condition. As we do not know this a priori,52 we shall leave this question open for now
and answer it soon. As such we shall assume that we have n+ hypermultiplets where the fun50
It is an open question whether each family is entirely disconnected from the other or rather whether somesub-maximal punctures are shared between the different families.
51
For the other three cases it seems the domain walls are more complicated. Specifically, we can form
US p(2N )− US p(2N ) tubes by gluing two SU(N +1)− USp(2N ) tubes along the SU(N +1) punctures. These tubes
will then contain gauge fields from the gluing. Tubes realizing the SU(N + 1) − SU(2)N or USp(2N ) − SU(2)N
combinations have not been worked out to our knowledge, though one should be able to do this using the resultsof [118].
52
Technically, we can choose the boundary conditions at the punctures to be what ever we want. However, tomaintain the similarity to the previous case, we want to choose the boundary conditions for the bulk hypermultiplets at the punctures to be the same as the one at the domain wall. The issue then stems from our lack ofknowledge regarding the boundary conditions on the domain wall for domain walls with only chiral fields livingon them.

92

SciPost Phys. Lect. Notes 78 (2024)

n+

1 t

X+
t

2

2(N+1)

N+1
y

Y+

1 tN

1 t

Y−

F
1/t
0

2

n−

B

N+1
z

X−

1 tN

W = X+ B Y− + Y+ B N X− + F B N+1
Figure 50: The basic tube theory for the compactifications of the minimal(DN +3 , DN +3 ) conformal matter theory. The SU(N + 1) symmetries correspond to thetwo punctures; the two collections of fundamental fields come from the bulk fields
in the two domains separated by the wall; and the bifundamental field with the flip
field come from degrees of freedom residing on the domain wall. The parameters n+
and n− are fixed to be 2N + 4 and 2 respectively. In the case of N = 1 we reproduce
the domain wall of Figure 45.
damental receives Neumann boundary condition on one side, and n− hypermultiplet where
the antifundamental receives Neumann boundary condition on the same side. Naturally, we
have that n+ + n− = 2N + 6.
Next we consider the domain wall. As before we shall make the assumption that the fields
living on the domain wall can be described using only chiral fields. On general grounds, we
expect there to be an SU(N +1)×SU(N +1) bifundamental chiral field. It should interact withthe fundamental and anti-fundamental chiral fields associated with the two punctures via thesuperpotentials:
W = X B Ỹ + Y B N X̃ ,(160)
where here B refers to the bifundamental chiral, and X , Y and X̃ , Ỹ refer to the fundamental
or anti-fundamental chiral fields associated with the punctures. There might be additional
fields associated with the domain wall. To get a better understanding of this, it is good toagain consider the anomalies of the tube. Specifically, we want to look at the Tr(U(1)3R ),Tr(U(1)R ) and Tr(U(1)2t × U(1)R ) anomalies, where U(1) t is defined as in Figure 50. These
anomalies only receive contributions from the punctures and not from integrating the anomaly
polynomial on the surface, and so can be evaluated from the data available to us. The puncture
contribution gives:(N 2 − 1) (N 2 − 1)−
= −N 2 + 1 ,2
2
Tr(U(1)2t × U(1)R ) = 0 .

Tr(U(1)3R ) = Tr(U(1)R ) = −(161)

However, the evaluation in the gauge theory gives:
Tr(U(1)3R ) = Tr(U(1)R ) = −N 2 ,Tr(U(1)2t × U(1)R ) = −4(N + 1)2 .
93

(162)

SciPost Phys. Lect. Notes 78 (2024)

1 t

n+

t

1

1

ty

0 1/t 2

N+1
1

N+1
0 1/t 2

t

N

n−

t

2N + 4−1

0 1/t 2

N+1

1

N

N

t y

(a)

1

N+2

1

ty

N+1
0 1/t 22

1

t /y N+2
N(b)

Figure 51: The torus theory for the minimal (DN +3 , DN +3 ) conformal matter theory
obtained by gluing together two tube theories.
This suggests that we need to add an additional field. The minimal possible addition
consistent with our assumption regarding the nature of the domain wall is a single free chiralfield with R-charge 2 and U(1) t charge 2(N + 1). This field has a natural interpretation,
as its charges are just right for it to flip the baryon made from the SU(N + 1) × SU(N + 1)
bifundamental. As such we conclude that our conjecture for the tube is the theory shown in
Figure 50.
Testing the tube
We next move to perform more stringent tests on the tube theory that can be used to determine
the flux and to provide supporting evidence that it is indeed a tube theory associated with theclaimed compactification. As before, it is convenient to do this by gluing two tubes together
to form a closed surface. For this case, this leads to the theory show in Figure 51 (a), which
we shall next analyze.
The first thing we need to make sure of is that the resulting theory is consistent. Specifically,as the theory is chiral, we need to worry about gauge anomalies. Indeed for generic values ofn+ , n− we have a non-zero gauge anomaly Tr(SU(3)3gauge ) = 2N + 2 + n− − n+ . Demandingthat this anomaly vanishes, together with the constraint n+ + n− = 2N + 6, forces n+ = 2N + 4,
n− = 2. We therefore see that the consistency of the theory uniquely fixes the tube theory.
Additionally, we would like to make sure that the U(1) t symmetry is anomaly free as we havemade use of it when matching the anomalies between the tube and the 6d expectation. Indeed
U(1) t is anomaly free precisely with these values of n+ and n− . This is already an indicationthat we are on the right track.
The concrete torus theory we need to study then is the one shown in Figure 51 (b). We
can next study this theory and compare the results with our 6d expectations. As before theanalysis consists of two steps. One is to compute the superconformal index and the other is
to compute the anomalies. These are then compared to the 6d expectations and are needed
to be consistent with them and with one another. We shall first start with the computationof the superconformal index, which can be used to determine the flux and the mapping of94

SciPost Phys. Lect. Notes 78 (2024)

symmetries, which facilitates the matching of anomalies. For simplicity, we shall here take thecase of N = 2 and also compute the index of the model without the flip fields, as it appears
the flip fields decouple in the IR. This is seen by looking for the expected superconformal
R-symmetry via a-maximization. We then find for the superconformal index:


5
3
1
4
3
3
1 + t (pq) 8 2SU(2) y 8SU(8) + 3 8SU(8) + 6 (pq) 4
(163)
y
t


5
1
+ (3SU(2) − 1)pq + t 3 (pq) 8 (p + q)2SU(2) y 3 8SU(8) + 3 8SU(8)
y


9
1
1
3
9
3
8
+ t (pq)
8SU(8) + 3 56SU(8) + y 56SU(8) + y 8SU(8) + . . .
y9yOne interesting thing to note about this index is that it forms characters ofSU(2) × SO(16).
Specifically, we have that 16SO(16) → y 3 8SU(8) + y13 8SU(8) ,128′ SO(16) → y19 8SU(8) + y13 56SU(8) + y 3 56SU(8) + y 9 8SU(8) , so in terms of characters the index
reads:
5

1 + t 3 (pq) 8 2SU(2) 16SO(16) +34(pq) 4 + (3SU(2) − 1)pq
6
t

(164)

9

5

+ t 3 (pq) 8 (p + q)2SU(2) 16SO(16) + t 3 (pq) 8 128′ SO(16) + . . .
This suggests that this theory should be associated with the minimal flux preserving aU(1) × SU(2) × SO(16) subgroup of SO(20), which is the global symmetry algebra of the 6d
SCFT for N = 2. We can as before introduce a flux basis for SO(4N + 12) where the roots arespanned by 21 (1, 1, 0, 0, . . . , 0) + permutations + even number of reflections. In this basis, we
can associate with this theory a flux given by (−2, −2, 0, 0, 0, 0, 0, 0, 0, 0).53 This term arises in
the gauge theory from the mesons from the fundamental and antifundamental chirals at thetwo sides.
For general N , we expect the enhancement of U(1) × SU(2N + 4) → SO(4N + 8).
This can be seen by again looking at the mesons and noting that we now get a state in
the t N +1 2SU(2) ( y N +1 2N + 4SU(2N +4) + y N1+1 2N + 4SU(2N +4) ). This fits with the minimal fluxpreserving a U(1) × SU(2) × SO(4N + 8) subgroup of SO(4N + 12), which is given by(−2, −2, 0, 0, . . . , 0, 0) in our flux basis. This also determines the embedding of the symmetries.
We can next compare anomalies. For this we first need the anomaly polynomial of the 6d
SCFT. For the case at hand, this was computed in [94] (see also [19]), and we can just use
their results. Alternatively, it can be computed from the gauge theory description similarly tothe computation in Lecture III, though now we also need to take into account the contributionof the hypermultiplets (see [94, 117] for the relevant expressions). Either way, we have that,
I DC M =N (10N + 3)C22 (R)N (2N + 9)N
p1 (T )C2 (R) − C2 (R)C2 (SO(4N + 12))4N+12
24
48
2(N + 2)(2N + 1) 2+
p1 (T )C2 (SO(4N + 12))4N+12 +C2 (SO(4N + 12))4N+12 (165)24247p12 (T ) − 4p2 (T )(N − 1)
C4 (SO(4N + 12))4N+12 + (29 + (N − 1)(2N + 13))
.
−
6
5760
−

Here it is important that the group is actually Spin(4N + 12)/Z2 so this gives the minimal flux preserving theU(1) × SU(2) × SO(16) subalgebra. Indeed this is further supported using the logic of [113] (which we do not5
review here) by noting that the index contain the states t 3 (pq) 8 2SU(2) 16SO(16) , which have the precise charges to53

1

be coming from the broken currents of SO(20) under that subgroup. We also expect states in the 2t 6 (pq) 4 fromthe broken currents. These are just the flip fields which we have not included in the index.

95

SciPost Phys. Lect. Notes 78 (2024)

Next we need to introduce the flux. As we mentioned, we expect the theory to be associatedwith compactifications having a unit of flux preserving the U(1)×SU(2)×SO(4N +8) subgroupof SO(4N + 12). From our study of the index, we can infer that the expected relation between
the 4d and 6d global symmetries should be,1
1
4N + 12SO(4N +12) → (t N +1 + N +1 )2SU(2) + y N +1 2N + 4SU(2N +4) + N +1 2N + 4SU(2N +4) . (166)tyThis induces the following relation among the characteristic classes:
C2 (SO(4N + 12))4N+12 = 2C2 (SU(2))2 + 2C2 (SU(2N + 4))2N+4 − 2(N + 1)2 C1 (U(1) t )2
− 2(N + 2)(N + 1)2 C1 (U(1) y )2 ,

(167)

C4 (SO(4N + 12))4N+12 = C22 (SU(2))2 + C22 (SU(2N + 4))2N+4+ 4C2 (SU(2))2 C2 (SU(2N + 4))2N+4 + (N + 1)4 C1 (U(1) t )4
+ 2(N + 1)2 C2 (SU(2))2 C1 (U(1) t )2
− 4(N + 1)2 C2 (SU(2N + 4))2N+4 C1 (U(1) t )2
− 4(N + 2)(N + 1)2 C2 (SU(2))2 C1 (U(1) y )2
− 2(2N + 1)(N + 1)2 C2 (SU(2N + 4))2N+4 C1 (U(1) y )2
+ 4(N + 2)(N + 1)4 C1 (U(1) y )2 C1 (U(1) t )2+ (N + 2)(2N + 3)(N + 1)4 C1 (U(1) y )4
+ 2C4 (SU(2N + 4))2N+4− 6(N + 1)C1 (U(1) y )C3 (SU(2N + 4))2N+4 .
Here the flux is inside U(1) t , so we need to take C1 (U(1) t ) = N 1+1 t̂ + C1 (U(1)4dt ), where
4d
C1 (U(1) t ) is the 4d part of the characteristic class. Note that the minimal charge under U(1) there is N +1, which leads to the normalization factor in front of t, which ensures that the flux isof unit value in a normalization where the minimal charge is 1. Inserting all of this into (165)
and integrating on the Riemann surface, we arrive at the expected 4d anomaly polynomial:
I4d =(N + 1)(N + 2)
2(N + 2)(N + 1)3 3C1 (U(1)4dp1 (T )C1 (U(1)4d
(168)
t )−
t )
3
6
3 2
4d
4d
− 2N (N + 1)C12 (U(1)6d
R )C1 (U(1) t ) + 2(N + 2)(N + 1) C1 (U(1) y )C1 (U(1) t )4d− 2N (N + 1)C2 (SU(2))2 C1 (U(1)4dt ) − 2(N + 1)C2 (SU(2N + 4))2N+4 C1 (U(1) t ) ,where here we used the decomposition C2 (R) → −C12 (U(1)6d
R ) for the characteristic classes of
the SU(2) R-symmetry bundle in terms of those of its U(1) Cartan.
We can next compare that against the results from the gauge theory. Specifically, we findthat:
Tr(U(1)3t ) = Tr(U(1) t × U(1)2y ) = 4(N + 1)3 (N + 2) ,Tr(U(1) t ) = 4(N + 1)(N + 2) ,Tr(U(1) t SU(2)2 ) = N (N + 1) ,2
Tr(U(1) t × (U(1)6dR ) ) = −4N (N + 1) ,Tr(U(1) t SU(2N + 4)2 ) = N + 1 ,(169)with the rest of the anomalies vanishing either trivially or non-trivially. This matches what we
expect from 6d, particularly, the anomalies encoded in (168).

6.4

From tubes to trinions

So far we discussed a method to conjecture and test 4d theories associated with the compactification of certain 6d SCFTs on tubes with flux. We have noted how we can capitalize on the96

SciPost Phys. Lect. Notes 78 (2024)

understanding of even just one tube, and build from it a wealth of other tubes leading to apotentially extensive understanding of the compactification of 6d SCFTs on flat surfaces with
flux, that is tubes and tori. While we shall not discuss this here, we can further extend ourunderstanding to compactifications on surfaces with positive Euler number, like spheres withno punctures, by closing down the punctures (See [136, 139–141].). This, however, leaves
the question of understanding the compactification on surfaces with negative Euler number,notably Riemann surfaces of genus g > 1, a question which we shall try to address next. The
discussion follows the results of [27] (See also [118, 142].).
It turns out that we can make surprising progress on tackling this problem by considering
an altogether different and seemingly unrelated problem. The specific problem turns out tobe understanding the relationship between RG flows of 6d SCFTs and related RG flows of the
4d theories resulting from their compactifications. As such, we shall next switch gears andconsider this problem, after which we shall elucidate on how this new understanding can be
put to use in tackling the problem of finding trinions.
A relationship between flows in different dimensions
The problem we wish to consider can be simply stated as follows. We have considered thecompactification of 6d SCFTs. These are field theories in 6d and can be related to one anothervia various flows. For flows preserving supersymmetry, we are limited to ones triggered bygiving vevs to various fields. This can be separated into two classes: Tensor branch flows and
Higgs branch flows. The former are done by giving a vev to a scalar in the tensor multiplet,
while the latter are done by giving a vev to a scalar in the hypermultiplet. There are no scalars
in the 6d (1, 0) vector multiplet leading to the lack of a “Coulomb” branch for 6d (1, 0) SCFTs.
The vevs break conformal invariance and initiate an RG flow starting from the 6d SCFT and
ending with a new theory, that in some cases is also a 6d SCFT. Let us assume that is the case
UVIRand denote the starting 6d SCFT as T6d, the final 6d SCFT as T6d, and the operator to which
we give the vev as O6d .
UVNow consider compactifying T6d
on a Riemann surface to 4d, potentially with flux in its
UV
global symmetry. This should result in a 4d N = 1 theory which we denote as T4d. We knowthat the 6d SCFT contains the operator O6d , and we expect that this operator might reduce
to a 4d operator, O4d . Of course this may not always be the case, but let’s for the moment
assume it does.54 Then we can try to give the 4d operator, O4d , a vev as well and thus initiate
a 4d RG flow. Again, there is the possibility that the operator O4d cannot be given a vev, but
we shall for the moment assume that this is not the case. We then expect this to cause a 4d RG
UVflow between the 4d theory T4d, and the end point of the flow, which is some 4d theory thatIRIR
we denote as T4d . It is then natural to ask does the theory T4d
also has a higher dimensionalIRinterpretation as a compactification of T6d and if so what is the compactification data? Inother words, instead of asking what happens to specific theories upon dimensional reduction,
it is also natural to ask what happens to RG flows upon dimensional reduction.
So far we have been somewhat general regarding our chosen deformation. However, next
we wish to concentrate on a specific class of deformations, the ones associated with Higgs
branch flows. The main reason for that is the special place occupied by the operators whose
vevs cause these flows. Specifically, such operators are short representation of the 6d (1, 0)
superconformal group, that is they are BPS operators, and are in fact the shortest possible
representations. As such, they form a chiral ring. This is in contrast with tensor multipletsthat don’t enjoy any such protection. This is useful as it suggests that the resulting 4d operator
may also be BPS, and as such enjoy some protection. This reduces the chances that these
operators disappear during the flow, and furthermore allows for their detection and study via
54

As in any RG flow the spectrum of operator in the IR does not have to be the same as in UV.

