SciPost Phys. 10, 097 (2021)

3d N = 4 Bootstrap and Mirror Symmetry
Chi-Ming Chang1,2 , Martin Fluder3,4,7 , Ying-Hsuan Lin4,5 ,Shu-Heng Shao6 and Yifan Wang7,8,9
1 Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
2 Center for Quantum Mathematics and Physics (QMAP),
University of California, Davis, CA 95616, USA
3 Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa 277-8583, Japan
4 Walter Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, USA
5 Hsinchu County Environmental Protection Bureau, Hsinchu, Taiwan
6 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
7 Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
8 Center of Mathematical Sciences and Applications,
Harvard University, Cambridge, MA 02138, USA
9 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Abstract
We investigate the non-BPS realm of 3d N = 4 superconformal field theory by uniting
the non-perturbative methods of the conformal bootstrap and supersymmetric localization, and utilizing special features of 3d N = 4 theories such as mirror symmetry anda protected sector described by topological quantum mechanics (TQM). Supersymmetric localization allows for the exact determination of the conformal and flavor central
charges, and the latter can be fed into the mini-bootstrap of the TQM to solve for a subsetof the OPE data. We examine the implications of the Z2 mirror action for the SCFT singleand mixed-branch crossing equations for the moment map operators, and apply numerical bootstrap to obtain universal constraints on OPE data for given flavor symmetrygroups. A key ingredient in applying the bootstrap analysis is the determination of themixed-branch superconformal blocks. Among other results, we show that the simplest
known self-mirror theory with SU(2) × SU(2) flavor symmetry saturates our bootstrap
bounds, which allows us to extract the non-BPS data and examine the self-mirror Z2
symmetry thereof.
Copyright C. Chang et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 21-10-2020
Accepted 16-04-2021
Check forupdatesPublished 30-04-2021
doi:10.21468/SciPostPhys.10.4.097

Contents
1 Introduction

3

2 CFT central charges
2.1 C T from squashed S 3 partition function
2.2 CJ from mass deformed S 3 partition function

6
7
8

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SciPost Phys. 10, 097 (2021)

2.3
2.4
2.5

Minimal theories with flavor symmetry G
T [G] theories and maximal nilpotent orbit
N = 8 theories

12
14
18

3 Superconformal blocks
3.1 Superconformal algebra, representations, and Casimir operator
3.2 Z2 outer automorphism
3.3 Single-branch superconformal blocks
3.4 Mixed-branch superconformal blocks
3.4.1 s-channel
3.4.2 t-channel

18
18
20
20
21
22
23

4 The protected sector
4.1 Mini-bootstrap
4.2 Deformation quantization

24
25
27

5 Superconformal bootstrap
5.1 Coulomb-Higgs mixed correlators and Z2 outer automorphism
5.1.1 U(1) × U(1) flavor symmetry
5.1.2 Mirror symmetry and the Z2 outer automorphism
5.1.3 GC × GH flavor symmetry
5.2 The linear functional method
5.3 Numerical bounds
5.3.1 Single branch
5.3.2 GC = GH = U(1)5.3.3 GC = GH = SU(2)5.3.4 GC = GH = SU(3)

30
30
30
31
32
33
34
34
35
35
36

6 Conclusion

37

A Localization computation of the central charges
A.1 Double-sine functions and derivatives
A.2 Squashed S 3 partition function
A.3 Computation of C T
A.4 Computation of CJ

39
39
41
41
46

B Derivation of the superconformal Casimir equations
B.1 s-channel
B.2 t-channel

49
49
51

C Crossing matrices (6j symbols)

52

D Crossing equations in matrix form

52

References

55

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SciPost Phys. 10, 097 (2021)

1

Introduction

Quantum field theory (QFT) in three spacetime dimensions has proven to be a promising arena
for studying strongly coupled physics. It shares close resemblance to four-dimensional QFT inproviding a rich array of non-perturbative phenomena such as confinement, chiral symmetry
breaking, and mass gaps. 3d QFT is also particularly intriguing in its own right due to thepresence of Chern-Simons couplings, fractional (anyonic) statistics, monopole operators, andhighly nontrivial duality mechanisms such as the bosonization dualities (analogous to 2d), allof which play important roles in our understanding of the quantum phase transitions of gauge
theories and condensed matter systems. A particularly interesting and subtle case is when thetransition is second order and thus mediated by a conformal field theory (CFT). Knowledge ofthe operator spectrum and correlation functions in the CFT is crucial to completing the phase
diagram.
There is a rich family of 3d CFTs constructed as the infrared fixed points of Chern-Simons
matter theories. The existence of such fixed points is supported by substantial evidence including a non-renormalization theorem for the Chern-Simons level [1, 2], perturbative analyses of the beta function for the matter couplings [3–5], exact computations of three-point
functions and thermal free energies in the ’t Hooft limit [6–10], the bulk gravity or higher
spin duals under the AdS/CFT correspondence [5, 6, 11–14], as well as intricate duality webs
and the ’t Hooft anomaly matching thereof [6–8, 15–18]. Their non-perturbative dynamics
are explored to some extent by approaches including AdS/CFT, bosonization dualities, andSchwinger-Dyson equations.
Progress on the non-perturbative dynamics of CFT has been made by two powerful tools– the conformal bootstrap and supersymmetric localization. The conformal bootstrap method
has been successful in solving CFT, including the famous 3d Ising CFT that describes critical phase transitions for water and magnets [19]. The inclusion of supersymmetry and themethod of supersymmetric localization [20, 21] have made analytic computations possible
in superconformal field theory (SCFT). The unison of these two tools have proven powerfulin [22–28].
In this paper, we apply the conformal bootstrap and supersymmetric localization to thespecific context of 3d SCFT with N = 4 supersymmetry. 3d N = 4 SCFT boasts special properties such as mirror symmetry [29,30] and an exactly solvable subsector of topological quantum
mechanics (TQM) [22, 31–34] that present interesting interplay with bootstrap and localization. We elaborate on these special properties after briefly discussing some generalities.
Most known 3d N = 4 SCFTs arise as the low energy descriptions of quiver gauge theories, and more generally of M2 branes probing transverse singular geometries in M-theory[12, 13, 35–43]. These theories typically have a vacuum moduli space that includes somecombination of a Coulomb branch, a Higgs branch, and possibly mixed branches. In terms of
the CFT operators, they are parametrized by the vevs of half-BPS operators subject to chiralring relations. Some of these operators, such as monopoles, are of the disorder type: they
are non-perturbative in nature and their correlation functions are inaccessible by traditional
methods. Fortunately, certain integrated correlators of the current multiplets and stress-tensor
multiplets can be obtained from supersymmetric localization on S 3 with mass and squashing
deformations, and in particular encode the flavor and conformal central charges of the CFT.
The flavor symmetry of 3d N = 4 SCFTs takes a product form GC ×GH , where the factors GC
and GH are realized by different types of flavor current multiplets, whose bottom components
are charged under the SU(2)C and SU(2)H R-symmetry, respectively (and uncharged under theother).1 We study the single- and mixed-branch four-point functions of the moment map op1

In this paper, we will not be concerned with the global properties of groups which do not affect the correlation
functions of local operators.