97

x5 = − a
A6 = −

x5 = 01ϕ
2πr6dx5 = a5d

SciPost Phys. Lect. Notes 78 (2024)𝒞F≠0∂⟨ϕ⟩ ≠ 0F≠0∂⟨ϕ⟩ ≠ 0F=0∂⟨ϕ⟩ = 0

Figure 52: We consider a 6d SCFT on surface C with some value of flux. We thengive a vev to an operator φ charged under a U(1) symmetry with the flux. Assuming
the non trivial gauge fields giving rise to the flux are localized on the surfaces we
can turn on a constant vev almost everywhere, but the regions with non trivial gauge
fields. The special loci on the surface with non-constant vev turn out to be equivalent
to inserting minimal punctures in the IR theory in 6d. The number of such loci isdetermined by the flux for the U(1) under which φ is charged.
the superconformal index, which is very convenient for our purposes as it is not too sensitive
to potential 4d strong coupling phenomena. As such, we shall concentrate here on the studyof the dimensional reduction of such Higgs branch flows.
Consider a 6d SCFT, which can be deformed to a 6d gauge theory by going on the tensor
branch. We can go on the Higgs branch by giving a vev to a scalar field, φ, which in the gauge
theory description appears as the lowest components of a hypermultiplet.55 Naturally, the vev
must be such that it does not change the energy. Due to the kinetic terms, this necessitatesthat ∂ < φ >= 0, that is the vev is a constant.
So far we have considered the theory in flat spacetime with no background fields. However,
we next want to consider the case where the 6d SCFT is compactified on a Riemann surface
coupled to background gauge fields. For simplicity, we shall first take the Riemann surface tobe a torus supporting non-trivial fluxes associated with global symmetries. The main differenceis that we now expect the standard partial derivatives in the kinetic term to be replaced withcovariant derivatives, ∇, and as such that the requirement that the vev does not change theenergy to be replaced with ∇ < φ >= 0. The latter, however, is in general not satisfied justby < φ > being a constant.
Naively then, the presence of the flux eliminates the vev, and we may wonder if such flows
actually have an image in the dimensionally reduced theory. However, we can try and examine
this in more detail by using the domain wall picture. Here we use a specific realization of theflux, such that it is localized at a discrete number of points, relying on the observation thatthe compactification result generally only depends on the total value of flux and not so muchon its explicit realization. We have seen how this can be used to better analyze the reduction
process, and we can employ it again for our purposes here.
The main advantage that we gain from this is that we can now choose the flux configuration
so that it is localized at only few points. Away from these points, the solution < φ >= constant
55
More abstractly, these would be lowest components of Higgs branch chiral ring operators. For simplicity, we
shall phrase things here mostly using the Lagrangian 6d gauge theory, though we expect things to hold also for theactual 6d SCFT.

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SciPost Phys. Lect. Notes 78 (2024)

QFT1

Vev for 𝒪6d

UV
QFT2

𝒞2A

QFT1

Vev for 𝒪4d

3d

QFT4

𝒞1 = 𝕊1

4d

QFT′1
IR

Figure 53: Commuting diagram of flows in 4d, in 6d triggered by vevs, and acrossdimensions triggered by geometry.

QFT′3should still be valid so we expect that we should be able to turn on such vevs, with potential
modification at the points were the flux is non-vanishing. Such solutions are actually quite
common in physics, and are known as vortex solutions. These are ubiquitous in spontaneously
broken U(1) gauge theories, and describe configurations in which the Higgs field, whose vev
triggers the breaking of the U(1) gauge symmetry, approaches its vev value at infinity, only
significantly deviating from it in a localized neighborhood. At this neighborhood the value ofthe Higgs field approaches zero. Additionally, the flux of the U(1) approaches zero at infinity,
only significantly deviating from this value at the localized neighborhood where the Higgs field
is nearly vanishing. This neighborhood is then where the flux is localized, and its size defines
the size of the vortex. The solution we are proposing then can be thought of as such a vortex
solution in the limit where the size of the vortex becomes zero. This analogy is useful as it
gives some idea on how these space dependent vevs would look like if one considers moregeneric flux configurations
We then expect that these vevs can still be turned on but now instead of being strictly
constant, they should look like a vortex solution, at least in the limit where the flux is localizedaround small disjoint patches in space. Now consider the resulting 4d theory. We can look at itfrom two different perspectives. Consider first performing the compactification and then giving
the vev. We shall also assume that the operator acquiring the vev desends to an operator ora collection of operators in the 4d theory. We would then naively expect that giving a vev tothis operator should mimic the effect of the 6d vev. See Figure 52 for an illustration.
Now let us consider the case where we give the vev already at the 6d level. We then flow
to a new 6d SCFT. We expect that the theory we got by giving a vev to the 4d theory to be also
realizable by the compactification of this 6d SCFT. The subtle issue here is the nature of thecompactification surface. It turns out that this surface is given by the initial surface with extra
punctures, which can be attributed to the points where the vev is not constant. In other words,
it seems that the non-constant vevs are manifested in this reduction as additional punctures.
In addition the flux of the compactification may change.
UVIR
UV
The picture emerging from this study is as follows. Consider the theories T6d
, T6d
, T4d
IR
UV
and T4dthat we mentioned previously. Recall that T4d
is the result of the compactification
UVIRof the theory T6d on some Riemann surface with flux. Then we have that T4d
is also theIRcompactification, of now theory T6d , on the same surface, but with potentially more punctures
and a different value of flux. This idea is illustrated in Figure 5 (which we reproduce here in
Figure 53 for convenience).
These results were motivated in [27] by the study of the compactification of the 6d SCFTs
living on N M5-branes probing a C2 /Zk singularity. Particularly, these SCFTs possess Higgs

99

QFT′2

QFT3

ℱB

QFT3

4d

6d

𝒞2B

ℱA

QFT2

SciPost Phys. Lect. Notes 78 (2024)

branch flows leading from theories with one value of k to ones with smaller values,56 and one
can study these flows. In this class of theories, a string theoretic picture can be used to argue
from the domain wall picture that one indeed gets extra minimal punctures in the theory after
the flow. Finally, this can be explicitly checked in various models. The results are that indeed
the two 4d theories are related as explained with the number of additional minimal punctures
being proportional to the flux felt by the operator receiving the vev.
Interestingly, this picture suggests a curious connection between punctures and and domain walls. Specifically, we noted that if the operator receiving a vev is charged under thesymmetry receiving a flux, which we can insert in 5d through a domain wall, then we should
get a puncture left at the location of the domain wall. We noted that the domain walls are a
specific tractable limit of representing fluxes, and there are in principle various deformations
one can take of them. These deformations usually don’t affect the IR, or affect it only throughmarginal operators. An example of these types of deformations is the moving around of thedomain walls, where we can even merge several domain walls together to form a different
domain wall. Indeed, we have used this previously to construct more complicated domain
walls from known ones. We can then wonder what these relations imply for the puncture.
Specifically, the motion and union of domain wall is something that also apply to puncture,
where we can move punctures around and even in some cases merge them together to form
different punctures. The exact relation between the two and what can it teach us about them issomething that has not been explored in detail yet. See for related discussion [24,25,116,118].
Application to the construction of trinions
So far we have discussed what happens to Higgs branch flows under the process of dimensional
reduction. We noted that if two 6d SCFTs are related by a Higgs branch flow then a similar
relation may exist for the compactified theory. Specifically, if the 4d theory, resulting from thecompactification of the 6d SCFT from which the flow is initiated, has a BPS operator whichcomes from the 6d Higgs branch chiral ring operator receiving the vev, and if such an operator
can receive a vev, then giving it a vev in 4d should produce the image of this flow in 4d.
This flow should then lead to a new 4d theory which also possesses a higher dimensional
interpretation as the compactification of a 6d SCFT, now the one at the end of the 6d RG flow.
However, the details of the relation are sensitive to the flux, specifically the one felt by theoperator. If the latter is zero then most of the compactification details are expected to remainunchanged, up to the fact that part of the symmetry is broken by the vev. On the other hand,
if the flux felt by the operator is non-zero then some of the details of the compactification
will change. Notably, the compactification surface will change by the addition of punctures,
number of which is expected to be the number of flux quanta felt by the operators, and whose
nature may depend on details involving the flux. Additionally, the flux associated with thecompctification will change due to the breaking and identification of symmetries caused by
the vev. This is motivated by the domain wall picture of the flux, and has been tested inseveral models, notably the family of A and D type conformal matter 6d SCFTs, although it isstill unclear whether it will hold for arbitrary 6d SCFTs.
This observation has an interesting application that we shall next consider. Say we have an
established conjecture for the 4d theories resulting from the tube compactification with global
symmetry fluxes of a family of 6d SCFTs, which are related to one another via Higgs branchflows. Such flows are triggered by a vev to a BPS operator in 6d. We can then consider looking
for the 4d operator expected to come from such 6d operator. This can be done by reading the
global symmetry charges of this operator, translating them to the charges of symmetries in the
56
This class of theories actually possess a very rich structure of Higgs branch flows. Specifically, besides the flows
reducing k, there are also flows that reduce N . Finally, there is a rich structure of additional Higgs branch flows
leading to new 6d SCFTs. We shall not consider such flows here.

100

SciPost Phys. Lect. Notes 78 (2024)

4d theories and looking for a chiral operator with the same charges.57 We can then give such
4d operator a vev, assuming this is indeed possible in the 4d theory. If so then this initiates an
RG flow in 4d which should be the image of the 6d RG flow.
This flow should end with a 4d theory associated with the compactification of a different
6d SCFT in this family, but now on a surface with different flux and more puncture. The
latter point is of special interest to us, as it allows us to gain insight into the compactificationof 6d SCFTs in this family on spheres with more than two punctures just from knowledgeof the compactification on spheres with two punctures. This is the main application of thisobservation to understanding the compactification on generic surfaces.
We next want to comment on the number and nature of the puncture. In the spirit of the
discussion in this section so far, we shall assume that the 4d theories resulting from the tube
compactification of this family of 6d SCFTs were derived using the domain wall method. This
would then suggest that the basic models here are associated with very simple domain walltheories, that is ones whose fields living on them are just chiral fields. These domain walls
are then thought of as being somewhat elementary on account of not being separable to two
simpler domain walls. This then first suggests that they should correspond to minimal flux,
and as such it should be possible to get a sphere with three puncture for some tube theory
in this family. Second, the simplicity of such domain walls would also lead us to expect theresulting puncture to be of minimal type. This is indeed generically what has been observedin this construction.
More complicated domain walls may lead to other types of punctures. However, in ourconstruction, we opted to construct such domain walls by chaining simpler domain walls together. This has lead to the interpretation where you can represent a flux by a single domain
wall, or by chaining of multiple domain walls corresponding to smaller flux. For the case athand, this suggests that the same may be applied to the punctures, that is that it may be possible to merge multiple punctures to form a puncture of a different type. This all relies on theobservation that only a handful of parameters are relevant in the IR theory.
This method can then be used to get theories corresponding to spheres with more than
two punctures just from knowledge of the theories associated to spheres with two punctures.
However, we don’t have that much control on what type of additional punctures would appear.
Generically, we expect to get two maximal punctures, coming from the original punctures of
the tube and a collection of minimal punctures, roughly associated with the number of basic
domain walls needed to engineer the flux. Nevertheless, it might be possible to merge thepunctures to form more general punctures, which would then be interpreted as coming froma single domain wall which is the merger of multiple domain walls. In some cases, we canactually form a maximal puncture using this construction, and thus achieve a derivation of atrinion theory [118, 142].
We shall next illustrate this method with an example.
From tubes to trinions for D type conformal matter theories
Let us briefly apply the general algorithm of deriving spheres with more than two punctures byflowing from (DN +3 , DN +3 ) minimal conformal matter tori to (DN +2 , DN +2 ) minimal conformal
matter tori with punctures. We will rederive the results of [25] using the technology described
57

As we previously mentioned, the Higgs branch chiral ring operators, to which we give a vev to start the flow,
are in short representations. Specifically, if the scalar primary is in the R dimensional representation of SU(2)R ,they obey a shortening condition where the application of the supercharge annihilates the state if the SU(2)Rrepresentation of the final state is of dimension R + 1, see [143] for the details. This suggests that the top (bottom)
SU(2)R component should be annihilated by the SU(2)R component of the supercharge that raises (lowers) theSU(2)R spin. These components should then descend to N = 1 chiral fields. Additionally, they should carry thesame charges under the flavor symmetry as the primary of the Higgs branch supermultiplet.

101

SciPost Phys. Lect. Notes 78 (2024)

2

1 Φ′

Φ 1

N+1

h −2

Q

1 Φ̃

2N+4

N+1

h a −1hah N a −N−2

0

h −2

q′

Φ̃′ 1

2

q

h N a N+2Q′0

W = Φ′(Q + Q′ )Φ̃′ + ΦΦ̃(Q + Q′) + Q N+1q + Q′N+1q′
N

N

Figure 54: Two tubes Φ-glued together to form a torus with flux (−2, −2, 0, 0, · · · ).
Some terms in the superpotential are irrelevant for general N .
above and following [118]. We begin by considering the flow in 6d from the (DN +3 , DN +3 )minimal conformal matter to (DN +2 , DN +2 ) minimal conformal matter in flat space. For this,we recall that we can think of the minimal DN +3 type conformal matter as the UV completion of
the 6d gauge theory with gauge group US p(2N − 2) and 2N + 6 fundamental hypermultiplets.
The Higgs branch flow in question is described in the gauge theory by giving a vev to two
fundamental hypermultiplets, or more correctly, to the quadratic gauge invariant made fromthem. This gauge invariant is indeed a Higgs branch chiral ring operator, which furthermore
is the conserved current supermultiplet.
Next, we look at the 4d theory we associated to the torus compactification of this theory.
Specifically, we take the simple torus model obtained by gluing together two tube theories,
here reproduced in Figure 54.
We next seek the 4d operator expected to descend from the 6d Higgs branch chiral ring
operator in question. In this case, as the operator is just part of the conserved current multiplet,
it is easy to identify a possible choice. Specifically, we noted previously that the operator Φ′ Φ,in the notations of Figure 54, matches the 6d expectation as being an operator descending
from the broken current multiplet, and as such we shall take this operator.
We then consider giving a vacuum expectation value to this operator. As it is a mesonicoperator, it breaks one of the SU(N +1) gauge nodes down to SU(N ). We can analyze the flow
triggered by the vev using a variety of techniques, e.g. the supersymmetric index, as was donein Section 5.4. We will not do so in full detail but just outline the initial important steps. TheN +1
weight of the operator Φ′ Φ in the index (which encodes its symmetry) is x = qp hab1 ε−1 .
Q2N +4
Here bi ( i=1 bi = 1) parametrize the SU(2N + 4) global symmetry and ε the SU(2) global
symmetry. We thus should set x = 1 e.g. by setting ε appropriately. We need then to findthe locus of the pinching
the SU(N + 1) contour integral. Parametrizing the SU(N + 1)
Qof
N +1
gauge fugacities by zi ( i=1 zi = 1) the contour integral setting x = 1 is pinched e.g. at
1

z1 = (qp) 2 h a b1 .58 Using these values for z1 and ε (following from x = 1) one obtains thequiver on the left side of Figure 55.59
It is useful to redefine the symmetries in the following way,
1

2N +3

1

1

u2N +4 = (a b1 )− N +1 h− N +1 (qp)− 2N +2 − 2N +4 ,
v

2N +4

−1 N +3=ha− 2N1+4

(qp)

,

1

(170)

1
εe = a N +2 b1−1 (qp)− 2 .

To be precise we parametrize the fields Φ′ as (qp) 2 hN a−N −2 zi ε±1 .
QN
159The preserved SU(N ) gauge symmetry is parametrized by ui so that i=1 ui = 1 and that zi = z N ui−1 for i > 1.
58

102

SciPost Phys. Lect. Notes 78 (2024)

1

Q′
Φ′+

1

1
3−
N

N

h 1− N +2N a −3− N −2N b11− N
1

1

1

1
1+
N

Φ

2N+3

1

2N + 1

−

1
N

Q′1

Q′1

h −1a b1Φ̃ 1
1
2N + 3

N+1

−

1
N

3 Φ̃′+

1h 2N+1a b1
1

Q′

M1

1

1

h a −1 b1

3+N(ah)1+ N b1N(2N + 3)

u 2N+4(ab1)− N h − N−11(u v )

Φ̃′− h a

N+1

N 2 −(2N+4)

M 1

2N+5

(v 2u −N−4) 2N + 5

Φ̃

N+1
N+2

(v 2u N+1)− 2N + 5

2N + 42N + 4

1
N+2

N+2

Q′2

1
N+2

v 2N+4

Q3

2
N+3
N+2

−1 2N+3 −1
b1

u

1

−(N+1)(2N+4)Φ̃′+

u 2N+4(ab1)− N h − N
1

1

1
N+2

2N + 1Figure 55: On the left the quiver obtained after the flow using the UV fugacities(dropping the gauge singlet fields). On the right the quiver with redefined (andsimplified) definitions of symmetries. We have also shifted N by one on the rightside to switch again to the notations with N = 1 corresponding to the E-string.
Here εe is a fugacity parametrizing an SU(2) symmetry appearing on the quiver on the rightside of Figure 55: The combination of U(1) symmetries giving εe enhances to SU(2) in the IRof the flow triggered by the vev. One should turn on all the superpotential terms consistent
with the symmetries detailed in Figure 55.
Now we want to interpret the theory on the right side of Figure 55 as a torus compactification of (DN +3 , DN +3 ) minimal conformal matter with a single SU(2) minimal puncture. In
particular we want to interpret it as a three punctured sphere, with two maximal SU(N + 1)
punctures and one minimal SU(2) puncture, Φ-glued to itself to form the torus. The quiverof Figure 55 is naturally interpreted as such. The fields M1 and M form 2N + 6-plet of fundamental fields which flip the moment maps when Φ-gluing. The SU(2)εe is the the symmetry
we will associate to the minimal puncture: Note that this is a combination of 6d symmetriesof the higher N minimal conformal theory we started with. The gauge symmetry SU(N + 1)
can be interpreted as gluing the two maximal SU(N + 1) punctures. This gives us the quiverof Figure 57 as a candidate for a three punctured sphere for (DN +3 , DN +3 ) minimal conformal
matter theory.
The trinion we obtain here for N = 1 is directly related to the one we discussed in Section
5.1. The trinion of Figure 35 is simply obtained by taking the trinion of Figure 57 and gluing
to it the two punctured sphere. The gluing is done by S-gluing all the moment maps except
for one of the mesonic ones of Figure 57 to one of the SU(2) moment maps of the tube, see
Figure 58. The fluxes of the various parts are related as,(−2, 0, 0, · · · ) + (1, −1, 0, · · · ) = (−1, −1, 0, · · · ) .

(171)

We thus obtain here a trinion theory for general values of N . For more details of the resulting
theory and nature of the minimal puncture see [25].

7

Discussion and comments

We will briefly discuss the salient features of the geometric constructions of 4d N = 1 QFTs in
4d by compactifying 6d SCFTs overviewing the relevant literature. We will also comment onsome research directions not covered in our Lectures, but which bare direct relevance to thesubject. We will end with some general remarks.

103

SciPost Phys. Lect. Notes 78 (2024)

Cutting a

Vev

Cycle

ℱN = (−2, − 2,0,⋯)ℱN−1 = (−2,0,0,⋯)ℱN−1 = (−2,0,0,⋯)Figure 56: The sequence of operations: We start with a (DN +3 , DN +3 ) torus with flux
FN = (−2, −2, 0, · · · ) preserving U(1) × SO(4N + 8) × SU(2) symmetry. We turn onthen a vev to an operator which is charged under all three factors and breaks thesymmetry to U(1) punc t. × [U(1) × SO(4N + 6)] and corresponds to the following fluxof the IR theory, FN −1 = (−2, 0, 0, · · · ). We obtain a torus with one puncture. The
puncture parametrized by a combination of fugacities of the D2N +6 symmetry of the
UV theory, U(1) punc t. , which are not parametrizing the D2N +4 symmetry of the IR 6dtheory. Finally, we cut the trous along a cycle to obtain a sphere with two maximaland one minimal punctures with the flux equal to the one of the torus.

Across dimension dualities
We have discussed two examples of across dimension dualities, the 6d minimal SU(3) SCFT
[24] and the compactifications of (DN +3 , DN +3 ) minimal conformal matter [19, 25, 26, 118],with the N = 1 E-string case being our main example. There are several other examples forwhich such dualities can be derived systematically. First, one can consider compactifications ofA1 (2, 0) theory analyzed in the seminal work of Gaiotto [8]. One can repeat our analysis in thiscase verbatim. For example in this case turning N = 2 preserving flux the 4d theories turn out
to be described by conformal Lagrangians and one can explicitly find them using the generaltechnology of Section 3.3. One can analyze domain walls in 5d and obtain tube theories
implementing N = 1 preserving flux from which (and the N = 2 trinions) N = 1 A1 class S
theories [128] can be constructed.60 The 5d domain walls, and the corresponding 4d theories
obtained by compactifying on two punctured spheres with some value of flux, of 6d (ADE, ADE)
conformal matter SCFTs [19, 126, 130] (see Figure 59 for an example), D-type (2, 0) theory
probing A type singularity [144], and higher rank E-string theory [132,136,145] have been also
analyzed giving rise to a large class of across dimension dualities. Using then flows between
these theories one can produce [27] models with two maximal and one minimal punctures[118, 142], and in particular class S theories with a minimal puncture can be derived in thisway.61 There are many other examples of across dimension dualities obtained using various
methods: e.g. N = 1 deformations of N = 2 Lagrangian theories leading to strongly coupled
N = 2 theories again [12, 146–148]; a large set of N = 2 theories which have been shownto have weakly coupled conformal Lagrangians, free or interacting, see e.g. [115, 149, 150];
a search over Lagrangians with restricted set of R-charges to describe theories with N = 2[22, 82] and N = 3 supersymmetry [83].
The evidence that one can derive for dualities across six and four dimensions is similar in
The tube models were discussed in [131, 139–141].
Of course historically these models were originally derived using very different methods [8, 125] and theseresults have been used as checks of the more sophisticated derivations.
60

61

104

SciPost Phys. Lect. Notes 78 (2024)

Φ̃

2N+5

(v u

F

N+1
N+2

)

+4
2 N+1 − 2N
2N + 5

Q′1
N+1

u

Φ̃′+

1
N+2

u

1
N+2

2N+4

Q′2
N+1

N+2

N+3
N+2

v 2N+41−(N+1)(2N+4)

1

u

2N+4

2

N+2

Q3

Figure 57: The three punctured sphere obtained by cutting the cycle of the one puncN +1
N +1
tured torus. We have a baryonic superpotential Φ̃′+ · (Q′ 1 + Q′ 2 ), which is irrelevant but the case of N = 1. The trinion is associated to flux (−2, 0, · · · ) which
preserves U(1) × SO(4N + 10) symmetry. The Figure is taken partially from [25].
nature to the evidence one typically gives for 4d IR duality. First one can match IR symmetries. As both start and end dimensions are even one can also match the ‘t Hooft anomalies
for various continuous symmetries. One can also study consistency of various deformations,and other field theoretic operations, with duality. For example, combining two theories bygauging some symmetries, such that the fluxes associated to the two theories are consistent
with bigger symmetry, we should observe the enhanced symmetry in the IR. Certain relevantsuperpotential deformations in 4d breaking some of the 6d global symmetries should produce
a theory with the flux for the broken symmetry vanishing [113, 117]. Finally one can try to
map operators across the flow [113,151]. Local operators in 4d come from local operators andsurface operators wrapping the surface in 6d. At least for low quantum numbers such a map
can be performed for local operators [113], and it is plausible that a more complete procedurecan be devised.
Although many 4d theories obtained in compactifications have an across dimensional dual
description, we do not have such a description for the most general compactification. For
this reason theories without known across dimension dual are referred to as being currently
non-Lagrangian. Nevertheless, many of these non-Lagrangian theories are connected to Lagrangian ones by gauging a subgroup of their global symmetry. The canonical example is
the Argyres-Seiberg duality [152, 153] relating N = 2 strongly coupled SCFTs to cusps ofconformal manifolds of N = 2 Lagrangians. Such N = 2 constructions have a geometricinterpretation in terms of pair of pants decompositions of a Riemann surface [8]. Similarly
the N = 2 strongly coupled SCFTs can appear at cusps of N = 1 Lagrangian theories [154].
Such relations between Lagrangian and strongly coupled theories, though not giving a description of the strongly coupled SCFT itself, often can be “inverted” to acquire a lot of useful
information about protected quantities, such as indices, of the SCFTs [155]. Such inversion
procedures have a field theoretical meaning of gauging symmetries emergent at strong coupling [156, 157], an important procedure we will soon discuss.

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SciPost Phys. Lect. Notes 78 (2024)

2N+4
N+1
N+2

N+1
N+2

1
N+2

N+1

N+2

N+3
N+2

1

2N+4

1

1
N+2

1
N+1

1

N+1
N+2

0
N+1

N+1

1
N+2

2

N+1

N+2
1
N+2

1
N+2

1
N+2

2

Figure 58: The relations between two trinions. On the left the trinion of Figure 57
glued to the two punctured sphere. The SU(N + 1) gauging has N + 2 flavors andthus it S-confines to give a WZ model of mesons anbaryons. The mesons charged
under SU(N + 1) puncture symmetry becomes a bifundamental under puncture andSU(N + 2) gauge symmetry. The meson built from the Φ-field (denoted by red line)
gives mass to one of the 2N + 5 fundamentals. The baryon gives mass to the fieldcharged under U(1) symmetry (denoted by shaded grey on the quiver). Finally, we
have a baryon built from the bifundamental and the Φ field denoted by red solid line
which gets a mass with the dashed red line. This theory then flows to the one on theright side. The Figure is taken from [25].

Atypical degeneration
Analyzing the interplay between 6d and 4d flows one can generate 4d theories corresponding
to compactifications on surfaces with additional minimal punctures as we discussed. However,an interesting question is whether one can also obtain surfaces with more general puncturesand in particular maximal ones. Maximal punctures are such that we can glue theories together along these punctures by gauging the symmetry associated to them. Thus this willallow to construct theories associated to arbitrary surfaces. A useful observation here is thatcollections of sub-maximal punctures can collide and form bigger punctures: This effect was
dubbed atypical degenerations in the case of class S in [116]. We have encountered suchan effect in Section 4.4 discussing compactifications of SU(3) minimal SCFT with triplets ofempty punctured building a maximal one. Here the empty punctures have no symmetry while
maximal ones have SU(3) symmetry. Similar effects were observed [24] studying compactifications of minimal SO(8) SCFTs: In both of these cases the rank of the symmetry of the
maximal puncture is emerging at special loci of the conformal manifolds.
In compactifications of (DN +3 , DN +3 ) minimal conformal matter collections of N minimalpunctures were argued to form a US p(2N ) maximal puncture [118]. Similarly, studying next
to minimal Ak conformal matter (2 M5 branes on Zk orbifold) it was argued that collections ofk minimal punctures form a maximal puncture [118] of a novel type: In both these cases therank of the maximal puncture is the same as the collection of colliding minimal punctures but
the non-abelian structure is enhanced. The new puncture in the Ak case can be obtained bystudying compactifications on a circle of the Ak conformal matter with no holonomies turnedon [17]. The resulting QFT in 5d can be thought of as a gauging of a strongly coupled SCFT
in 5d. In turn this SCFT has a real mass deformation leading to a gauge theory description.
Thus in terms of the 5d Lagrangian constructions this puncture corresponds to gauging an
emergent UV symmetry. This underscores the need to understand and classify all possible 5d
106

SciPost Phys. Lect. Notes 78 (2024)

1

2

3
2

2

1

1

Figure 59: Compactification of minimal (E6 , E6 ) conformal matter theory on a torus
with 1/6 unit of flux breaking E6 × E6 6d symmetry down to U(1) × E6 results in the
depicted conformal quiver theory [126]. This is in fact N = 2 quiver shaped as theaffine Dynkin diagram of E6 with four decoupled free chiral fields (denoted by thecrosses).

Figure 60: Atypical degenerations of the surface. In numerous examples in certain
limits of collision of some types of punctures one can obtain a new type of a puncture.
The symmetry of the new “composite” puncture might be different than the symmetry of the punctures building it. Conversely, there are examples of exactly marginal
deformations breaking the puncture symmetry and not corresponding to naive geometric deformations of the surface.
effective descriptions of 6d SCFTs when studying compactifications to 4d. To build higher
genus surfaces gauging symmetries of colliding minimal punctures we thus need to gauge
emergent symmetries on special loci of the conformal manifold (or in the IR).
Conversely considering a theory with a puncture (maximal or non-maximal) there might beexactly marginal deformations which break subgroups of the puncture symmetry. We have seen
an example of this in Appendix D: A2 class S compactifications with minimal punctures haveexactly marginal deformations which break the puncture symmetries. These exactly marginal
deformations preserve only N = 1 supersymmetry and do not correspond to complex structure
moduli or the flat connections on the surface.62 It would be interesting to understand whether
these directions on the conformal manifold can be understood geometrically.
For more general AN >1 class S compactifications one also has N = 1 preserving deformations when certain
non maximal punctures appear on the surface with the puncture symmetries broken to a subgroup on a general
locus of the conformal manifold [158].
62

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SciPost Phys. Lect. Notes 78 (2024)

Gauging emergent symmetries
An important procedure which appeared several times in our discussion is the utility of gauging emergent symmetries. In particular some symmetries enhance in the IR of an RG flow
or at some strong coupling loci of the conformal manifold allowing to consider coupling the
emergent conserved currents to dynamical gauge fields. This is a manifestly strongly coupledprocedure. Although the gauge coupling might be weak the starting point of the construction
is strongly coupled. Nevertheless, if the rank of the symmetry is manifest in the UV, or at weakcoupling of the conformal manifold, a lot of the protected information of the gauge theory canbe explicitly computed. This follows from the fact that the supersymmetric partition functions
related to counting problems are independent of the RG flow and the continuous parametersof the theory. Several of the 4d theory constructions corresponding to compactifications of
6d SCFTs obtained till now involve such gaugings [24, 26, 117, 118, 125, 132, 156, 157]. In
some cases the emergent symmetry in one duality frame might be explicit symmetry of the
Lagrangian in another duality frame and thus the gauging would be completely weakly coupled. An example is (D5 , D5 ) minimal conformal matter where the trinion is constructed as afour punctured sphere with two maximal SU(3) punctures and two minimal SU(2) puncturescolliding to form a US p(4) maximal puncture somewhere on the conformal manifold [118].
However, using a sequence of Seiberg dualities one can obtain a description with the the two
SU(3) symmetries and the US p(4) symmetry manifest [127]. An interesting question in thiscontext is whether there are any fundamental obstructions for having a weakly coupled description of a given theory with a given symmetry manifest. We have discussed several examples of theories with weakly coupled descriptions and emergent symmetry in 4d such that the
6d dual description has the symmetry manifest but is not weakly coupled; and the questionis whether we can or cannot find a description with manifest symmetry in 4d. For N = 2
theories in 4d many examples of theories with no Lagrangians with manifest symmetry are
known just by listing all the possible manifestly N = 2 supersymmetric Lagrangians [9]. A
related question would be then under which conditions we can find say N = 1 Lagrangians
with manifest global symmetry for such models.63

General types of punctures: 5d UV dualities
A useful knob in constructing 4d theories corresponding to a surface is through gluing theories
corresponding to smaller surfaces with the gluing done along a puncture. A given 6d SCFT
might have a variety of types of maximal punctures along which we can glue.64 Such maximal
punctures correspond as we have discussed to circle compactifications of the 6d SCFT, i.e. 5dgauge theories UV completed by the 6d SCFT. Compactifying on a circle, if the 6d (1, 0) SCFT
has a global symmetry, one has a choice of a holonomy in this global symmetry to be turnedon around the circle. Different such holonomies might lead to different 5d descriptions, seee.g. [96–98, 112, 164–169]. Such different 5d effective descriptions are UV dual to each otherin the sense that they are different deformations of the same UV SCFT. We have mentioned
above the relatively simple constructions of three punctured spheres e.g. for the (DN +3 , DN +3 )minimal conformal matter with two SU(N +1) maximal punctures and one US p(2N ) puncture,
although the trinions with three punctures of the same type are more complicated (These canbe obtained from the former by gluing in two punctured spheres.). The 5d UV dualities canlead to 4d IR dualities by constructing same surfaces from building blocks having different
63

Such questions can be phrased also in other dimensions. For example let us mention here a simple example
in 3d where there are (at least) three different descriptions manifesting only two of three symmetries: N = 2
supersymmetry, SU(3) global symmetry, or time reversal symmetry [159–163].
64
One might have also more general “fixtures” connecting different types of maximal and non-maximal punctures. See e.g. [149].