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SciPost Phys. 10, 097 (2021)

erators (scalar primaries in the flavor current multiplets for GC and GH ), exploiting constraints
from superconformal symmetry and crossing using the numerical bootstrap methods.2 This is
made possible by first determining the mixed-branch 3d N = 4 superconformal blocks (thesingle-branch blocks were determined in [25, 44, 45]). By comparison with certain OPE data
computable by supersymmetric localization, we observe that all known candidate theories are
consistent with the numerical bounds. Furthermore, in some cases, such as GC = GH = SU(2),the bound (see Figure 3) is saturated by a known theory, namely the SQED with 2 flavors, andwe use the extremal functional method to extract the more general, non-BPS spectrum, thatappears in its moment map OPE.
Mirror symmetry
Mirror symmetry in 3d [29, 30] bears close relation to 4d S-duality and 2d mirror symmetry.
3d mirror symmetry is an infrared duality: two UV gauge theory descriptions flow to the same
conformal fixed point. In particular, the notions of Coulomb and Higgs branches swap between
the two descriptions. The Coulomb branch is highly quantum as it embodies the complicatedgauge theory dynamics that involves monopoles, yet the Higgs branch is protected against
such effects by supersymmetric non-renormalization theorems [29]. The key insight of mirror
symmetry is that the strongly coupled Coulomb branch in one UV description has an alternative
simple description as the protected Higgs branch of another UV description, which leads tohighly-nontrivial predictions on observables in the Coulomb branch of the first description.
Mirror symmetry also has important consequences for non-supersymmetric dualities in 3d. By
turning on deformations that break the supersymmetry, it gives nontrivial evidence for the 3dbosonization duality mentioned above [46, 47], the zero mass case of which was otherwise
unsettled from just anomaly matching arguments.3
However, despite much progress in understanding mirror symmetry at the level of BPS
data, little is known in the non-BPS realm, which is crucial for analyzing the non-supersymmetricbosonization duality from deformations. We make progress towards analyzing the non-BPS
aspect of mirror symmetry by applying the bootstrap method to the simplest interacting 3d
N = 4 SCFT, namely the SQED with 2 flavors, given by the IR fixed point of SQED with
two hypermultiplets of unit charge. In this particular case, the UV description is self-mirror
in the sense that the Higgs and Coulomb branches are simply exchanged within a single UV
description under the mirror map, and the theory enjoys an additional Z2 global symmetrythat simultaneously acts on the N = 4 superconformal algebra as an outer-automorphism andexchanges the two factors of GC × GH . As usual, such symmetries will leave signatures on the
fixed point OPE data which we explore by analyzing a system of mixed correlators involving
both Coulomb and Higgs branch moment map operators. As we will see, this theory saturates
the bootstrap bound, and by extracting the non-BPS OPE data we are able to provide nontrivial
evidence for the (self-)mirror symmetry.
Topological quantum mechanics
Moreover, 3d N = 4 SCFT is also special in that it contains a one-dimensional protected subsector, known as the topological quantum mechanics (TQM). In general, there are two TQMs,
one associated with the Coulomb branch, and the other with the Higgs branch [22, 31–34].
2

Due to supersymmetry, the four-point functions of the moment map operators contain all the dynamical information of the four-point functions of the operators in the flavor current multiplets.
3
In this case, starting from the mirror symmetry between a free hypermultiplet and the N = 4 SQED with
one charged hypermultiplet, the authors of [46] considered supersymmetry-breaking deformations to deduce thenon-supersymmetric bosonization duality between the Dirac fermion and the 3d scalar QED with a level-1 ChernSimons term.

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SciPost Phys. 10, 097 (2021)

The operator algebra in the TQM is an associative algebra given by a non-commutative deformation of the Higgs or Coulomb branch chiral ring. The OPE data in the TQM can be solved
or constrained using the conformal bootstrap, i.e. imposing the associativity and unitarity ofthe operator algebra, which is dubbed the mini-bootstrap because it is a closed subsystem of
the full bootstrap equations. In particular, the single-branch four-point function of (twisted)
moment map operators restricted to a line reduces to a four-point function in the TQM.4 As wewill see in Section 4, when the flavor symmetry group GC or GH is simple, the mini-bootstrap
constrains the OPE data in this four-point function down to a two-dimensional convex quadrilateral parametrized by the flavor central charge and another OPE coefficient. The OPE datain the TQM can also be computed by supersymmetric localization [32–34].
Brief summary of the results
We briefly summarize the main results of this paper.
• We compute the conformal and flavor central charges for several classes of 3d N = 4
SCFTs, by taking derivatives with respect to the mass and squashing parameters of thedeformed S 3 free energy. The results are (2.51) and (2.52) for the SQED, (2.53) and(2.54) for the SQCD, (A.32) and (A.59) for the T [SU(3)], (2.55) for the T3 , (A.46) and(A.65) for the II k (3) Chern-Simons matter theory. Some of these theories are dubbed as
the minimal 3d N = 4 SCFTs with flavor symmetry G and their the central charges are
summarized in Table 1.
• We compute the superconformal blocks for the four-point function of two Coulomb andtwo Higgs branch moment map operators.5 The results for the s-channel superconformal
blocks are in (3.24), (3.25) and (3.4.1), and the t-channel superconformal blocks canbe found in (3.28), (3.29), (3.30) and (3.31).
• We explore the analytic bootstrap constraints in the protected topological quantum mechanics sector of the SCFT to solve or constrain the flavor central charge and certain OPE
coefficients. The results are summarized in Table 5 and Figure 2. The minimal nilpotent
theories sit at the kinks, and the maximal nilpotent theories sit at the boundaries of theallowed regions (see Section 2.3 and 2.4 for definitions of these theories).
• Finally, we employ the numerical bootstrap technology to map out the space of consistent unitary 3d N = 4 SCFTs, for several flavor symmetry groups and assuming or notassuming the minimal/maximal nilpotent orbit condition on the protected operators.
The key results are in Table 7, Figures 3 and 5, and we find free theories and T [SU(2)]to sit at the boundary of consistency. We also present a diagnosis on bootstrap data forthe self-mirror property, and explicitly verify it in T [SU(2)].
Organization of the paper
We start by reviewing the notion of conformal and flavor central charges in 3d CFT in Section 2,
and determine their values for an array of 3d N = 4 SCFTs that realize minimal and maximal
nilpotent orbits as their Higgs branch moduli spaces. Such data will later be compared with thenumerical bootstrap bounds. In Section 3, we review the 3d N = 4 superconformal algebra
and its representations, and determine the superconformal blocks when the external operators are either Higgs or Coulomb branch chiral primaries. In Section 4, we present analytic
4