108

SciPost Phys. Lect. Notes 78 (2024)

types of maximal punctures, see Figure 61 for an example. It is thus very useful to completely
map out all the possible 5d theories UV completed by 6d SCFTs and interrelations between
these.

Integrable models
Yet another interconnection between physics in different dimensions goes through the appearance of quantum mechanical integrable models in various counting problems. The integrable
models have a long history of interconnections with gauge theories in various dimensions,see e.g. [170]. The connection most relevant to us here goes as follows. One considers a 6d
(1, 0) SCFT (eight supercharges), compactifies to 4d breaking half of the supercharges, andconsiders surface defects in the 4d theory. In particular one can count protected operators (theS3 × S1 partition function, the index) of the 4d theories in the presence of such defects. Using
general considerations the resulting partition function can be obtained by acting on the partition function of the theory without the defect with an analytic difference operator [119].65 We
have different operators labeled by the surface defect we want to introduce. These operators
can be thought of in general as Hamiltonians of an integrable relativistic quantum mechanicalsystem. The choices of the defects we are considering are determined by the 6d SCFT and are
the same for all the 4d theories obtained by compactifications of this SCFT. In particular these
integrable models are thus labeled by a theory with eight supercharges, which bares a direct
relation to the constructions of [172]. These integrable systems can be explicitly derived bystudying analytic structure of the indices. In particular they are obtained by computing theresidues of poles the index can have in presence of minimal punctures and are acting on the
fugacities of a maximal puncture. As such these operators are then also labeled by the choiceof the maximal puncture i.e. the choice of the 5d effective gauge theory description.
The various duality properties of the 4d theories obtained in compactifications imply mathematical properties of the operators introducing defects [119].66 For example, the operators
have to commute and the indices themselves give Kernel functions of these operators,(6d C F T ;5d QF T 1 )Oℓ(6d C F T ;5d QF T 2 )(x15d |y6d ) · I (x15d , x25d |y6d ) = Oℓ

(x25d |y6d ) · I (x15d , x25d |y6d ) .

(6d C F T ;5d QF T 1 )Here Oℓ(x15d |y6d ) is the operator corresponding to the introduction of a defectlabeled by ℓ acting on a maximal puncture coming from the 5d QF T 1 description of the 6d C F T
on a circle. The supersymmetric index I (x15d , x25d |y6d ) corresponds to a surface with at least
two maximal punctures of types (5d QF T 1 ) and (5d QF T 2 ), and y6d label the Cartan of the 6d
global symmetry. Such Kernel functions relations are highly non trivial and thus give importantmathematical checks of the conjectured across dimension dualities.
Explicitly for compactifications of the ADE (2, 0) theories the relevant models turn out
to be elliptic ADE Ruijsenaars-Schneider systems [119, 173]; for the rank N E-string this is
the BCN van Diejen model [127, 174] (this was shown explicitly for N = 1 using the index
methods and conjectured for higher N ); for minimal 6d SCFTs with SU(3) and SO(8) gauge
groups these were discussed in [120, 121]; for A type conformal matter the operators werediscussed in [119, 175]; for minimal D type conformal matter the operators were computedin [127]. These operators can be obtained independently by studying Seiberg-Witten curves[176, 177] directly in 6d [178]. One way to understand this correspondence between indicesand integrable models is as a version of AGT correspondence where we view the 6d theory on
S3 × S1 × C (with C being a Riemann surface) and take either C small to obtain the index in
4d, or the S3 × S1 to be small to obtain a TQFT on C (which should be related to the integrable
65
One can also construct in principle such operators by explicitly coupling 2d degrees of freedom to 4d ones[171].
66
See [127] for a recent overview.

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SciPost Phys. Lect. Notes 78 (2024)

model) [179]. In the case of (2, 0) theories and a particularly simple version of the index,
the Schur index [180, 181], the TQFT turns out to be the q-deformed YM [180, 182–187].
The Schur index then has further relations to beautiful mathematical structures such as chiralalgebras in 2d [188].67

6d dualities
Studying compcatifications of different 6d SCFTs on different surfaces to 4d one at times obtains same IR 4d theories, or more generally same 4d theories up to decoupled free fields.
Such 6d dualities were analyzed for the relations between (2, 0) compactifications on punctured spheres and (1, 0) on tori with no flux [16–18]. Let us give here three examples of suchdualities between two (1, 0) theories.
First, let us discuss the example mentioned in the bulk of the paper: rank one E-string on
a genus two surface with no flux is the same as minimal SU(3) (1, 0) 6d SCFT on a sphere withfour maximal punctures (and Z3 twist lines). The resulting theory in 4d has a large conformal
manifold and the statement of the duality is that the two theories reside on the same manifold.
In particular, considering the E-string compactification the number of the marginal operators
minus the conserved currents is given by,
3g1 − 3 + 248 E8 (g1 − 1) + 1 g =2
1→

dim Mc = 252 .

(172)

Here g1 = 2 is the compactification genus. The first two terms are the expected marginal
operators minus the current coming from complex structure moduli and the flat connections
for the E8 symmetry. The last deformation is accidental for this compactification. On the other
hand considering the same quantity for the SU(3) minimal SCFT on the four punctured sphere
we obtain,
3g2 − 3 + s2 +s2
X
(10i − 8i ) + 3 × (31 , 32 , 33 , 34 )i=1

→ dim Mc = 252 .

(173)

g2 =0, s2 =4Here Ri denotes the representation R of the ith SU(3) puncture symmetry group. The firstterm comes from the complex structure moduli and the second from the punctures, while thelast one is accidental for this compactification. Going along this large conformal manifold,likely including the accidental exactly marginal directions in the two cases, the two theories
can be then connected.
Second example is as follows [19].
Compactifications of conformal matter theories residing on 2 M5 branes probing Zz(N +1) singularity on a sphere with N + 1 minimal and two maximal punctures and flux F = z N 2+1 in the U(1) t subgroup of theSU(k)×SU(k)×U(1) t ⊂ SU(2k)symmetry of the 6d theory is dual (up to N (N +1) decoupled freefields) to the compactification of (DN +3 , DN +3 ) minimal conformal matter theory on a torus withflux z breaking the SO(4N +12) symmetry of the 6d SCFT to U(1)×SU(N +1)×SO(10+2N ) for
integer z. See Figure 61 This duality produces field theories which are 6d dual to each otherup to free fields. The simplest case of this duality is taking N = 1 and z = 12 where N = 2
SU(2) N f = 4 SCFT with a decoupled chiral field, i.e. A1 (2, 0) theory on a four punctured
sphere (plus a decoupled field), is equivalent to the rank one E-string on a torus with half aunit of flux breaking E8 to U(1) × SO(8) ⊂ U(1) × E7 [26].68
Let us mention here that in the case that an N = 1 theory is not chiral and has rationally quantized R-charges
one can define a simplified version of the N = 1 index [154] which coincides with the Schur index for the N = 2
theories.
68
This compactifcation of the E-string is actually naively consistent with SO(8) exchanged with any rank four
subgroup of E7 , e.g. G = F4 , US p(8) [26]. This is in particular responsible for the observations that the protected
spectrum of the theory forming irreps of either of these groups, see e.g. [189].
67

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SciPost Phys. Lect. Notes 78 (2024)

2

2

2

2

2

2

2

2

2

2

2

2

4

2N

4

N+1

N+5

N+1

N+1

N+1

N+5

2N

N

Figure 61: On the left we have the theory obtained by compactifying (DN +3 , DN +3 )minimal conformal matter theory on a torus with one unit of flux breaking theSO(4N + 12) symmetry of the 6d SCFT to U(1) × SU(N + 1) × SO(10 + 2N ). On
the right we have a dual theory [19] (double circles correspond to US p groups andthe lines starting and ending on the same vertex are in antisymmetric representation). We assume that N is an odd integer. Note the two dual theories manifest
different subgroups of the global symmetry group. The model on the right is obtained by gluing two tubes with US p(2N ) and SU(N + 1) punctures while on the left
we glue tubes with SU(2)N punctures. This duality follows from a 5d UV duality.
The theory on the left (removing the free flip fields) can be alternatively obtained bycompactifying the SCFT on two M5 branes probing ZN +1 singularity on a sphere withtwo maximal and N + 1 minimal punctures: This is thus an example of a 6d duality.
The last example is as follows. Taking AN −1 (2, 0) theory on a sphere with two maximal andtwo minimal punctures (half positive and half negative in the notations of [128]) and zero flux isdual to compactification of (DN +1 , DN +1 ) minimal conformal matter theory on a sphere with two
maximal and a minimal puncture, and half a unit of flux breaking SO(4 + 4N ) 6d symmetry to
U(1)×SU(2)×SO(4N ). In both of these cases we obtain N = 1 SU(N ) SQCD with 2N flavors.
Again the explicit symmetry of the theories in the two compactifications is different: In former
the symmetry is SU(N )2 × U(1)3 and in the latter it is SU(N − 1)2 × SU(2) × SU(2N ) × U(1)3 ,and the two compactifications are dual to each other exploring the large conformal manifold.
It will be extremely interesting to understand these and other 6d dualities by embedding
them for example in string theory.

More comments
Let us briefly mention several additional related avenues of research. First, one can consider
compactification scenarios in the holographic regime. In particular when the 6d SCFTs are
obtained on collections of M5 branes one can consider taking the number of branes to be large
and study the corresponding supergravity backgrounds. This for example was done in theclass S context in [128,190–192] and studying branes on orbifolds in [137,193,194]. Anotherproperty which we did not address in this review is the higher form/higher group/categoricalsymmetries [28, 29, 195] of the 4d theories derived in compactifications. For some studies ofthese issues see [194, 196–199].
It is interesting to consider compactifications to lower dimensions (3d and 2d). This can bedone in a variety of ways. For example, one can consider compactifications on a circle of the
4d theories obtained by first considering 6d SCFTs on a surface C . The dualities following fromthe compactifciations in 4d will then give rise to a rich set dualities in 3d [13, 200]. Moreover
the resulting 3d theories might have novel 3d dual descriptions which might not trivially lift111

SciPost Phys. Lect. Notes 78 (2024)

to 4d.69 These typically take the form of mirror duality [201]. The point is that 3d dynamics isricher than the 4d one: For example abelian gauge theories typically lead to interesting SCFTs
and there is no upper bound on the matter representations to lead to interacting theories in
the IR. Although for compactifications of (2, 0) theories such mirror dualities are extensively
studied, see e.g. [141, 201–206], for compactifications of (1, 0) theories on C × S1 not much is
known.
One can also study compactifications on surfaces starting from 5d [207, 208] or 4d [209–211]. Similarly we can consider compactifying on circles starting from 5d [212] and 3d [213–216]. One can motivate through these constructions various dualities in 3d. See e.g. [217].
Finally one can consider compactifications on higher dimensional surfaces. Starting from 6d
we can compactify on three manifolds [15, 218–222] or four manifolds [223, 224].

Final remarks
Let us make several more philosophical remarks. Our discussion of 4d supersymmetric dynamics has been performed using standard Lagrangian techniques in 4d and 5d. The starting
point in 6d is a strongly coupled SCFT, and although some field theoretic Lagrangian tools(such as tensor branch descriptions) can be utilized, at the moment to fully understand suchmodels one usually includes string theoretic constructions (see [10] for a review). One candraw several general lesson from our considerations. First, it is often the case that if one wants
to understand a weakly coupled UV description of a given IR physics it is useful to give up someof the symmetry and let it emerge only in the IR. In our discussion the symmetry was typicallycontinuous global symmetry, but this can be also supersymmetry. Related to this there arevarious suggestions to understand 6d SCFTs themselves by giving up some symmetry, in thiscase space-time symmetry. These include engineering 6d SCFTs as limits of 4d quiver theories through the procedure of deconstruction [225, 226], trying to decode the full 6d physics
from its circle compactifications [108, 109], or DLCQ matrix model [227, 228].70 Giving up
symmetry we can gain many insights and it will be interesting to understand the limits of suchan approach. A complimentary way of thinking, at least about the conformal fixed points, isgiving up description in terms of weakly coupled fields completely and making an emphasis on
extracting all the possible physics from symmetries, including the conformal symmetry. Thisbootstrap approach has led to many beautiful results in recent years, see e.g. [230–232]. Another more abstract approach applicable to theories with well defined Coulomb branches, suchas N = 2 theories, is to study the Coulomb branches abstractly without use of a Lagrangian
using the Seiberg-Witten curve associated to them [233–235].71 The program of systematically studying the curves and extracting physics from them is pursued e.g. in [242–245]. Even
using the geometric techniques one can search for general expressions of various quantities
describing SCFTs such that the geometry will be manifest and no use of Lagrangians made.
Examples of this are the AGT correspondence [44] (see [246] for a review) relating the S4 partitions function of N = 2 theories to 2d CFT correlators and the relation of the supersymmetricindex to integrable models and to 2d TQFTs [179] we discussed above.72
69

However, some mirror dualities have a surprizing lift to 4d. For example, a canonical example of mirror selfdual
theory, the 3d T [SU(N )] model residing on S-duality domain wall of N = 4 SU(N ) SYM in 4d, can be uplifted
to 4d [132, 145]. In fact the 4d theory is the one obtained by compactifying rank N E-string on a two punctured
sphere: The mirror duality exchanges the two punctures. One of the puncture symmetries is emergent in the IR
and thus the exchange of punctures is not a trivial operation from the point of view of the UV Lagrangian.
70
See [229] discussing all three approaches.
71
This approach can be applied in certain cases also for N = 1 theories [236], which was studied for theories
obtained from M5 branes on A type orbifolds in [237–241].
72
Sometimes these types of relations are referred to as BPS/CFT correspondence, see e.g. [247, 248].

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SciPost Phys. Lect. Notes 78 (2024)

Another comment we want to make is that although all of our discussion is N = 1 supersymmetric in 4d, that is relies on having four supercharges, the starting point in 6d has eightsupercharges, that is has N = 2 supersymmetry in 4d counting. By considering such theories
on surfaces and turning on various knobs and levers we can derive a plethora of N = 1 dynamics. An interesting question is then to what extent all supersymmetric phenomena havetheir origin in theories with eight supercharges. This question can be of course extended byadding more supercharges and imbedding everything in string theory, if one is willing to leave
the cradle of purely QFT discussion.73 A way to phrase a similar question is to what extent allnon trivial dynamics in lower dimensions follows from QFT constructions involving geometricdeformations starting in 6d, and/or going beyond QFT in string theory: e.g. are there CFTs inlower dimensions which cannot be constructed as limits and deformations of supersymmetric
CFTs in 6d and/or of string theoretic setups?

Acknowledgments
We would like to thank our numerous amazing collaborators for sharing with us their ideas
and insights. SSR is grateful to the organizers of the “Cargese Summer School: Quantum
Gravity, Strings and Fields” (2018), “ICTP spring school on superstring theories and related
topics” (2019), and “The 15th Kavli Asian Winter School on Strings, Particles and Cosmology”(2021) where parts of the material discussed here were presented.
Funding information The research of SSR, OS, and ES was supported in part by Israel Science Foundation under grant no. 2289/18, by I-CORE Program of the Planning and Budgeting
Committee, by a Grant No. I-1515-303./2019 from the GIF, the German-Israeli Foundation for
Scientific Research and Development, and by BSF grant no. 2018204. The research of SSR
was also supported by the IBM Einstein fellowship of the Institute of Advanced Study andby the Ambrose Monell Foundation. The research of ES was also supported by the European
Union’s Horizon 2020 Framework: ERC grant 682608 and the “Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics”. The research of OS was also supported by
the Clore Scholars Program and by the Mani L. Bhaumik Institute for Theoretical Physics at
UCLA. GZ is supported in part by the ERC-STG grant 637844-HBQFTNCER, by the INFN andby the Simons Foundation grant 815892.

A

Superconformal algebra in 4d

In this appendix we consider the spacetime symmetry algebra sl (4|1) of four dimensional
conformal field theories with N = 1 supersymmetry, and its representations.
We begin with the bosonic conformal subalgebra. Written using spinorial notation withα, α̇ = ±, ±̇, the nonvanishing commutators are given by(J1 )αβ , (J1 )γδ = δγβ (J1 )αδ − δαδ (J1 )γβ ,


γ̇
γ̇
γ̇
(J2 )α̇β̇ , (J2 ) = δα̇ (J2 ) − δ (J2 )α̇δ̇ ,
(A.1)
δ̇
δ̇
β̇
β̇


1
(J1 )αβ , Pγγ̇ = δγβ Pαγ̇ − δαβ Pγγ̇ ,2Note that this is a different question than the one discussed in the context of the swampland program [249–251]. In the swampland context one constrains D-dimensional low energy QFTs coupled in the UV to D-dimensional
gravity. Here we allow D-dimensional QFTs to be coupled to higher dimensional gravity in the string constructions.
73

113

SciPost Phys. Lect. Notes 78 (2024)
1(J2 )α̇β̇ , Pγγ̇ = δγ̇α̇ Pγβ̇ − δα̇ Pγγ̇ ,
2 β̇


1
(J1 )αβ , K γ̇γ = −δαγ K γ̇β + δαβ K γ̇γ ,21 α̇ γ̇γγ̇ α̇γα̇γ̇γ(J2 ) β̇ , K= −δ K + δ K ,β̇
2 β̇
[H, Pαα̇ ] = Pαα̇ ,H, K α̇α = −K α̇α ,K α̇α , Pβ β̇ = δβα δα̇ H + δβα (J2 )α̇β̇ + δα̇ (J1 )βα .


β̇

β̇

We next turn to the part of the algebra involving fermionic generators. There are eightsuch generators, the four Poincare supercharges Q α , Q̃ α̇ and the four conformal supercharges
S α , S̃ α̇ , with the following nonvanishing anticommutators,Q α , Q̃ α̇ = Pαα̇ , α α̇
S , S̃ = K α̇α ,



1 β
3
β
(A.2)
Q α , S = δα H + R + (J1 )αβ ,
2
2

¦
© 1 
3
β̇
β̇
β̇
Q̃ α̇ , S̃ = δα̇ H − R + (J2 ) α̇ .
22We are left with the following commutators of the fermionic generators with the bosonic ones,
1(J1 )αβ , Q γ = δγβ Q α − δαβ Q γ ,21(J2 )α̇β̇ , Q̃ γ̇ = δγ̇α̇Q̃ β̇ − δα̇ Q̃ γ̇ ,
2 β̇


1
(J1 )αβ , S γ = −δαγ S β + δαβ S γ ,
2


1
γ̇
(J2 )α̇β̇ , S̃ γ̇ = −δ S̃ α̇ + δα̇ S̃ γ̇ ,
β̇
2 β̇

 1
1
[H, Q α ] = Q α ,H, Q̃ α̇ = Q̃ α̇ ,
2
2


1
1
[H, S α ] = − S α ,H, S̃ α̇ = − S̃ α̇ ,22[R, Q α ] = −Q α ,R, Q̃ α̇ = Q̃ α̇ ,[R, S α ] = S α ,R, S̃ α̇ = −S̃ α̇ , α̇αK , Q β = δβα S̃ α̇ ,K α̇α , Q̃ β̇ = δα̇ S α ,β̇βPαα̇ , S = −δαβ Q̃ α̇ ,β̇Pαα̇ , S̃ β̇ = −δα̇ Q α .


(A.3)

Once the superconformal algebra is known, the next natural thing to examine is its representation theory. That is, we wish to classify the possible multiplets and understand theirproperties.
A generic multiplet of the algebra is characterized by the charges (∆, R, j1 , j2 ) of the superconformal primary state with respect to the generators (H, R, j1 , j2 ), where j1 and j2 are thefollowing Cartans:
j1 = (J1 )++ = − (J1 )−− , j2 = (J2 )+̇+̇ = − (J2 )−̇−̇ .
(A.4)
114

SciPost Phys. Lect. Notes 78 (2024)

The primary state is annihilated by the conformal supercharges S α , S̃ α̇ ,74 and the multiplet
is constructed by acting on it with the Poincare supercharges Q α , Q̃ α̇ . In some cases, a combination of these supercharges annihilates the primary as well, resulting in a short multiplet
and a relation between the charges of the primary. These relations are just a saturation of thecorresponding unitarity bounds, derived from requiring the absence of negative-norm statesin the multiplet, and determine the conformal dimension ∆ in terms of the other charges ( j1 ,j2 and R). This, in turn, means that the conformal dimension is protected against changing
the parameters of the theory. We list the possible such shortening conditions (classified by thecorresponding null state in the multiplet) and their common names in the table below, as wellas the associated unitarity bounds [20, 41, 143].75
Shortening Condition

Name

Primary

Null State

Unitarity Bound–∆ > 2 + 2 j1 − 32 R–

∆ > 2 + 2 j2 + 32 R(R−1)
j1 − 21 ; j2 ∆+1/2∆ = 2 + 2 j1 − 32 R

[0; j2 ]∆+1
(R+1)
j1 ; j2 − 12 ∆+1/2∆ = 2 − 32 R

(R)
[ j1 ; j2 ]∆(R)[ j1 ; j2 ]∆

–

L

–

L

εαβ Q α |Primary〉β = 0

A1[ j1 ; j2 ]∆ , j1 ≥ 12(Q)2 |Primary〉 = 0

A2[0; j2 ]∆εα̇β̇ Q̃ α̇ |Primary〉β̇ = 0

A1[ j1 ; j2 ]∆ , j2 ≥ 12

(Q̃)2 |Primary〉 = 0

A2[ j1 ; 0]∆Q α |Primary〉 = 0

B1[0; j2 ]∆Q̃ α̇ |Primary〉 = 0

B1[ j1 ; 0]∆

(R)

(R)

(R−2)

(R)

(R)

(R+2)

[ j1 ; 0]∆+1
 1 (R−1)
2 ; j2 ∆+1/2
 1 (R+1)
j1 ; 2 ∆+1/2

(R)
(R)

∆ = 2 + 2 j2 + 32 R
∆ = 2 + 32 R
∆ = − 32 R
∆ = 32 R

As can be seen, unbarred letters denote shortening conditions with respect to the supercharges Q α while barred letters are with respect to Q̃ α̇ . Moreover, the letters L and L correspond to the absence of shortening conditions with respect to Q α and Q̃ α̇ , respectively.
To completely specify a multiplet, one needs to impose both Q α and Q̃ α̇ shortening conditions, and as a result each kind of multiplet is denoted by two letters – an unbarred one(R)
and a barred one. For example, L L [ j1 ; j2 ]∆ corresponds to a long multiplet with charges(R)(∆, R, j1 , j2 ) for the primary state, while LB 1 [ j1 ; 0]∆ to a multiplet with a chiral primary.76
The constraints on the charges of the primary state resulting from the various combinations ofshortening conditions are detailed in the following table [143].
LL∆ > 2 + max 2 j1 − 32 R, 2 j2 + 32 R

A1

∆ = 2 + 2 j1 − 32 R
j1 ≥ 21 , R < 23 ( j1 − j2 )

A2
B1



A1

A2

B1

∆ = 2 + 2 j2 + 32 R

∆ = 2 + 32 R

∆ = 32 R

j2 ≥ 21 , R > 23 ( j1 − j2 )j2 = 0 , R > 23 j1

j2 = 0 , R > 23 ( j1 + 1)

∆ = 2 + j1 + j2

∆ = 2 + j1

∆ = 1 + j1

j1 , j2 ≥ 21 , R = 32 ( j1 − j2 )j1 ≥ 12 , j2 = 0 , R = 23 j1

j1 ≥ 12 , j2 = 0 , R = 23 ( j1 + 1)

∆ = 2 − 32 R

∆ = 2 + j2

∆=2

∆=1

j1 = 0 , R < − 23 j2

j1 = 0 , j2 ≥ 12 , R = − 23 j2

j1 = j2 = 0 , R = 0

j1 = j2 = 0 , R = 23

∆ = − 23 R

∆ = 1 + j2

∆=1

∆=0

j1 = 0 , R < − 23 ( j2 + 1)

j1 = 0 , j2 ≥ 12 , R = − 23 ( j2 + 1)

j1 = j2 = 0 , R = − 23

j1 = j2 = R = 0

To illustrate how these constraints between the charges are obtained from the algebra,(R)
let us consider as an example a multiplet of the type LB 1 [0; 0]∆ . Here, the superconformal
primary state |χ〉 has j1 = j2 = 0 and satisfiesQ̃ α̇ |χ〉 = S α |χ〉 = S̃ α̇ |χ〉 = 0 .

(A.5)

Notice that as a result, the primary state is also annihilated by K α̇α .
Notice that we use different conventions for j1 and j2 in comparison to [20, 143], such that jthere = 2 jhere .
76
Note that a chiral free field multiplet is represented by Ak B 1 [ j1 ; 0](R)
.
∆

74
75

115

SciPost Phys. Lect. Notes 78 (2024)

As a result, the expectation value of the fourth equation in (A.2) is given by


1 β̇
3
β̇
〈χ|{Q̃ α̇ , S̃ }|χ〉 = 0 = δα̇ ∆ − R ,
2
2

(A.6)

implying ∆ = 32 R. In addition, since the dimension ∆ is greater than 1 (otherwise it would be(R)equal to the free-state value 1, corresponding to the primary of the multiplet A2 B 1 [0; 0]∆ ),we have R > 23 .
Let us next note that even though the short multiplets are protected in the sense we mentioned before, most of them are not absolutely protected since they can recombine as wechange the parameters of the theory with other short multiplets to form a long multiplet,
which is no longer protected. This can happen e.g. as we move along a conformal manifoldfrom one superconformal field theory to another, and short multiplets recombine to form along multiplet such that the operators residing in the short multiplets are not protected in the
new theory. In the other direction, a long multiplet at one point of the conformal manifold
can hit the unitarity bound at another point, decomposing into a collection of short multiplets.
(R=0)
As an example, let us consider the long multiplet L L [0; 0]∆=2+ε as ε → 0, where it hits theunitarity bound. It then splits into three short multiplets in the following way,
(R=0)

(0)

(−2)

L L [0; 0]∆=2 → A2 A2 [0; 0]2 ⊕ B1 L [0; 0]3
(0)

(2)

⊕ LB 1 [0; 0]3 ,
(−2)

(A.7)
(2)

where A2 A2 [0; 0]2 contains a conserved current while B1 L [0; 0]3 and LB 1 [0; 0]3 containa marginal operator.77 This recombination corresponds to the fact that a marginal operator
can fail to be exactly marginal only if it combines with a conserved current corresponding to abroken global symmetry [40], and is the reason that marginal operators and conserved currents
contribute to the superconformal index with opposite signs (see Eq. (56)). In particular,
the difference between the numbers of conserved-current and marginal-operator multiplets isinvariant under a change in the parameters of the theory which might be accompanied by therecombination (A.7).
For each of the multiplets listed above, one can compute the superconformal index anduse it to extract information about the operator content of a given theory from its index. The
multiplets that contribute nontrivially to the index are collected in the following table, alongwith the corresponding expressions for the index.
77

Note that in this discussion we refer to operators instead of states, which is possible due to the state/operator
correspondence in theories with conformal symmetry.

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SciPost Phys. Lect. Notes 78 (2024)

Multiplet

Superconformal IndexpR(pq) 2 + j2 +1 χ j1 ( p/q)(−1)2( j1 + j2 )+1(1−p)(1−q)
p
R +1
2
(pq)
χ j1 ( p/q)(−1)2 j1 +1(1−p)(1−q)
pR
2 j1 (pq) 2 χ j1 ( p/q)(−1)(1−p)(1−q)
p
1 ( j +2 j )+1
2
3 1
(pq)
χ j1 ( p/q)(−1)2( j1 + j2 )+1(1−p)(1−q)
p
1 j +13 1 χj (
(pq)
p/q)
1
(−1)2 j1 +1(1−p)(1−q)(R)

LA1 [ j1 ; j2 ]∆(R)

LA2 [ j1 ; 0]∆(R)

LB 1 [ j1 ; 0]∆(R)

A1 A1 [ j1 ; j2 ]∆(R)

A1 A2 [ j1 ; 0]∆2(pq) 3 j2 +1(R)

A2 A1 [0; j2 ]∆(−1)2 j2 +1 (1−p)(1−q)

A2 A2 [0; 0]∆A1 B 1 [ j1 ; 0]∆− (1−p)(1−q)
1 ( j +1)pp1(−1)2 j1 (pq) 3 1[χ j1 ( p/q) − (pq) 2 χ j1 − 1 ( p/q)](1−p)(1−q)(R)
A2 B 1 [0; 0]∆(pq) 3(1−p)(1−q)

(R)
(R)

pq

2

1

2

(pq) 3 ( j2 +1)(R)

B1 A1 [0; j2 ]∆− (1−p)(1−q)2(pq) 3(R)

B1 A2 [0; 0]∆− (1−p)(1−q)

Here, χ j (z) is the character of the spin- j representation of SU(2). We can immediately see
from this table that as stated in the two bullets around Eq. (56), the only operators that cancontribute at order (pq)n with n < 1 are relevant operators (corresponding to primaries of(R)
LB 1 [0; 0]∆ multiplets with R < 2) and that at order pq the only operators which can contribute are marginal ones (contributing with a positive sign) and certain fermionic components(R)from the conserved current multiplet A2 A2 [0; 0]∆ , which contribute with a negative sign. As
mentioned above, the signs of these contributions at order pq correspond to the recombination
rule (A.7). Note, in addition, that there is no recombination rule involving relevant operators,(R)meaning that the multiplets LB 1 [0; 0]∆ with R < 2 are absolutely protected. Let us finally emphasize that this discussion about the contributions of various operators to the index excludes
free multiplets, that is we concentrate only on interacting theories.

B

Leigh-Strassler argument for conformal manifolds

Let us here detail briefly yet another method to study the conformal manifold introduced in aseminal paper by Leigh and Strassler [51]. This method is very Lagrangian in nature on one
hand but on another hand it is rather intuitive. The basic observation is that as the superpotential is not renormalized due to holomprphy arguments [1] in a supersymmetric theory, thebeta-function of the superpotential coupling is determined by the anomalous dimensions of
the fields in the superpotential,ni1X
βλi ({λ, g}) ∼ ni − 3 +γα ( j) ({λ, g}) ,2 j=1 i

W=

L
X
i=1

λi

n
Y

Q αi ( j) .

(B.1)

j=1

Here αi ∈ S L is some permutation of the L chiral superfields Q i . Moreover also the gauge beta
function is related to the anomalous dimensions of the fields transforming under the gauge

117

SciPost Phys. Lect. Notes 78 (2024)

symmetry [252, 253],78!
β g i ∝ − 3C2 (Gi ) −X

T (R k ) +

k∈hi

X

T (R k )γk ({λ, g})

,

(B.2)

k∈hi

where the proportionality coefficient depends on couplings, C2 (G) is the quadratic Casimir of
the adjoint, T (R) is quadratic Casimir of irrep R, and the set hk is the set of representativesof fields Q i from every non trivial irrep of Gk . Now say we have L superpotential couplings
λ and M gauge
P couplings g such that the one loop beta functions for the gauge fields vanish,
3C2 (Gi ) − k∈hi T (R k ) = 0, and the superpotentials are marginal, ni = 3. Then the vanishingof the beta functions are in general L + M equations for L + M independent variables. As suchthey will typically have only isolated solutions. However this is not the case if the equations arelinearly dependent: And this can happen easily here as they are written in terms of anomalous
dimensions corresponding to fields which might be related by symmetries. The dimensionof the conformal manifold is thus given by the number of couplings minus the number of
independent beta functions (modulo global symmetry).
As an example consider the SU(3) SQCD with N f = 9 with the marginal superpotential(29). because of the symmetry of the superpotential the anomalous dimensions of all the(anti)fundamental fields are the same. Thus all of the beta functions are proportional to thisanomalous dimension which is a function of two couplings λ and the gauge coupling. We have
one function depending on two parameters which vanishes at the origin, and thus we expect
it in general to have a line of solutions. This gives a one dimensional conformal manifold on
which some symmetry is preserved.

C

Computing Kähler quotients with Hilbert series

One can compute in principle in a straight-forward way the Kähler quotients needed for thedetermination of the conformal manifold using the Hilbert series techniques.79 Let us assume
that we have an SCFT with global symmetry (commuting with any gauge symmetry, including
anomalous symmetries) G F and that all marginal operators, including gauge couplings charged
under the anomalous symmetry, are in representation R of G F . Let χR ({a}) be the characterof R, the complex conjugate representation of R, and {a} being a set of complex parameters
corresponding to the Cartan generators of G F . We define a plethystic exponential of a functionf ({x}) ({x} is some collection of parameters),
∞

X1
n
PE [ f {x}] = exp
f ({x }) .
(C.1)
n
n=1
Then we compute the following integral,
HSMc (x) ≜I Rank
YGF
i=1



d ai
(G F )∆H aar({a}) PE x · χR ({a}) .
2πiai

(C.2)

(G )

F
Here ∆H aar({a}) is the Haar measure of G F and x is some complex parameter with |x| < 1.
The function HSMc (x) is the Hilbert series associated with the conformal manifold and assuch in particular it captures its dimension. The dimension can be extracted by taking the78
Note that for the free theory the R-charges of all the fields 32 and the one loop contribution in this formula isprecisely −3 Tr R Gi2 .
79
See e.g. [254] for uses of the Hilbert series in supersymmetric QFTs.