The four-point function only involves one type of moment map operators, Coulomb or Higgs.
The superconformal blocks for the four-point function of four Coulomb or four Higgs branch moment map
operators were computed in [25, 44].
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SciPost Phys. 10, 097 (2021)

solutions to the mini-bootstrap problem for the protected sector of 3d N = 4 SCFTs, and explain the relation to the deformation quantization of the Coulomb or Higgs branch. Equippedwith the exact protected OPE data and the superconformal blocks, we setup and implement
the bootstrap analysis in Section 5 to extract bounds on the non-BPS spectrum. We end by asummary of the main results and a discussion of future directions in Section 6. In Appendix A,
we present the details of the computations of the conformal and flavor central charges using
supersymmetric localization. In Appendix B, we derive the superconformal Casimir equations
used for deriving the superconformal blocks. In Appendix C, we state our conventions for theClebsch-Gordan coefficients and the crossing matrices (6j symbols). Finally, in Appendix D,
we summarize the crossing equations rewritten in matrix form.

2

CFT central charges

For any CFT in d dimensions, the two-point function of the stress tensor Tµν takes the formTµν (x)Tσρ (0) =C T Iµν,σρ (x),x 2d
Vb2d−1(2.1)

S

where VbSd−1 = 2πd/2 /Γ (d/2) is the volume of a (d − 1)-dimensional unit sphere. Similarly, if
the CFT has global symmetry G, the two-point functions of the conserved currents Jµa are givenby¶¬Iµν (x)CJ
Jµa (x)Jνb (0) = 2 δ ab 2(d−1) .
(2.2)
x
Vbd−1
S

The tensor structures Iµν,σρ (x) and Iµν (x) in (2.1) and (2.2) are completely fixed by conformal symmetry
 11Iµσ (x)Iνρ (x) + Iµρ (x)Iνσ (x) − δµν δσρ ,
2
d
xµ xν
Iµν (x) = δµν − 2 2 .
x

Iµν,σρ (x) =(2.3)

The coefficients C T and CJ , with appropriate normalization, capture dynamical information about the particular CFT. We refer to them as the conformal and flavor central charges,
respectively. We canonically normalize the stress tensor Tµν by the Ward identity. For symmetry currents Jµa associated to a simple Lie group G, we adopt the normalizationJµa (x)Jνb (0) ⊃1
VbSd−1

i f abcxµJ c (0),|x|d ν

(2.4)

with f a bc totally antisymmetric, purely imaginary and satisfying
1 aed bdeff
= δ ab ,
2h∨

(2.5)

where h∨ denotes the dual Coxeter number of G.
For the convenience of the reader, we record the values of C T and CJ for free fields in
d = 3; namely our convention is such that C T = 23 for either a real massless scalar or a
Majorana fermion. Thus, a free N = 4 hypermultiplet has conformal central charge
C T (hyper) = 8 ×
6

3
= 12.
2

(2.6)

SciPost Phys. 10, 097 (2021)

The flavor central charge CJ for the SU(2) flavor symmetry of a free N = 4 hypermultiplet
can be worked out explicitly from the OPE (see the next section), and takes the value
CJSU(2) (hyper) = 4.

(2.7)

Here and in the following, we denote the flavor central charge with respect to the symmetrygroup G of a SCFT T by CJG (T ).
We now outline the derivation of a formula that allows us to explicitly compute the conformal and flavor central charges for 3d N = 4 superconformal field theories. Namely, on the
one hand, we relate the conformal central charge, C T , of a given theory to its metric deformed(squashed) S 3 partition function. On the other hand, we derive a precise correspondence between the flavor central charge CJ of a theory and its mass-deformed S 3 partition function.
Such formulae were originally derived in [48, 49] for 3d N = 2 SCFTs, and we re-derive themhere in order to precisely connect with our choice of N = 4 convention. By means of supersymmetric localization [20, 21], these formulae allow for the explicit evaluation of the central
charges in a large class of theories with (UV) Lagrangian descriptions. We refer to Appendix A
for the required ingredients for the localization computation as well as some examples.

2.1

C T from squashed S 3 partition function(0;0)

The 3d N = 4 stress tensor multiplet A[0]1
(1;1)

(0;0)

→ [1]3/2

Φ

→

[0]1

ΨαAȦ

(2;0)

consists of conformal primaries [50, 51]

(0;2)

(0;0)

⊕ [0]2→ [2]2⊕ [2]2→jµAB ⊕ jµȦḂ ⊕ U(1;1)→ [3]5/2
AȦ
Sµα

→

(0,0)

→ [4]3
→

Tµν ,

(2.8)

(R ;R )

where each entry, [2`]∆ H C , labels the representation content of the corresponding conformal
primary under the bosonic subgroup of OS p(4|4); ` and ∆ denote the spacetime spin andscaling dimension, while RH and RC label the SU(2)H and SU(2)C spins, respectively. We useµ and α for spacetime vector and spinor indices, and reserve A, B (resp. Ȧ, Ḃ) for SU(2)H(resp. SU(2)C ) doublet indices. The arrows in (2.8) represent (anti)commutators with thesupercharge(1;1)
QAαȦ ∈ [1]1/2 ,(2.9)
which generates the entire multiplet starting from the superconformal primary Φ, which is adimension ∆ = 1 scalar uncharged under the R-symmetries.
By supersymmetry, the two-point functions of the conformal primaries in (2.8) are all determined in terms of the stress tensor Tµν ,Tµν (x)Tσρ (0) =C T Iµν,σρ (x)
.
16π2
|x|6

(2.10)

In particular, the R-symmetry currents, normalized as in (2.4) for the SU(2)H × SU(2)C symmetry, have two-point functions,6〈 jµAB (x) jνC D (0)〉 =

CT ε
96π2

ε

C(A B)D

|x|4

Iµν

,

〈 jµȦḂ (x) jνĊ Ḋ (0)〉 =

CT ε
96π2

ε

Ċ(Ȧ Ḃ) Ḋ

|x|4

Iµν

.

(2.11)As explained in [48], an efficient way to compute the conformal central charge C T forinteracting SCFTs (with at least N = 2 supersymmetry) is to couple the SCFT to a certainp
Note that the SU(2)H currents are given by JµAB ≡ Jµa (σa )AB and our normalization requires f abc = i 2εabc inequation (2.4). Similarly for the SU(2)C currents.
6

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SciPost Phys. 10, 097 (2021)

family of supersymmetric curved background, where the spacetime manifold is a squashed S 3 ,i.e.
ω21 |z1 |2 + ω22 |z2 |2 = 1, zi ∈ C,
(2.12)
with squashing parameters ωi . Conformal invariance further implies that the partition function of the SCFT only depends on
ω1
b2 ≡
.
(2.13)
ω2
On such backgrounds, the corresponding partition function Z b is protected by supersymmetric
non-renormalization theorems, and thus can be computed reliably given a UV supersymmetricgauge theory description. Localization techniques then reduce the path integral to a finite
dimensional matrix model over the Coulomb branch moduli of the gauge theory [21,52–57].7
Thus, the central charge C T is conveniently encoded in the dependence of the partition
function Z b on the squashing parameter b. Defining the squashed S 3 free energy by
F b = − log Z b ,(2.14)

the precise formula relating C T to F b is given as follows [48]8
CT =

2.2

48 ∂ 2 F b.
π2 ∂ b2 b=1(2.15)

CJ from mass deformed S 3 partition function

Having the explicit formula for the conformal central charges in hand, let us now turn to(2;0)
the flavor central charges. The 3d N = 4 Higgs branch flavor-symmetry multiplet B [0]1
consists of conformal primaries
(2;0)

[0]1

a
LAB

(1;1)

(0;0)

→ [1]3/2→→ [2]2

ϕa

→

αAȦ

(0;2)

⊕ [0]2(2.16)

Jµi ⊕ N a ,ȦḂ

where a denotes the adjoint index of the flavor symmetry group G. The Coulomb branch flavor(0;2)
symmetry multiplet B [0]1 is similar but with the quantum numbers RH and RC interchanged.
For the simplicity of discussion, we assume that G is simple in this section. By supersymmetry, the two-point functions of the operators in (2.16) are all determined in terms of thecurrent two-point function¶¬
CJ ab Iµν (x)(2.17)Jµa (x)Jνb (0) =
δ
.
16π2
x4
µ

In particular, it follows from the N = 4 algebra, {QAαȦ, Q BβḂ } = −iεAB εȦḂ γαβ ∂µ that the twopoint functions of L a and N a , normalized asaJµa = εȦḂ QAȦγµQ B Ḃ LAB,a
NȦaḂ = QA(ȦQ B Ḃ) LAB,(2.18)

where γi are the Pauli matrices, depend on CJ as follows
CJ ab εC(AεB)DCJ a b εC(AεB)Da(2.19)〈LAB(x)L Cb D (0)〉 =δ, 〈NȦaḂ (x)NĊbḊ (0)〉 =
δ
.
2
2
64π
x
32π2
x4
Similar to the case of C T , we can obtain the flavor central charge CJ for flavor symmetry
G by coupling the SCFT to certain supergravity backgrounds that involve mass deformations
for G, and compute the supersymmetric partition function.
See also [58] for a nice review on 3d supersymmetric localization, and appendix A for the relevant ingredientsfor our purposes.
C
8
The notation here is related to that of [48] by τ r r = 24T and jµR = p12 ( jµ12 + jµ1̇2̇ ), where jµR denotes the U(1)Rcurrent for the N = 2 subalgebra. This is in order for the supercharges in the N = 2 subalgebra to have U(1)R
charge ±1.
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SciPost Phys. 10, 097 (2021)

Massive supersymmetric background on S 3(2;0)

The Higgs branch multiplet B [0]1

couples to background vector fields with field contenta
DAB, λAaȦ, Aaµ , φȦaḂ .

(2.20)

To linearized order, the mass deformation of the SCFT amounts to a linear deformation of theaction byZp aδS = d 3 x g D aAB LAB+ φ aȦḂ NȦaḂ + Aaµ J aµ + fermionic terms .
(2.21)For our purpose, it is convenient to take the spacetime manifold to be S 3 .
To specify the background, we begin with the round S 3 with no mass deformation. The
osp(4|4) symmetry of the SCFT on the round S 3 is generated by Killing spinors satisfying(see [32])1(2.22)∇µ ξAȦ = γµ ξ0AȦ, ∇µ ξ0AȦ = − 2 γµ ξAȦ,
4r
where r is the radius of S 3 . In the mass-deformed theory, the maximal subalgebra we canpreserve on S 3 issu(2|1) ⊕ su(2|1) ⊂ osp(4|4),(2.23)
where the embedding of u(1) × u(1) is specified by two generators, one for each of su(2) andsu(2),(2.24)
hAB ∈ su(2), h̄ȦḂ ∈ su(2),satisfyingh̄ȦḂ h̄Ḃ Ċ = δȦĊ ,hAB hB C = δAC ,(2.25)and the supercharges generating the subalgebra (2.23) are specified by the restriction
ξ0AȦ =ihAB ξB Ḃ h̄Ḃ Ȧ.
2r(2.26)

Without loss of generality, and akin to [32], we choose
h = −σ2 ,h̄ = −σ3 ,(2.27)

where σ i are the Pauli matrices. The fields in the background (abelian) vector multiplet thentake the following values in order to preserve the su(2|1) ⊕ su(2|1) supersymmetry:Aaµ = λAaȦ = 0,

1
a
2
DAB
= − M a σAB,r

φȦaḂ = M a σȦ3Ḃ ,(2.28)

where a is chosen to take values a ∈ {1, 2, . . . , n}, such that T a generate the Cartan subalgebraof G, and M a denote the mass parameters for the flavor symmetry group G of the SCFT.
For this background, we define Z(M a ) to be the mass-deformed partition function on S 3
and F (M a ) ≡− log Z(M a ) to be the free energy of the SCFT. Following the argument in [49], while keeping
track of our choice of normalization of the operators, the flavor central charge can be extracted
from the S 3 free energy as
F (M a )| M 2 =π2 CJG
16

9

δab M a M b ,(2.29)

SciPost Phys. 10, 097 (2021)

where the superscript G is used to emphasize the fact that the flavor central charge CJG is withrespect to the flavor symmetry group G.
If G is a subgroup of a larger simple symmetry group G 0 of the SCFT, then0

CJG = I G,→G 0 CJG ,(2.30)

where I G,→G 0 denotes the Dynkin index of the embedding.9
The formula for CJ from the gauge theory free energy
In practice, the mass-deformed free energy of an N = 4 SCFT is computed by localization
using its UV gauge theory (Lagrangian) description, and the mass parameters M a for the flavor
symmetry are related to the gauge invariant masses of hypermultiplets.10 However, the fullsymmetry G can be emergent from the RG flow, and consequently only a subset of massdeformations is accessible from the UV description. Therefore, in order to compute CJ for thefull symmetry group G using (2.29), we need to know the precise relation between the UV and
IR mass parameters as well as how the UV symmetry group GUV embeds into the IR symmetrygroup G.
We start by recalling the matter sector of a general 3d N = 4 gauge theory, which is
described by n hypermultiplets
qAi , ψαi Ȧ,(2.32)
with i = 1, 2, . . . , 2n transforming in the fundamental representation of US p(2n). The Euclidean action of hypermultiplets coupled to vector multiplets (both dynamical and background) on S 3 is given by
Z