118

SciPost Phys. Lect. Notes 78 (2024)

limit x → 1 and is equal to the degree of divergence of HSMc (x) in this limit. By definition
the plethystic exponent generates all the symmetric products of the argument function, andthe integrations with the Haar measure project on G F invariants. Thus HSMc (x) is just given
by a sum of all the symmetrized invariants of the couplings weighed by powers of x. This sum
is generated by the independent such invariants and there might be algebraic relations. The
result thus can in general be written as a rational function in x. The denominator which is
product of terms of the form 1 − x in with ni being a power in which some singlet can appear.
The numerator is a polynomial encoding algebraic relations between the various generators.
Although this procedure is mathematically straightforward, technically when the group
G F is large, it is hard to implement. Let us give an example. Consider we have a marginal
deformation in 10 of SU(3). This is a three index completely symmetric representation. First
we compute its character. To compute this we note that the coefficient of x n in the expansionof PE [x · χR ({a})] is precisely the character of the nth symmetric power of R.80 Taking thecharacter of fundamental of SU(3) to be χ3 = a1 + a2 + 1/(a1 a2 ) we then obtain,2
a1 a2 a1 a1+++ a23 .
++++
a13 a23 a1 a22 a12 a2 a2 a1 a2 a22

(C.3)



d ai 1 Y11(1 − ai /a j ), PE x · χ10 (a1−1 , a2−1 ) =.
4
2πiai 6 i̸= j
1 − x 1 − x6
i=1

(C.4)

χ10 (a1 , a2 ) = 1 + a13 +

1

1

1

Then we compute,
I Y
2Here a3 = 1/(a1 a2 ). We find thus that we have a two dimensional conformal manifold as
the degree of divergence is two. We have one singlet in fourth symmetric product and one insixths and there are no algebraic relations involving them.
Another example we will compute here is the case of 6, two index symmetric, of SU(3).
Following same procedure as above we have,χ6 (a1 , a2 ) =1
1
1++ a1 a2 + a12 + a22 + 2 2 .
a1 a2
a1 a2

(C.5)

Then we compute,
I Y
2



d ai 1 Y
1(1 − ai /a j ), PE x · χ6 (a1−1 , a2−1 ) =.
2πiai 6 i̸= j
1 − x3i=1(C.6)

We get a single invariant which is the determinant of the two-index symmetric matrix definition of 6.

DClass S interpretation of the exercise

The theory discussed in the exercise of Section 3.4 has a class S origin. We will discuss thedefinition of this theory here. We can consider class S theory of type A2 , that is three M 5 branes
compactified on a Riemann surface. These models have N = 2 supersymmetry. The building
block of the four dimensional theories of this kind is the so called T3 model corresponding tocompactification on a sphere with three maximal punctures. The symmetry associated to every
puncture is SU(3) (as 5d compactification of A2 (2, 0) theory gives maximally supersymmetric80

The character of nth antisymmetric power is the coefficient of (−1)n x n in expansion of PE [−x · χR ({a})].

119

SciPost Phys. Lect. Notes 78 (2024)

Q

3
3

3

𝒳

2
3

5

SU(4)

Q̃

𝒴 43

1
3

W = 𝒳2Q Q̃ + 𝒴Q Q̃
Figure 62: The T3 model. On the1 right we have a N = 1 non-conformal Lagrangian3
flowing to T3 obtained in [82]. (See[255] for a different
Lagrangian description.)2Q̃
3 of SU(4), while Y is adjoint
The fields X form a doublet of Q
two-index
antisymmetrics
q SU(5)×q̃
plus a singlet of SU(5). The Lagrangian has manifest
U(1)×SU(2) subgroup
2
SU(4)
SU(3)
3emerge in the IR.
of E6 manifest
in
the
UV
.
The
U(1)
symmetryisconjectured
to
t
𝒳
3

H 4
3
H′ 43

SU

(2

)

YM theory with gauge group SU(3)) and for the ˜
T3 model the symmetry enhances to E6 such
𝒴′ 𝒴′
that,
4
Q′1, 1)Q̃ ′+ (1, 8, 1) + (1,3 1, 8) + (3, 3, 3) + (3̄, 3̄, 3̄) ,78 E6 → (8,
(D.1)
1
2
3
where on the right we have the decomposition in terms of SU(3)3 symmetry. This theory has
an additional U(1) t symmetry in N = 1 language
𝒴 43 coming from the extended R-symmetry of
N = 2. The T3 theory has relevant operators M with N = 1 R-charge 34 , U(1) t charge +1,
and in 78 E6 : These are the moment map operators. In addition to these there is a marginal
˜ Q̃ ′ + Q Q̃ qq̃ + HQ Q̃ + H′Q′Q̃ ′WCoulomb= 𝒳2(QbranchQ̃ + Q′
Q̃ ′) + 𝒴Q′
Q̃ U(1)′ + Q𝒴′Q′ + Q̃ 𝒴′
operator
X witht charge −3 and R-charge 2. These are the only marginaland relevant operators of T3 . One constructs more general theories corresponding to surfaces
with maximal punctures by combining together T3 models gauging diagonal combinations ofpuncture SU(3) symmetries with N = 2 vector multiplets. In N = 1 language in addition to
the N = 1 vector multiplet one adds a chiral adjoint superfield Φ coupling through a superpotential to the moment maps of the glued punctures, W = ΦM − ΦM ′ . This discussion is the(2, 0) version of the Φ-gluing.
An additional building block of theories in this class is the free trinion, which can be obtained from T3 by an RG flow giving a vacuum expectation value to one of the SU(3) moment
maps breaking the SU(3) down to a U(1). We thus obtain a theory corresponding to a spherewith two maximal SU(3) punctures and one U(1) puncture. The theory is just a bifundamental hypermultiplet of two SU(3) symmetries with the U(1) symmetry charging oppositely thetwo half-hypermultiplets. Let us denote these half-hypermultiplets q and eq.
Gluing the T3 theories together the resulting models have the N = 2 preserving conformal
manifolds parametrized by the N = 2 gauge couplings used to glue the spheres together. If
the surface has at least one minimal puncture one can build also N = 1 preserving conformal
manifolds on which the U(1) t symmetry and the symmetry of one minimal puncture is broken. We go to a duality frame in which we glue the free trinion to a generic class S theory oftype A2 . Then, let us analyze the conformal manifold. The marginal operators involving theglued puncture operators and the free trinion fields are: The N = 2 gauging (which is exactly
marginal), the Coulomb branch operator Φ3 charged −3 under U(1) t , and the baryons q3 andeq3 charged + 32 under U(1) t and ±1 under the minimal puncture U(1) by definition. Thus wecan build a Kähler quotient as (Φ3 )(q3 )(eq3 ) is not charged under any symmetry. This adds a
one dimensional N = 1 preserving conformal manifold. Moreover as U(1) t is now broken thedimension three Coulomb branch operators coming from the adjoint chirals involved in gaug120

SciPost Phys. Lect. Notes 78 (2024)

2
3

3

q

+1

3

1

q̃

3

2
3

3

𝒳

S

−1

W
Figure 63: The free trinion model corresponding to a sphere with two maximal andone minimal punctures.

1
3

SU

(2

)

Q Q̃
ings used to construct the theory we glued to the free trinion become also exactly marginal.
q
Finally in special cases the conformal manifold is even richer. Let us consider
oneSU(4)T3 glued2
SU(3)4
to one free trinion. We obtain a sphere3 𝒳 punctures. The
3 with three maximal and one minimal
H
17
43
conformal anomalies of this theory are a = 15
4 and c = 4 . As in the general discussion above H′
1
we have exactly marginal deformations corresponding
to the N = 2 gluing, X , and the N = 1 3
˜
𝒴′ 𝒴′
3
4
deformation involving the baryons. However here we have additional deformations. We can
Q′ Q̃ ′
3
1
take the component of the E6 moment3 map charged (3, 3, 3) and contract it with eq building
a 2
3gauge invariant operator H in (3, 3, 3) of the three maximal punctures of the four punctured
sphere which is marginal. In a same manner by contracting (3̄, 3̄, 3̄) of the moment map
𝒴 43
e in representation (3̄, 3̄, 3̄) of the three maximal puncturewith q we get a marginal operator Hesymmetries. In fact as H H is a singlet of all the symmetries we have additional directions of the˜
2 2(Q Q̃ + Q′Q̃ ′) + 𝒴Q′Q̃−2=𝒳
conformal manifold on which SU(3)3 is broken to SU(2)3 ×W
U(1)such that 3i → 2i ai + 1ai ′ +, Q𝒴′Q′ + Q̃ 𝒴′Q̃Q
38i → 3 + 2(ai3 + ai−3 ) + 1, and i=1 ai = 1. On this conformal manifold we have operators in
representation,
4 (1, 1, 1)0,0 + 2 (2, 2, 2)0,0 + (1, 2, 2)±3,0 + (2, 1, 2)0,±3 + (2, 2, 1)±1,±1 .

(D.2)

We can next turn on the last deformations breaking the two U(1)s to a diagonal one and locking
the two SU(2) symmetries onto each other. This adds another dimension to the conformal
manifold and leaves us with SU(2)2 × U(1) symmetry and marginal deformations,
5 (1, 1)0 + 2 (2, 3)0 + 2 (2, 1)0 + 2 (2, 2)±3 + (3, 1)0 .

(D.3)

We can break the first SU(2) to the Cartan using the last deformation preserving SU(2)×U(1)2
with marginals being,
6 10,0 + 2 30,±1 + 2 10,±1 + 2 2±1 3,±2 1 .

(D.4)

Next we break the Cartan U(1) using the next to last operators preserving SU(2) × U(1) andhaving marginal operators in,
9 10 + 4 30 + 4 2±3 .

(D.5)

We use the second operator to break the remaining SU(2) to the Cartan preserving U(1)2 andhaving marginal operators in,
13 10,0 + 3 10,±2 + 4 1±1 3,±2 1 .

(D.6)

Next second type of the operators is turned on to break the second U(1) with only one U(1)preserved and marginal operators being,
18 10 + 8 1±3 ,
121

(D.7)

Q
𝒳

2
3

Q̃
1
3

SciPost Phys. Lect. Notes 78 (2024)

W = 𝒳2Q Q̃ + 𝒴Q Q̃

Q
2
3

3

𝒳

SU(4)

Q′ Q̃ ′

Q̃

q
SU(3)

q̃

3

4
3

2

SU

2
3

˜
𝒴′ 𝒴′

(2
)

1
3

3

1
3

H 4
3
H′ 43

1

3

𝒴 43

5

SU(4)

𝒴 43

˜ Q̃ ′ + Q Q̃ qq̃ + HQ Q̃ + H′Q′Q̃ ′W = 𝒳2(Q Q̃ + Q′Q̃ ′) + 𝒴Q′Q̃ ′ + Q𝒴′Q′ + Q̃ 𝒴′

Figure 64: The theory corresponding to a sphere with three maximal punctures andone minimal puncture obtained by gluing to a T3 the free trinion. On the right wehave a non-conformal description of the theory using the Lagrangian of [82] for the
T3 . The fields H and H ′ are gauge singlets, and Y is adjoint of SU(2). One maximal
puncture SU(3) and the minimal puncture U(1) are manifest in the UV, while theremaining two maximal puncture SU(3) symmetries emerge in the IR enhancing an
SU(2)2 × U(1)2 symmetry of the Lagrangian. The U(1) t symmetry is conjectured toemerge in the IR. One can further turn on marginal deformations consistent with the
R-symmetry assignment in the Figure breaking completely all the global symmetry.
and at last the second operators can be used to construct additional 15 exactly marginal operators breaking all the symmetry to obtain 33 dimensional conformal manifold.
The relevant deformations of the model are the moment maps of the two maximal punctures of the T3 , the operator Tr Φ2 , and qe
q: This gives rise to 8 + 8 + 1 + 9 = 26 relevant
operators. The conformal Lagrangian we derive in the exercise exactly reproduces the features of this strongly coupled model. We can compute the supersymmetric index either using
the theory of Figure 64,(q; q)5 (p; p)5IA =3! × 4!
×4 YΓe ((qp)

1
6

I Y
2

dzi
2πizi
i=1
4 
2 Y

u±1
a )

a=1

I Y
3

dua
2πiua
a=1

Q3

3 
2
1
2
Γe ((qp) 3 zi±1 )Γe ((qp) 3 zi±1 )
Q3
Q4
Γ
(z
/z
)
e
i
j
i̸= j
a̸= b Γe (ua /u b )


i=1

(D.8)

4 Y
3 
2 Y

5
1
2
Γe ((qp) ua u b )Γe ((qp) 6 (ua /zi )±1 ) Γe ((qp) 3 ) ,1
3

a=1 i=1

a<bor the one of Figure 21,
6 
2
Q2 
1
1
I 2
I
3 ±1
Γe ((qp) 3 zi±1 u±1 )(q; q)3 (p; p)3 Y dzi
dui=1 Γe ((qp) zi )
IB =
(D.9)
8×2
2πizi
2πiu
Γe (u±2 )Γe (z1±2 )Γe (z2±2 )Γ2 (z1±1 z2±1 )
i=1

4 
3 
 

2 
1
1
1
1
1
× Γe ((qp) 3 u±1 )Γe ((qp) 3 z1±1 z2±1 )Γe ((qp) 3 ) Γe (qp) 3 z1±2 Γe (qp) 3 z2±2 ,and obtain the same result,
2

1

2

1

IA = IB = 1 + 26(qp) 3 − (q + p)(qp) 3 + 33qp + 25(q + p)(qp) 3 + (348 − q2 − p2 )(qp) 3 + · · · .
(D.10)
We do not have a proof of the equality but have checked it to the stated order in expansion
in q and p. As a side remark, we can also compute the so called Schur limit of the index[180, 181]. The Schur index of the N = 2 theory is a limit of the supersymmetric index
122

SciPost Phys. Lect. Notes 78 (2024)

1

computed using N = 1 superconformal R-symmetry by setting p = q 2 [22, 154]. This index
has an alternative description using the geometric data, A2 theory on a sphere with threemaximal and one minimal puncture in our case. The expression is [181],∞ λ1 (dim(λ1 , λ2 )) dim 1 (λ1 , λ2 )(q2 ; q)2 (q3 ; q) X X
q2
IC =
.
3
(dimq (λ1 , λ2 ))2(q; q)25 (q 2 ; q)2 λ =0 λ =0
3

1

2

Here,
q−λ1 (1 − q1+λ2 )(1 − q2+λ1 )(1 − q1+λ1 −λ2 ),(1 − q2 )(1 − q)21
dim(λ1 , λ2 ) = lim dimq (λ1 , λ2 ) = (1 + λ2 )(2 + λ1 )(1 + λ1 − λ2 ) .
q→1
2

dimq (λ1 , λ2 ) =(D.11)The sum here is over the finite dimensional representations of A2 ; three factors of dim(λ1 , λ2 )in the numerator come from maximal punctures and the factor of dim 1 (λ1 , λ2 ) comes fromq2

the minimal puncture; the power of dimq (λ1 , λ2 ) in the numerator is 2g−2+s = 2×0−2+4 = 2
with g the genus and s the number
The overall factor involving the qQ∞ of punctures.
jPocchhammer symbols ((z; q) =
(1
−
zq
))
also
is
detemined by the geometric data.
j=0
Using these expressions and either one of the Lagrangians we obtain in the Schur limit,
1

1

IC = IA(p = q 2 ) = IB (p = q 2 ) = 1 + 25q + 56q3/2 + 403q2 + 1348q5/2 + 5844q3(D.12)+ 19756q7/2 + 70702q4 + 225640q9/2 + 719961q5 + 2155220q11/2 + 6338818q6 + · · · .
Here we do not have a proof of the equality but have checked it to the stated order in expansion
in q. These are additional indications that the proposed duality is correct.