3 AB
j
j
3 p
/ j +
Shyper = d x g εAB Ωi j Dµ qAi Dµ qB − iεȦḂ Ωi j ψȦi Dψε Ωi j qAi qB
2Ḃ
4r
 (2.33)
1 aȦḂ b AB
a Ḃ
a b
iAB i
aȦḂ aA a
− φ φȦḂ ε qA T T qB − iqA D T qB + iψȦφ T ψḂ − 2iq λAḂ T ψ ,2where Tiaj are generators of usp(2n). Note that the scalars are subject to the reality condition11j(qAi )∗ = qiA = εAB Ωi j qB .

(2.34)On the supersymmetric background (2.28), the hypermultiplet action becomes
Z

3 AB
p
j
j
/ j +
Shyper = d 3 x g εAB Ωi j ∇µ qAi ∇µ qB − iεȦḂ Ωi j ψȦi ∇ψε Ωi j qAi qB
Ḃ
4r 2

i
j
j
j
AB i
2
AB i
ȦḂ i+ ε q(A(M )i j qB) + (σ2 ) qAMi j qB + i(σ3 ) ψȦMi j ψ ,Ḃr(2.35)
wherea ai
Mi j = MUV
T j,(2.36)
9

The Dynkin index of an embedding G ⊂ G 0 between simple Lie groups is defined byP
i T (ri )I G,→G 0 ≡,T (r)(2.31)

where r denotes a representation of G 0 which decomposes into ⊕i ri under G 0 → G, and T (·) denotes the quadratic
index of the representation. This definition does not depend on the choice of r.
10
The IR mass parameters M a associated to Coulomb branch symmetries come from the FI parameters in the UV
gauge theory instead. The generalization is straightfoward.
11 iq ≡ q1i , q̃i ≡ qi2 for i = 1, · · · , n have the same RH3 charge and represent N = 2 chiral multiplet scalars [31].

10SciPost Phys. 10, 097 (2021)

with a restricted to a = 1, 2, . . . , n labelling the Cartans of usp(2n), encodes the UV mass
parameters (or vector multiplet scalar vevs).
When a subgroup H ⊂ US p(2n) is gauged, the residual flavor symmetry is given by thecommutant which we will refer to as GUV . In the cases we are interested in here, the hypermultiplets transform in a single representaton (R GUV , RH ) of GUV × H ⊂ US p(2n), with indices(i,ı̂) and dim(R GUV ) dim(RH ) = 2n. The mass matrix decomposes asa aiMiı̂ j̂ = MUV
T j K ı̂ ̂ + φ â T̂ âı̂ ̂ I i j ,(2.37)

where I i j and K ı̂ ̂ are invariant tensors of GUV and H, respectively, satisfying

I i j I j k = δi k ,K ı̂ ̂ K ̂ k̂ = δ ı̂ k̂ .

(2.38)

We still denote the adjoint index of GUV by a and use â for the adjoint index for the gaugeagroup H. In the gauge theory, φ â are dynamical while MUV
are mass parameters.
We claim that the UV and IR mass parameters for GUV are simply identified asa
MUV= M a.

(2.39)

This is equivalent to checking the normalization of the conserved current multiplets in the UV
gauge theory description. Comparing (2.33) with (2.21), we identify the currents for GUV 12
1 ȦḂ
j
AB
ai
a
j
αβ
Jµ = 2T j iε qAi ∂µ qB − ε ψαȦi γµ ψβ Ḃ ,(2.40)
2and the moment map operatora= −T ai j q(Ai qB) j .
LAB(2.41)

In flat space, the two-point functions of free massless hypermultiplet fields in (2.35) are
given byj〈qAi (x)qB ( y)〉 =1 Ωi j εAB,
8π |x − y|jḂ〈ψȦi (x)ψ ( y)〉 =γµ (x µ − y µ )iεȦḂ Ωi j.
8π
|x − y|3(2.42)

Performing explicit Wick contractions, it is easy to see that (2.40) indeed satisfies (2.4). We
can also compute the UV flavor central charge (central charge for the free theory) from thetwo-point function of (2.40),〈Jµa (x)Jνb ( y)〉 =4δ abIµν dim(RH ),
16π2(2.43)

which gives13G

CJ UV (hypern ) = 4 dim(RH ).

(2.44)

To summarize, for UV flavor symmetry GUV ⊂ GUV × H ⊂ US p(2n) carried by gauged hypermultiplets, we define MUV as the reduced mass matrix such that the full mass matrix for thehypermultiplets is
M = MUV ⊗ K ⊕ vector multiplet scalar vevs,(2.45)
where K is a normalized invariant tensor for the gauge group H introduced in (2.37). Then theflavor central charge of the SCFT for GUV is determined by the gauge theory supersymmetricfree energy via
G

F |M2 =

π2 CJ UV16

Tr M2UV .

(2.46)

12
Here and below we focus on the gauge-invariant currents and suppress all gauge indices ı,  which have been
contracted with the invariant tensor I ı  .
USp(2n)13Note that for n free hypermultiplets, CJ(hypern ) = 4. The extra factor of dim(RH ) comes from the embedding index for the subgroup GUV ⊂ GUV × H ⊂ USp(2n) (see (2.31)).

11

SciPost Phys. 10, 097 (2021)

Table 1: Conformal and flavor central charges C T and CJ for candidate minimal
SCFTs with Higgs branches given by the minimal nilpotent orbits of G, as computedby localization.

2.3

G

CJ

CT

SU(2)

16
3

64
16
3 + π2SU(3)654 − 192
π2G

Candidate minimal theory

SU(4)

32
5

256
48
5 − π2

SU

SQED

SU(5)

20
3

100 − 3712
9π2SO

SU(2) SQCD

SO(5)

4

54 − π642

US pFree hypers

SO(6)

32
5

272
48
5 + π2

E6

T3

SO(7)

8

156 − 12416
15π2

SO(8)

64
7

736
200
7 − π2

US p(2n)

4

12n

E6

192
13

160.246

Minimal theories with flavor symmetry G

Given a simple Lie group G, there is a notion of minimality for 3d N = 4 SCFTs with flavor
symmetry G: Assuming that the symmetry G is generated by the Higgs branch current multiplets, G naturally acts on the Higgs branch moduli space of the theory. A special class ofsuch Higgs branch geometries are given by (the closure of) the nilpotent orbits of G, among
which the minimal nilpotent orbit Omin (G) has complex dimension 2(h∨ (G) − 1) and is the
one-instanton moduli space of G on R4 . A “minimal” theory is one whose Higgs branch moduli space is (the closure of) the minimal nilpotent orbit, and whose conformal central charge
C T is the smallest.
The minimal nilpotent orbit Omin (G) has a simple description as a holomorphic-symplectic
variety embedded in the Lie algebra g of G,

Omin (G) = {g | I2 = 0},(2.47)

where I2 denotes the Joseph ideal in the polynomial algebra of g defined by(adj ⊗ adj)S = 2adj ⊕ I2 .