E

Basics of 6d SCFTs

In this appendix we will review some aspects of 6d N = (1, 0) theories. First, we note that due
to the fact that spinors are pseudo real in 6d, supercharges of different chiralities are inequivalent. Thus, we have several possible supersymmetries including the chiral N = (1, 0), (2, 0)with eight and 16 real supercharges and the non-chiral N = (1, 1) with 16 real supercharges.
The N = (1, 0), (2, 0) can be extended to include also conformal symmetry to superconformal algebras [85]. For a detailed discussion of 6d superconformal algebra we refer the reader
to [86] as well as the more modern exposition in [143].
Let us summarize the basic facts we need for our discussions. The (2, 0) algebra is a special
case of (1, 0) and we will focus only on the latter. Six dimensional N = (1, 0) theories haveeight real supercharges and contain the following field multiplets,• Hypermultiplet: The bosonic part of the hypermultiplet contains four real scalar (or twocomplex scalars q, eq† transforming in the doublet of SU(2)R ) that parameterize the Higgs
branch of vacua. In addition, its fermionic part contains a Weyl fermion ψα transforming
as a spinor of SO(5, 1).
• Vector multiplet: The bosonic part of the vector multiplet contains only a vector field
A transforming as a vector of SO(5, 1); therefore, these theories don’t have a Coulomb
α̇branch of vacua. Its fermionic part contains a single Weyl fermion λ transforming as aco-spinor of SO(5, 1) and the doublet of SU(2)R .

123

SciPost Phys. Lect. Notes 78 (2024)• Tensor multiplet: The bosonic part of the tensor multiplet contains a two-form B with aself dual field strength H = d B, H = ⋆H. In addition there is a real scalar φ that parameterizes the tensor branch of vacua. Both the scalar and the two-form are invariant under
the R-symmetry. The fermionic part contains a single Weyl fermion χα transforming asa spinor of SO(5, 1) and the doublet of SU(2)R .
All Lorentz invariant Lagrangians we can write are IR free in 6d. Nevertheless mainly due
to string theoretic constructions there is strong evidence that 6d interacting SCFTs exist [89](see [10] for a review). Let us consider a gauge theory with a simple gauge group, hyper
multiplets, and tensor multiplets. The standard kinetic term for the gauge fields,Z
1
tr (F ∧ ⋆F ) ,(E.1)
g Y2 M
in 6d implies that the gauge coupling g Y M scales as length, meaning this theory is trivial in
the IR. Regarding tensor multiplet it is problematic to write the kinetic term for the self dual
field strength H as,ZZZ
H ∧ ⋆H =H∧H =−

H∧H,

(E.2)

implying the simple kinetic term vanishes. Nevertheless, we can treat this theory as if it hasa Lagrangian describing the interactions with the constraint H = ⋆H. In this description thescalar in the tensor multiplet can couple to the gauge field as,Zc

φ tr (F ∧ ⋆F ) ,(E.3)

with constant c. This term allows us to absorb the gauge coupling into a vacuum expectation
value of φ with the effective gauge coupling of the theory given by,
1
g e2f f

= c〈 φ 〉 ,(E.4)

meaning the gauge coupling as well as the BPS instanton tension are controlled by the tensor
modulus and are non-vanishing on the tensor branch. The interpretation of these observations,mainly coming from string theoretic constructions, is that the underlying UV theory is a 6d
SCFT that has been deformed by a non-zero vev to the scalar in its tensor multiplet. Thisinitiates an RG flow leading at low-energies to a 6d gauge theory with inverse coupling squared
proportional to the size of the vev.

E.1

The Green-Schwarz mechanism in 6d

The Green-Schwartz mechanism in 6d arises from the interaction term coupling the tensor
multiplet B and the field strength FZc

B ∧ tr (F ∧ F ) .

(E.5)

This term is implied form the coupling of φ in (E.3) and the fact that φ is a part of the tensor
multiplet. The related B equation of motion is81d ⋆ H = c tr (F ∧ F ) .
81

(E.6)

This equation of motion can be derived using the interaction term together with the kinetic term of H as if itexists.

124

SciPost Phys. Lect. Notes 78 (2024)

Using the self-duality constraint then modifies the Bianchi identity of H asd H = c tr (F ∧ F ) ,(E.7)

meaning the instantons are charged under the two-form B. Using the the Dirac quantization
condition for higher forms one finds that c 2 must be quantized.
It’s important to note that the B-field is not invariant under gauge transformations. It ise = d B − c I3 (A, F ) to be gauge invariant where I3 is the Chern-Simons 3-form.
required from H
The descent equations together with the modification of the Bianchi identity gived I3 = I4 = tr (F ∧ F ) ,δI3 = d I21 .

(E.8)

e requires
We find that the invariance of Hd(δB) = c δI3 = c d I21 ,(E.9)

meaning δB = −c I21 . The contribution of the modified Bianchi identity to the anomaly polynomial eight-form is then given byI6 ≡ δ (c B ∧ tr (F ∧ F )) = c δB ∧ tr (F ∧ F ) = c 2 I21 ∧ tr (F ∧ F ) ,I7 ≡ d I6 = c 2 d I21 ∧ tr (F ∧ F ) = c 2 δI3 ∧ tr (F ∧ F ) ,I8 ≡ d I7 = c 2 d I3 ∧ tr (F ∧ F ) = c 2 δI4 ∧ tr (F ∧ F ) = c 2 (tr (F ∧ F ))2 .

(E.10)

This means that gauge theories with non-vanishing quadratic part of the 1-loop anomaly polynomial can be made gauge anomaly free by the addition of a tensor multiplet as long as thecoefficient of the quadratic part is negative definite. In addition, since c 2 needs to be properly
quantized, so is the coefficient of the quadratic part of the 1-loop anomaly polynomial.

E.2

The 6d anomaly polynomial eight-form

One way to compute the 6d anomaly polynomial is by considering the description of the 6dtheory on its tensor branch. Such descriptions are given by specifying the gauge symmetry(with the various GS terms) and the matter content which can consist of three types of multiplets: Vectors, hypermultiplets, and tensor multiplets. Let us here describe the contributionsof each of the matter multiplets. As usual to do so we will need to identify the contributionsof the Weyl fermions residing in each multiplet.
The computation proceeds as follows.82 The contribution of the Weyl fermion in 6d to theanomaly polynomial can be shown to be given by the 8-form part ofÂ(T ) ch (B) .

(E.11)

Here Â(T ) is the so called Dirac A-roof genus. This has the following general structure,
p1 (T ) 7p1 (T ) − 4p2 (T )
+
+ ...
24
5760
2

Â(T ) = 1 −(E.12)

The characteristic classes appearing in the expression, pi (T ), are the i-th Pontryagin classesof the tangent bundle T . Concretely, the first two Pontryagin classes are given by

 
2
4
1
2
4
2 2
trR
−
p1 (T ) = −
trR
,
p
(T
)
=
−
trR
.
(E.13)
2
2
(4π)4
(4π)2
82

See e.g. [30, 94, 256] for more comprehensive discussion. We mainly follow the notations of [117].

125

SciPost Phys. Lect. Notes 78 (2024)

These are given in terms of the curvature two form Rab= 12 Rabµν d x ν d x µ defining trR2n=Raa12 . . .R c2n1 .
Finally, ch (B) appearing in (E.11) is the Chern character of the entire bundle for all the gauge
and global symmetries that are not included in the Dirac A-roof genus. The Chern charactercan be expanded in terms of the B bundle Chern classes as followsach (B) = rank (B) + C1 (B) ++C12 (B) − 2C2 (B)2+C13 (B) − 3C1 (B) C2 (B) + 3C3 (b)6

C14 (B) + 4C1 (B) C3 (B) − 4C12 (B) C2 (B) + 2C22 (B) − 4C4 (B)24Here Cn (B) are the n-th Chern classes of the bundle B defined by,


X
itF
k
Cn (B) t = det+I2πn= I +i+ ··· .

(E.14)(E.15)

trF
trF 2 − (trF )2 2
2trF 3 − 3trF trF 2 + (trF )3 3t+t−it + ...
(2π)
2 (2π)26 (2π)3

Here F is the curvature form of B. The Chern character obeys
ch (B1 ⊕ B2 ) = ch (B1 ) + ch (B2 ) ,ch (B1 ⊗ B2 ) = ch (B1 ) ch (B2 ) .

(E.16)We are then ready to write the polynomial for each one of the three types of multiplets.
First, we need to understand the global symmetries of the 6d (1, 0) SCFTs. These include
the SU(2)R R-symmetry and global symmetry which we will denote by F6d . In addition wewill also need the contribution of the gauge symmetries, which we will denote by G6d . Thus,
according to (E.16) we decomposech(B) = ch(R) ch(G6d ) ch(F6d ) .
Let us start with the hypermultiplet H. This is specified by its representation under the
global symmetry, r F6d , and the gauge symmetry, rG6d . The Weyl fermion residing in the hypermultiplet is a singlet of the 6d R-symmetry. The anomaly polynomial is then given by,
7p1 (T )2 − 4p2 (T ) d rF6d 
I8H = d rG d rF+
C2 (G6d )2r − 2C4 (G6d ) rG
G6d
6d
6d
6d
5760
12


p1 (T ) C2 (G6d ) rG
p1 (T )
6d
+ d rF− d rGC1 (F6d )2r − 2C2 (F6d ) rF
F6d
6d
6d
6d
24
48

 1
1
− C2 (G6d ) rG
C1 (F6d )2r − 2C2 (F6d ) rF
+ C1 (F6d ) rF C3 (G6d ) rG
F6d
6d
6d
6d
6d
2
2

d rG
6d
C1 (F6d )4r + 4C1 (F6d ) rF C3 (F6d ) rF − 4C1 (F6d )2r C2 (F6d ) rF
+
F6d
F6d
6d
6d
6d
24

+2C2 (F6d )2r − 4C4 (F6d ) rF.
(E.17)
F6d

6d

The rB index of the Chern classes indicates the representation in which the corresponding
traces in (E.15) are computed, while d rB is the dimension of rB . Note that all the gauge groups
in all our constructions are simple and thus the first Chern class is always zero for us.
Second, let us discuss the vector multiplet V . The Weyl fermion in this multiplet is now
a doublet of the 6d R-symmetry and its chirality is opposite to the one in the hypermultiplet(and the one in the tensor multiplet to be discussed next). We then compute the polynomial
to be,
C2 (R)22
7p1 (T )2 − 4p2 (T )1 −
C2 (G6d )2Ad j − 2C4 (G6d )Ad j − dAd j
5760
12
24
p1 (T ) C2 (G6d )Ad j
p1 (T ) C2 (R)2 C2 (R)2 C2 (G6d )Ad j−− dAd j−.
(E.18)
24
48
2

I8V = −dAd j

126

SciPost Phys. Lect. Notes 78 (2024)

Note that here the Chern classes for the gauge bundles are computed in adjoint representation.
In addition, note that since we have a single Weyl fermion in the doublet of SU(2)R and not
two Weyl fermions that form a doublet the contribution is divided by 2.
Finally, we compute the contribution of a tensor multiplet T . The tensor contains a single Weyl fermion in the doublet of the R-symmetry with the same chirality as the one in thehypermultiplet.83 The tensor multiplet also contains a self dual tensor which is chiral. Suchchiral fields in general also contribute to gravitational anomaly [257]. Collecting all the contributions we obtain,
I8T =23p1 (T )2 − 116p2 (T ) C2 (R)2 p1 (T ) C2 (R)
+
+
.
5760
24
48

(E.19)As discussed in the previous sub-section to specify the anomaly polynomial we will also needto account for the contributions of the Green-Schwartz terms.

E.3

Example: The rank one E-string anomaly polynomial

Here we look at the simple, but important example of the rank one E-string theory anomaly
polynomial. The rank Q E-string theory is the low-energy theory on Q M5-branes on top of
the end-of-the-world E8 brane. It was shown in [94] that the anomaly polynomial of the rank
one E-string theory is given by
E−st r ing

I8

=

11
1
1
13
C2 (R)22 −C2 (R)2 p1 (T ) −C2 (R)2 C2 (E8 )248 +C2 (E8 )2248
24
48
60
7200
29 (7p1 (T ) − 4p2 (T ))1+
p1 (T ) C2 (E8 )248 +.
(E.20)
240
5760

The tensor branch description of this theory contains only one tensor multiplet. Thus, we canwrite this anomaly polynomial as the contribution of a single tensor multiplet and a GreenSchwartz term involving global and spacetime symmetries [258],
E−st r ing

I8F= I8t ensor +


2
1
1
1
C2 (R) −C2 (E8 )248 + p1 (T ) .
2
60
4(E.21)

Twisting, fluxes, and integrating anomaly polynomials

In this appendix we will show how one can compactify 6d SCFTs to 4d theories preserving half
the supersymmetry. In general in order to preserve all the supersymmetry when compactifying a supersymmetric theory the compactification surface needs to be flat. Another option topreserve supersymmetry is to perform a topological twist that preserves part of the supersymmetry.

F.1

The topological twist and compactification

Topological twisting [259] of a d dimensional supersymmetric theory is performed by a choiceof subgroup of the rotation symmetry SO(d) and mapping it into the R-symmetry group. Then
given a choice of a supersymmetry generator Q which is invariant under the combined actionof SO(d) rotations and the R-symmetry we can consider the subspace of observables which
are in the kernel of Q. This subspace defines a topological field theory.
83

As in the vector multiplet case we will need to divide its contribution by 2.

127

SciPost Phys. Lect. Notes 78 (2024)

In our context we will use topological twisting on the compactifying surface to preserve part of the supersymmetry.
We decompose the supersymmetry spinors underSO(d − 1, 1) L × GR → SO(d − p − 1, 1) L × SO(p) M × GR and look for a choice of subgroup
inclusion of SO(p) M into GR that will leave some of the SO(d − p − 1, 1) L spinors invariant
under the new twisted SO(p) symmetry. Specifically for compactifications of 6d N = (1, 0)
theories with OS p(6, 2|2) superconformal group the bosonic subgroup is the 6d N = (1, 0) conformal group times US p(2) ≃ SU(2) R-symmetry. In particular, the supercharges Q transformas (4, 2) under SO(5, 1) L × SU(2)R . Decomposing the supercharge in terms of 4d symmetries
we obtain,
SO(5, 1) L × SU(2)R → SO(3, 1) L × SU(2)R × SO(2) M ,Q : (4, 2) → (2, 1; 2)+ 1 ⊕ (1, 2; 2)− 1 .
2

(F.1)

2

In this case in order to preserve half the supersymmetry we break SU(2)R → U(1)R and thentwist the SO(2) M = U(1) M as U(1) M ′ = U(1) M − U(1)R . Under the original decomposition
and the twist we findSO(5, 1) L × SU(2)R → SO(3, 1) L × U(1) M ′ × U(1)R ,Q : (4, 2) → (2, 1)+1,+ 1 ⊕ (2, 1)0,+ 1 ⊕ (1, 2)0,− 1 ⊕ (1, 2)−1,− 1 .
2

2

2

2

(F.2)

Thus, the preserved supercharges are Q = (2, 1)0,+ 1 with conjugate Q† = (1, 2)0,− 1 which22means we preserve N = 1 supersymmetry. The 4d R-symmetry is identified with twice U(1)R .
When compactifying the 6d (1, 0) theories to 4d there are additional knobs and levers one
can utilize due to the fact that many (1, 0) theories have flavor symmetries. Specifically we
can turn on background configurations for gauge fields for the flavor symmetries supported
on the Riemann surface in a way that preserves N = 1 supersymmetry in 4d. One can demonstrate this for example by considering a compactification of a free hypermultiplet in 6d on atorus. Since this is a free field theory the computation can be performed very explicitly. Specifically, one reduces the theory first to 5d on a circle with a holonomy for the global symmetryaround the compactification circle that varies along the fifth dimension. Such a configuration
amounts to turning on flux for the global symmetry supported on the torus and allows analyzing explicitly the preserved supersymmetry. This analysis was performed by Chan, Ganor,and Krogh in [260] (Section 2), to which we refer the reader interested in the details of suchcomputations.