(2.48)For this reason, we refer to the condition in (2.47) as the Joseph relations. We expect suchconditions to impose strong constraints. Indeed as we will see in Section 4, the Higgs branchprotected operator algebra (1d TQM) [22, 31–34] can be solved completely and uniquely (except in the case of G = A1 ) by certain generalized higher-spin algebra for G.
Below we present a list of candidate minimal theories for each simple Lie group G, andprovide their conformal and flavor central charges computed using the formulae presented in
the previous sections. The results are also summarized in Table 1. We leave the details of thecomputations to Appendices A.3 and A.4.

12

SciPost Phys. 10, 097 (2021)

n Free hypermultiplets
Consider n free hypermultiplets with the following Z2 symmetry gauged:14
Z2 : qAi → −qAi .

(2.49)

This is precisely the ADHM gauge theory for the 1-instanton moduli space of US p(2n) on R4 .
The conformal and flavor central charges are then given by (see (A.9) and (A.48))
C T (hypern ) = 12n,USp(2n)CJ(2.50)(hypern ) = 4.

N = 4 SQED with n ≥ 2 unit-charge hypermultiplets
The 3d N = 4 SQED coupled to n unit-charge hypermultiplets flows in the IR to an interacting
SCFT for n ≥ 2. This is precisely the ADHM gauge theory for the one-instanton moduli spaceof SU(n) on R4 . The theory has a mirror-dual description as an affine An−1 type quiver theory,
which is a cyclic quiver consisting of n U(1) gauge-nodes15 and n bifundamental hypermultiplets. From the SQED description, we can extract the conformal and flavor central charges,
given by

12 2n2 ψ(1) 2n + 1 + π2 n + 4 n + 4(2.51)C T (SQEDn ) =,π2 (n + 1)
Γ 0 (z)dwhere ψ(1) (z) ≡ dz
Γ (z) is the z-derivative of the digamma function, andCJSU(n) (SQEDn ) =

8n
.
n+1

(2.52)

There is an interesting family of U(1)k × U(1)−k Chern-Simons matter theories with SU(n)flavor symmetry: the II k (n + 1) theories of [59]. We compute the values of C T and CJ for the
II k (3) series in Appendix A.3 and A.4. The minimal k = 2 theories have enhanced SU(n + 1)
symmetry and are dual to the SQED with n + 1 unit-charge hypermultiplets.

N = 4 SQCD with SU(2) gauge group and n ≥ 5 fundamental half-hypermultiplets
The 3d N = 4 SQCD with SU(2) gauge group coupled to n fundamental half-hypermultiplets
flows in the IR to an interacting SCFT for n ≥ 5. This is precisely the ADHM gauge theory forthe one-instanton moduli space of SO(n) on R4 .
When n is even, the theory has a known mirror-dual description in terms of a 3d N = 4
affine Dn/2 shaped quiver. From the SQCD description, we can extract the conformal and flavor
central charges, given by
C T (SQCDn ) =
12 n π2 (n − 4)(n − 2) + 12n − 20 (n − 4) + 2(n − 2)(3n − 2)(n − 4)2 ψ(1)π2 (n − 4)(n − 2)(n − 1)andCJSO(n) (SQCDn ) =1416(n − 4)
.
n−1

n−2
2




+ 48

,

(2.53)(2.54)

One can gauge the Z2 symmetry by coupling to a nontrivial 3d Z2 gauge theory, but this will not make adifference on the local operator spectrum. In fact, even if we do not gauge Z2 , although the full (free) theory does
not satisfy the minimal nilpotent orbit condition, the OPE of Z2 singlets is still identical to the gauged versions,and suffice for mere comparisons with bootstrap.
15
The diagonal U(1) vector multiplet decouples.

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SciPost Phys. 10, 097 (2021)

E6,7,8 theories
The 4d N = 2 En Minahan-Nemeschansky theories [60, 61] have Higgs branches realized by
the one-instanton moduli spaces of En . Their S 1 reduction naturally gives rise to 3d N = 4 En
theories that inherit the 4d Higgs branch. Unlike their 4d parents, these 3d theories do have(mirror) Lagrangian descriptions given by quivers of affine E6,7,8 types, which we can use tocompute the conformal and flavor central charges.
In the case of E6 (also known as T3 theory), we carry out the computation in Appendix A.3,
and obtain
C T (T3 ) = 160.246,(2.55)E
CJ 6 (T3 ) = 4.
In later sections, we give an independent derivation of CJ by solving the 1d TQM associatedto the Higgs branch.
G2 , F4 theories
These are the most mysterious 3d N = 4 SCFTs in this class whose existence are purely conjectural. There are no known string/M-theory constructions for them. However, if they exist, thenwe can determine CJ unambiguously by solving the 1d TQM associated to the Higgs branch ofa would-be G2 or F4 minimal theory (see Section 4 and Table 5).

2.4

T [G] theories and maximal nilpotent orbit

Let us now discuss a special class of theories, dubbed T [G]. They were originally introduced
in [43] as the S-dual [62] (under the S-tranformation of the full S-duality group S L(2, Z))of particular (Dirichlet for the 4d N = 2 vector multiplets and Neumann for the adjoint hypermultipelts) boundary conditions of the 4d N = 4 supersymmetric Yang-Mills theory withgauge group G.16 More precisely, the 4d S-dual configuration consists of the 3d T [G] theory
coupled to the 4d bulk gauge fields.
The T [G] theories are 3d N = 4 SCFTs in the infrared, and carry (IR) G ∨ × G (G ∨ is the
GNO or Langlands dual of G [64]) flavor symmetry, with G ∨ and G realized on the Coulomb
branch and Higgs branch, respectively. The G ∨ and G act faithfully only through their adjoint
forms, and thus the distinction (for observables that only involve local operators) between
them is only relevant when they have different Lie algebras. The Higgs branch is given by the
maximal nilpotent orbit Ng of G with complex dimensiond ≡ dimC Higgs = dim(g) − rank(g).

(2.56)The Coulomb branch is given by the maximal nilpotent orbit Ng∨ of G ∨ with the same complex
dimension.17 Under 3d N = 4 mirror symmetry [29, 30], T [G] is mapped to T [G ∨ ] and viceversa, with Coulomb and Higgs branches interchanged.
These facts can be observed by realizing that the theory T [G] is given by the S-dual to
Dirichlet boundary conditions in N = 4 super-Yang-Mills theory with gauge group G (see [43]for more details). Consider the N = 4 super-Yang-Mills theory with gauge group G coupled
to T [G]. The coupling lifts the Higgs branch and we are left with the Coulomb branch. Then,
upon performing an S-transformation, we obtain N = 4 super-Yang-Mills with gauge group G ∨and Dirichlet boundary conditions, whose moduli space is given by the nilpotent cone Ng∨ ofG ∨ (found by solving Nahm’s equation) [65]. We conclude that this is the Coulomb branch of
Recall that for G = G2 and G = F4 , even though the Langlands dual pairs have identical Lie algebras, the Sduality group is not a subgroup of S L(2, Z) [63]. In particular, the S-transformation involves a nontrivial involution
on the vacuum moduli space.
17
The full vacuum moduli space also contains mixed branches.
16

14

SciPost Phys. 10, 097 (2021)

T [G]. Similarly, one can couple the 4d theory to Dirichlet boundary conditions on the left andon the right of the 3d T [G] theory domain wall. This ungauges (freezes) both gauge groups G
and G ∨ . Then, being a symmetric setup under exchange of left and right, one concludes that
T [G] is mirror dual to T [G ∨ ].
In the case of the T [SU(N )] theory, there is a convenient description in terms of a D3D5-NS5 brane system in type IIB string theory [43, 66]. As the example of T [SU(2)] theory
plays a major role in the following, let us elaborate some more on the brane constructionsof T [SU(N )] theories (we refer to the original paper for a more exhaustive discussion [43]).
We start with a set of D3 branes, spanned by x 0 , . . . x 3 inside 10d Minkowski space (withcoordinates x 0 , . . . x 9 ), whose worldvolume theory leads to the N = 4 super-Yang-Mills theory
in the bulk. To describe half-BPS boundary conditions, we may add D5 branes, spanned by
x 0 , x 1 , x 2 , and x 4 , x 5 , x 6 , as well as NS5 branes, spanned by x 0 , x 1 , x 2 , and x 7 , x 8 , x 9 , onwhich the D3 branes may end (along x 3 ), see Table 2. In Figure 1, we illustrate this for theexamples of the T [SU(5)]-theory.
The brane construction makes the mirror self-duality manifest: Famously, S-duality – orrather the S-transformation inside the full S-duality group – acts by exchanging NS5- and D5
branes. Thus, starting with the brane setup in Figure 1, applying S-duality, and then repeatedly
performing Hanany-Witten moves [66] (now moving four NS5 branes to the left of all theD5’s and one to the right), it is straightforward to see, that the resulting brane configuration
precisely reduces to the one we started with.
Table 2: The D3-D5-NS5 brane system in type IIB string theory, describing boundary
conditions of 4d N = 4 super-Yang-Mills theory (for unitary groups).
0

1

2

3

D3

×

×

×

×

D5

×

×

×

NS5

×

×

D3

NS5
x 456

NS5

4

5

6

7

8

9

×

×

×

×

×

×

×

D3

D3

NS5

x 789

NS5

NS5
D5

x3
Figure 1: The D3-D5-NS5 brane system in type IIB string theory, describing the 3d
N = 4 T [SU(5)] theory. Here, the D5 branes can be moved beyond the right-mostNS5 brane by standard Hanany-Witten moves. The SCFT is reached by taking thelimit in which the separations along x 3 are taken to zero, i.e. we are moving onto
the origin of the moduli space.
By separating the NS5 branes in the x 3 direction (which corresponds to tunning the FI
parameters in the field theory), this brane construction naturally provides a UV Lagrangian15

SciPost Phys. 10, 097 (2021)

description for the T [SU(N )] theories, given by the quiverU(1)U(N − 2)U(2)SU(N )

U(N − 1)

(2.57)

where (as usual) the circular nodes are gauge-groups connected by bifundamental hypermultiplets, while the squares denote flavor symmetry factors. In the IR, the SU(N ) flavor symmetryof the Higgs branch is manifest, while the SU(N ) flavor symmetry of the Coulomb branchcomes from IR enhancement due to monopole operator.18
Similarly, by including various types of O3 planes in the brane setup [67], one may write
down the UV quiver gauge theory for T [SO(2N )],SO(2)S p(2)

SO(2N − 2)SO(4)Sp(2N − 2)SO(2N )

(2.58)
and for T [S p(2N )] theories [43],SO(2)S p(2)

SO(4)

SO(2N − 2)Sp(2N − 2)SO(2N )

Sp(2N )

(2.59)We remark that for the remaining cases, there are no – or only “bad” in the sense of [43] –quiver descriptions. However, below we conjecture the general S 3 partition function even for
those cases.
The Higgs branch chiral ring is generated by the moment map operators, which are dimension1 scalars in the adjoint representation, denoted collectively by the matrix-valued operator N .
They are subject to the chiral ring relationsεi (N ) = 0,(2.60)

where εi label the fundamental Casimirs of g. For G = SU(n), this is just the statement that
Tr N i = 0 for i = 2, 3, . . . , n. In general, this means that the singlet representation is absent
from symmetric tensor products of the moment map operators. Compared to the minimal
nilpotent orbit theories, these theories sit at the opposite end of 3d theories carrying G flavor
symmetry: by tuning Higgs branch vevs, one can flow from T [G] to any of the 3d N = 4
theories with nilpotent sub-orbits as Higgs branches.
Let g be the Lie algebra of a simple Lie group G. The S 3 partition function for T [G] theories
wasPl(w) 2πi(w(M)·M̃)ew∈W (G) (−1)
T [G](2.61)
Z(M; M̃) = d Q.
Q
i 2 α∈∆+ 2 sinh(π(α · M)) α∈∆+ 2 sinh(π(α · M̃))Here, d is the Higgs branch dimension (2.56), M, M̃ are Cartan elements of the Lie algebra
g representing the mass and FI parameters for the G × G ∨ flavor symmetry, ∆+ denotes the18
Note that for the T [SU(2)] theory (i.e. the SQED with 2 flavors), there is an alternative description as a ChernSimons theory [59]. There, the U(1) × U(1) ⊂ SU(2) × SU(2) flavor symmetry is manifest, and furthermore the
Higgs and Coulomb branches are put on equal footing.