F.2

Integrating the anomaly polynomial

When considering the general compactification case on a genus g Riemann surface with non
trivial fluxes to abelian subgroups of the 6d flavor symmetry we need to break the flavor
symmetry as G → U(1)k × H with H a subgroup of G commuting with all the U(1)s we give
flux to.
Decomposing the group means we need to decompose a representation of the group R thatappears in the 6d anomaly polynomial as Ci (G)R , to the representations of U(1)k × H.
X  (q i ,...,q i )
1
k
Rg →
Rhi
,
(F.3)
i

where Rhi and qai are the i-th summand representation of H and charge of U(1)a , respectively,in the decomposition sum.
Writing the Chern character of the 6d global symmetries up to eight forms using (E.14) we
get


1
1 
ch (gR ) = dim Rg − C2 (g)Rg + C3 (g)Rg +C2 (g)2R − 2C4 (g)Rg ,
(F.4)
g
2
12
128

SciPost Phys. Lect. Notes 78 (2024)

where we set C1 (g)Rg = 0 as the global symmetries we are interested in 6d have semisimple
Lie algebras. Using the properties in (E.16) we can write the Chern character after the above
decomposition asch (gR ) → ch


M



=

u(1)q i ⊗ · · · ⊗ u(1)q i ⊗ hR i
1

i

h

k

X

k
 Y


ch hR ich u(1)qai
h

a=1

i




X
 
1
1
2
i
=
C2 (h)R i − 2C4 (h)R i + . . .
dim Rh − C2 (h)R i + C3 (h)R i +
h
h
h
h
2
12
i
k X
∞ 
Y
n 
1 i
×
,
(F.5)
qa C1 (u(1)a )n!
a=1 n=0
where C1 (h)R i = 0 as H has a semisimple Lie algebra. Comparing forms of equal dimensionhwe find how the Chern classes decompose
C2 (g)Rg→ −C3 (g)Rg

→

k
X
 
1 X X
dim Rhi qai q ib C1 (u(1)a ) C1 (u(1) b ) +C2 (h)R i ,h
2 a,b=1 i
i

k
 
1 X X
dim Rhi qai q ib qci C1 (u(1)a ) C1 (u(1) b ) C1 (u(1)c )3 a,b,c=1 i−2C4 (g)Rg

→ −

k X
X

qai C1 (u(1)a ) C2 (h)R i +X

C3 (h)R i ,
h

h

a=1 i
k
X

i

X
 
1
dim Rhi qai q ib qci qdi C1 (u(1)a ) C1 (u(1) b ) C1 (u(1)c ) C1 (u(1)d )4 a,b,c,d=1 i

k X
X
 
1
+3
1 − dim Rhi qai q ib C1 (u(1)a ) C1 (u(1) b ) C2 (h)R ih6a,b=1 i−3

k X
X

qai C1 (u(1)a ) C3 (h)R i −
h

a=1 i

+

k
1 X X

1
−
2
2 a,b=1X
1X
C2 (h)2R i +C4 (h)R i
h
h
2 i
i
!2





dim Rhi qai q ib C1 (u(1)a ) C1 (u(1) b ) +X

C2 (h)R ih. (F.6)ii

Next we wish to fix the flux on the compactification Riemann surface Σ g of genus g for theselected U(1) subgroups. Thus we set
Z
C1 (U(1)a ) = −za .

(F.7)

Σg

In addition, we perform the aforementioned topological twist which translates to setting the
R-symmetry Chern class as t 2C2 (R) → − C1 R′ −
,
2

(F.8)

where t is a two form related to the compactification surface symmetry and integrates toZt = 2(1 − g) .

Σg

129

(F.9)

SciPost Phys. Lect. Notes 78 (2024)

We also set the Pontryagin classes
p1 (T6d ) → p1 T4d + t 2 ,
p2 (T6d ) → p2 T4d + p1 T4d t 2 ,(F.10)and to meet the flux constraint
C1 (U(1)i ) → −zit+ εi C1 (U(1)R ) + C1 U(1) Fi ,2(1 − g)(F.11)

where the first term gives the required flux, the second term takes into account possible mixingof the U(1) symmetry with the superconformal R-symmetry, where ε is a parameter to be
determined via a-maximization [76], and the third term is the 4d curvature term of the U(1)symmetry. Note that when we compactify the theory on a generic Riemann surface, only terms
linear in t will contribute to the 4d anomaly polynomial six-form.84
Finally, with the 4d anomaly polynomial at hand one can calculate the anomalies of various
symmetries. For U(1) symmetries one can extract the anomalies from the anomaly polynomialusing the relations
Tr (U(1) x ) = −24∂C1 (U(1) x ) ∂ p1 (T4 ) I6 ,
Tr U(1) x U(1) y U(1)z = ∂C1 (U(1) x ) ∂C1 (U(1) y ) ∂C1 (U(1)z ) I6 .
For mixed or cubic non-abelian anomalies one finds the relations

Tr G 3 = 2Cr ∂C3 (G)r I6 ,
Tr G 2 U(1) x = −Tr ∂C1 (U(1) x ) ∂C2 (G)r I6 ,(F.12)(F.13)

where Tr and Cr denote the Dynkin and cubic index of the representation r, respectively. These
relations can be derived by comparing the contribution of Weyl fermions to the anomalies on
one side and to the anomaly polynomial on the other side.

F.3

ExamplesHere we will examine the E-string theory and its compactification possibilities preserving 4d
N = 1 supersymmtry discussed in this appendix. We start with a simple compactification ona genus g Riemann surface with no fluxes. We first write the full 6d anomaly polynomial fora rank Q E-string theory
Q 4Q2 + 6Q + 3 2(Q − 1) 4Q2 − 2Q + 1 2
E−st r ing
I8=
C2 (R) +C2 (L)24 
24
Q Q2 − 1(Q − 1) (6Q + 1)−
C2 (R) C2 (L) −C2 (L) p1 (T )348
Q (6Q + 5)
Q (Q − 1)−
C2 (R) p1 (T ) +C2 (L) C2 (E8 )248
48
120
Q (Q + 1)
Q−
C2 (R) C2 (E8 )248 +p1 (T ) C2 (E8 )248120
240
7p12 (T ) − 4p2 (T )Q+
C2 (E8 )2248 + (30Q − 1),
(F.14)
7200
5760
where in our notations Ci (G)rG denoted the i-th Chern class of the G-bundle in the rG representation, C2 (R) ≡ C2 (R)2 and C2 (L) ≡ C2 (L)2 , and pi (T ) is the i-th Pontryagin class of thetangent bundle.
84
Terms with higher powers of t can still contribute to other anomalies. Notable examples include: Anomalies
in symmetries originating from isometries of the Riemann surface [137], and anomalies associated with couplingson the conformal manifold [261].

130

SciPost Phys. Lect. Notes 78 (2024)

Since we wish to compactify on a genus g Riemann surface we need to use a topologicaltwist to preserve half the supersymmetry as shown previously. This translates to the assignment t 2C2 (R) → − C1 R′ −.
2
In addition, we need to set the Pontryagin classes
p1 (T6d ) → p1 T4d + t 2 , p2 (T6d ) → p2 T4d + p1 T4d t 2 ,(F.15)(F.16)

and finally setZt = 2(1 − g) .

(F.17)

Σg

Effectively this means that in the resulting 4d anomaly polynomial only the terms linear in t
survive. This results in
Q 4Q2 + 6Q + 3
Q (6Q + 5)
E−st r ing(g − 1) C13 (R) +(g − 1) C1 (R) p1 T4
=
I6
6
24

2
2Q Q − 1
Q (Q + 1)+(g − 1) C1 (R) C2 (L) +(g − 1) C1 (R) C2 (E8 )248 .
360(F.18)
From the anomaly polynomial we can extract that the only nontrivial anomalies are
Tr (U(1)R ) = −Q (6Q + 5) (g − 1) ,Tr U(1)3R = Q 4Q2 + 6Q + 3 (g − 1) ,
2Q Q −1
Q (Q + 1)
Tr U(1)R SU(2)2L = −(g − 1) ,Tr U(1)R E82 = −(g − 1) .
3
2(F.19)In the next example we will again compactify the E-string theory only this time on atorus with flux preserving a U(1) × E7 subgroup of the E8 group. This requires breaking
E8 → E7 × U(1) using the branching rule
248 → 133 ⊕ 56+1 ⊕ 56−1 ⊕ 1+2 ⊕ 1 ⊕ 1−2 .
This translates to the following Chern class assignments using (F.6)

C2 (E8 )248 → C2 E7 133 + 2C2 E7 56 − 60C1 (U(1))2 = 5C2 E7 56 − 60C1 (U(1))2 ,(F.20)(F.21)

where in the second equality we used the relation implied by (F.13). Since we are dealing witha torus, no topological twist is required, but due to the flux we still break half the supersymmetry. This is consistent with the assignments we used in the former example when taking
g = 1. In addition we need to set in accordance with the flux
C1 (U(1)) → −zt
+ εC1 (U(1)R ) + C1 (U(1) F ) ,2(1 − g)where here we will take the limit of g → 1 after the compactification.


 
Q
Q
E−st r ing
I6= z (C1 (F ) + εC1 (R))
p1 T4 + Q (Q − 1) C2 (L) + C2 E7 56
2
62+ Q (Q + 1) z (C1 (F ) + εC1 (R)) C1 (R) − 2Qz (C1 (F ) + εC1 (R))3 .

131(F.22)(F.23)

SciPost Phys. Lect. Notes 78 (2024)

In this case since we have an additional U(1) flavor symmetry which can mix with the Rsymmetry in the IR fixed point. The mixing can be calculated using a-maximization, where
the trial a-anomaly is
a t r ial =
318Qε Q − 2ε2 + 1 z + 12Qεz .
32(F.24)

This anomaly is maximized forε=v
t 3Q + 5
18

sign (z) .

(F.25)For our final example we will consider a compactification on a genus g Riemann surface
with flux preserving U(1)×E7 ⊂ E8 . In this case we use the same process as the former example
only without setting the genus to one. The resulting anomaly polynomial setting the mixing
coefficient to zero for brevity is
Q 4Q2 + 6Q + 3
Q (6Q + 5)
E−st r ing=(g − 1) C13 (R) +(g − 1) C1 (R) p1 T4
I6
6
24


2Q Q2 − 1
Q (Q + 1)+(g − 1) C1 (R) C2 (L) +(g − 1) C1 (R) C2 E7 56
3122+ Q (Q + 1) z t C1 (t) C1 (R) + Q (Q + 1) (g − 1) C12 (t) C1 (R) − 2Qz t C13 (t)QQ+ z t C1 (t) p1 T4 + Q (Q − 1) z t C1 (t) C2 (L) + z t C1 (t) C2 E7 56 .
(F.26)
2
6
As before one can find the mixing of the U(1) with the R-symmetry in the IR.

GSupersymmetric boundary conditions

In this section we focus on five dimensional quantum field theories with N = 1 supersymmetry
and discuss the most basic boundary conditions that one can impose while preserving half ofthe supersymmetry.
Five dimensional N = 1 gauge theories have eight real supercharges and two kinds ofmultiplets, a vector multiplet containing a real scalar Φ, fermions Ψ and a vector field A, and a
hypermultiplet containing four real scalars φ and fermions ψ. The Lorentz group is given by
SO(4, 1) and there is an SU(2)R R-symmetry. In fact, there is an SO(4) symmetry rotating thefour real scalars in the hypermultiplet which is the product of SU(2)R and an additional SU(2) F
flavor symmetry. Denoting the representations under the bosonic SO(4, 1) × SU(2)R × SU(2) F
symmetry by (RSO(4,1) , RSU(2)R , RSU(2)F ), the supercharges Q iα transform as (4, 2, 1), the scalars
φ ia and fermions ψαa of the hypermultiplet as (1, 2, 2) and (4, 1, 2) (respectively), and thefermions Ψαi of the vector multiplet as (4, 2, 1) (with the real scalar and the vector field transforming trivially under SU(2)R × SU(2) F ). Here the indices i, a = 1, 2 and α = 1, . . . , 4 correspond to SU(2)R , SU(2) F and SO(4, 1), respectively.
Our goal is to find the most simple boundary conditions one can impose in a 5d N = 1
gauge theory such that half of the supersymmetry, corresponding to N = 1 in four dimensions,is preserved. Concretely, we will focus on free hyper and vector multiplets which appear in
the low-energy limit of such gauge theories.
We begin with addressing the case of a free hypermultiplet, where the theory is consideredin the domain x 4 < 0 and a boundary condition is given at x 4 = 0 (where the coordinates arex M with M = 0, . . . , 4). In order to find which kinds of simple boundary conditions preserve
half of the supersymmetry, we first describe the theory in terms of four dimensional N = 1
132

SciPost Phys. Lect. Notes 78 (2024)

multiplets. Then, finding boundary conditions that respect this multiplet structure (and therefore preserve the desired supersymmetry) will be natural.85
The action of a free hypermultiplet is given by
Z


1
1
5
ia M j b
a M αβb
S = d x − εi j εa b ∂ M φ ∂ φ + εab ψα Γ
∂ M ψβ ,(G.1)
42
where Γ M are the Gamma matrices in five dimensions. When written using the 4d N = 1 subsupersymmetry, such a hypermultiplet decomposes into an SU(2) F doublet of chiral multiplets(X , Y ). The Cartan of SU(2)R becomes the U(1)R R-symmetry in four dimensions, under which
both X and Y have R-charge 1. We would like to write the action (G.1) in terms of the chiralmultiplets X and Y . It is easy to see that in this description we should include a superpotential,
since the standard kinetic terms,Z
d x ⊥ d 4 x d 4θ X † X + Y † Y(G.2)(where we denote x ⊥ = x 4 ), do not reproduce the full action (G.1). As an example, noticethat only the εi j εa b ∂µ φ ia ∂ µ φ j b part of the scalar-field kinetic term, where µ = 0, . . . , 3, isrecovered. In order to obtain the missing piece, εi j εab ∂⊥ φ ia ∂ ⊥ φ j b , it is easy to see that weshould add the following superpotential,Zd x ⊥ d 4 x d 2 θ Y ∂⊥ X ,(G.3)

that includes it as well as the missing part of the rest of the action (G.1) . Indeed, the superpotential (G.3) sets the F -term of the X multiplet to be (up to a coefficient) F(X ) = ∂⊥ Y , where
Y is the scalar component of the anti-chiral multiplet Y . Similarly, it sets F(Y ) = ∂⊥ X such thatthe total contribution of the superpotential (G.3) to the scalar-field potential is given by the
missing part, which involves the derivatives of the scalars in the x ⊥ direction.
Now that we have expressed the action (G.1) in terms of the chiral multiplets X and Y , wecan look for simple boundary conditions that preserve the 4d N = 1 sub-supersymmetry. A
common choice is to give the scalar of X a Dirichlet boundary condition, setting it equal to zeroat x ⊥ = 0. By (4d N = 1) supersymmetry, this implies that the rest of the X multiplet vanishes
at the boundary, including the F -term ∂⊥ Y . This enforces Neumann boundary conditions for
Y , resulting in
X | x ⊥ =0 = 0 , ∂⊥ Y | x ⊥ =0 = 0 .
(G.4)
In a similar way, one can give Dirichlet boundary conditions for Y , enforcing Neumannboundary conditions for X ,Y | x ⊥ =0 = 0 , ∂⊥ X | x ⊥ =0 = 0 .
(G.5)
Let us next turn to discuss vector multiplets, focusing for simplicity on a U(1) gauge group.
As in the case of the hypermultiplet, we would first like to express the theory in terms of 4d
N = 1 multiplets. In this case, the 5d vector multiplet decomposes into a 4d vector multiplet
V and a chiral multiplet S, such that the bottom component of the chiral multiplet is givenby the complex scalar field Φ + iA⊥ . Recall that we do not consider the 5d vector multiplet as
coupled to matter in an interacting theory; however, in such a case the covariant derivative inthe x ⊥ direction is given by ∂⊥ −S. Notice also that the chiral multiplet S has an unusual gaugetransformation due to the appearance of A⊥ in its scalar component, given by S → S + i∂⊥ Λ
Notice that the full 5d N = 1 supersymmetry cannot be preserved since the anti-commutator of some of thesupercharges yields a translation in the x 4 direction, which is broken by the boundary condition at x 4 = 0.
85

133

SciPost Phys. Lect. Notes 78 (2024)

for an abelian gauge transformation with chiral-multiplet parameter Λ. The 5d action is reproduced by the standard kinetic term for V , plus a slightly-modified kinetic term for S thattakes into account its unusual gauge transformation. In contrast to the previous case of thehypermultiplet, no superpotential for S is needed here (and therefore its F field vanishes).
The kinetic terms of V and S set the D field of V to be (up to a coefficient) D = ∂⊥ Φ, resulting
in a contribution to the potential of Φ that reproduces the part of its kinetic term involving
derivatives in the x ⊥ direction.
Turning to possible boundary conditions that preserve the 4d N = 1 sub-supersymmetry, asimple choice would be to give the gauge field A a Dirichlet boundary condition, setting Aµ = 0
at x ⊥ = 0 (where recall that µ = 0, . . . , 3). Since this condition is only preserved by gaugetransformations with parameters which are constant along the boundary x ⊥ = 0, the gauge
symmetry actually breaks to a global symmetry there.86 The (4d N = 1) supersymmetriccompletion of this boundary condition simply sets the entire 4d vector multiplet V to zero on
the boundary, including its D-term. This results in the boundary conditions
V | x ⊥ =0 = 0 ⇒ Aµ x ⊥ =0 = 0 ,λσ | x ⊥ =0 = 0 , ∂⊥ Φ| x ⊥ =0 = 0 ,(G.6)

where we denoted the fermion of V by λσ .

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