16

SciPost Phys. 10, 097 (2021)

set of positive roots, W (G) is the Weyl group of G, and l(w) denotes the length of the Weyl
element w. The product ( · ) is the standard Killing form on h or h∗ .
This conjectured form of the S 3 partition function of the T [G] theory is motivated as a
generalization of the case of G = An−1 , which was proven by induction in [68, 69]. Furthermore, it passes a variety of non-trivial checks: first, it is manifestly mirror symmetric, i.e. we
end up with the conjectured answer for T [G ∨ ] upon exchanging the mass and FI parameters,
corresponding to a mirror symmetry transformation; second, it behaves appropriately under
the addition of line defects (Wilson and vortex lines).
Let us briefly elaborate on this and sketch the logic: It is possible to decorate the T [G]theories with a set of line defects. For instance, given a UV Lagrangian description, we cancompute the partition function with the addition of (flavor) Wilson lines (along the large circle
in S 3 ) in a representation R of G [70]. This will not enter the supersymmetric localization
computation of the S 3 partition function apart from a multiplicative factor, given by the addition of a Schur character in the mass-parameters of the representation R of G [21], i.e. theWilson loop vev is given by〈WR 〉 T [G] = tr R exp (2πi M) .

(2.62)On the other hand, we can consider adding (flavor) vortex lines VR to T [G ∨ ], which arelabeled by a representation R of (G ∨ )∨ = G. As opposed to Wilson lines, vortex lines are
disorder operators characterized by a vortex-like singularity for the fields in a theory near acurve in spacetime.19 It is known that vortex lines are mirror dual to Wilson lines [72–74]. In
particular, this means that their VEVs ought to agree upon adding them to mirror dual theories.
The vortex lines act on the integrand of the S 3 partition function by a sum over shifts of the
mass/FI parameters, i.e. up to an overall factor,
X
Z(m + i(m, ρ)),VR ◦ Z(m) ∝(2.63)
ρ∈R

where the sum is over the weights ρ of the representation R, and we have collectively denotedby m the masses (or equivalently FI-parameters).20 Thus, we can first evaluate the integral,
obtaining Z in (2.61), and then act with the vortex line operators by shifts [72, 74, 77]. By
doing so and using the fact that weights are invariant under the action of the Weyl group, one
can show that indeed〈VR 〉 T [G ∨ ] = 〈WR 〉 T [G] ,(2.64)
adding further evidence for the conjecture in (2.61).
Now, let us proceed to compute the conformal central charge CJ for T [G] theories of asingle flavor factor G using (2.46). We first remark that we only need the denominator of(2.61), since terms from the numerator necessarily mix G and G ∨ . Let us expand the partition
function in small M (and finite M̃) and only keep track of the sinh(πα · M) factor in the
denominator. We findZ T [G] (M; M̃)|M2
π2 X=−(α · M)2 ,(2.65)
6 α∈∆+Z T [G] (M; M̃)|0
where A|Mp means the p-th term in the small M expansion of A. Next, we use the following
completeness relation,
1 X(λ · α)(α · µ) ,(λ · µ) = ∨(2.66)h α∈∆+Such operators are somewhat analogously defined to 4d Gukov-Witten type surface defects [71]. Alternatively,
they can be defined by coupling the 3d theory to a 1d (supersymmetric) quantum mechanics, see e.g. [72].
20
In fact, they can be viewed as dimensional reductions of Macdonald q-difference operators [75–77], which canbe defined for general root systems.
19

17

SciPost Phys. 10, 097 (2021)

where λ and µ are arbitrary weights, to arrive at a compact formula for CJ of T [G] with respect
to the symmetry group G:
8(2.67)CJG (T [G]) = h∨ (G) .
3
The C T of T [G] for classical simple Lie group G can be computed using the formula (2.15)
and the quiver descriptions (2.57), (2.58), and (2.59). In Appendix A.3, we explicitly evaluated the C T of T [SU(3)]. The result is
C TSU(3) (T [SU(3)]) = 75.5329.

2.5(2.68)

N = 8 theories

There are three classes of N = 8 theories: the U(N )k × U(N )−k ABJM theories for k = 1, 2, theU(N +1)2 ×U(N )−2 ABJ theories, and the SU(2)k ×SU(2)−k BLG theories [12,13,35–40]. Viewing these as N = 4 theories, the SO(8) R-symmetry decomposes into the SU(2)C × SU(2)H Rsymmetry and the SU(2)fC ×SU(2)fH flavor symmetry. Among these theories, the U(1)k ×U(1)−k
ABJM theories for k = 1, 2 and the U(2)2 ×U(1)−2 ABJ theory have minimal nilpotent Coulomb
and Higgs branches. The central charges of them were computed in [22], and we summarize
the results in Table 3.
Table 3: The conformal and flavor central charges of the N = 8 theories with minimal
nilpotent Coulomb and Higgs branches when viewed as N = 4 theories.

theoriesCT

CJSU(2)

U(1)1 × U(1)−1 ABJM

24

4

U(1)2 × U(1)−2 ABJM

24

4

32

16
3

U(2)2 × U(1)−2 ABJ

The central charges of another infinite class of N = 8 theories were computed in [22].
They are the U(2)k × U(2)−k ABJM theories for k = 1, 2, and the SU(2)k × SU(2)−k BLG
theories for all positive integer k. The formulae are
Z∞


I4
tanhn (π y), In =dy y
c T = 32 2 −,I2sinh(πk y)−∞
3
1
C T = c T , CJSU(2) = c T .
2
4

3Superconformal blocks

3.1

Superconformal algebra, representations, and Casimir operator(2.69)The three-dimensional N = 4 superconformal algebra is OS p(4|4), which contains the bosonicconformal algebra SO(3, 2) with generators Pµ , Kµ , D, and the R-symmetry group SU(2)C ×SU(2)Hwith generators RCI and RHI . The fermionic generators of OS p(4|4) are the Poincaré Q-supercharges
Q αAȦ and superconformal S-supercharges S αAȦ. The (anti)commutators in the superconformal

18

SciPost Phys. 10, 097 (2021)

algebra that involve fermionic generators are{Q αAȦ, Q β B Ḃ } = εAB εȦḂ (σµ )αβ Pµ ,{S αAȦ, S β B Ḃ } = −εAB εȦḂ (σµ )αβ Kµ ,[Kµ , Q αAȦ] = εAB εȦḂ (σµ )αβ S β B Ḃ ,[Pµ , S αAȦ] = εAB εȦḂ (σµ )αβ Q β B Ḃ ,[Mµν , Q αAȦ] = (mµν )β αQ βAȦ, [Mµν , S αAȦ] = −(mµν )α β S βAȦ,1
1[RCI , Q αAȦ] = (σ I )B AQ αBȦ, [RCI , S αAȦ] = − (σ I )AB S αBȦ,22
1
1[RHİ , Q αAȦ] = (σ İ )Ḃ ȦQ αAḂ , [RHİ , S αAȦ] = − (σ İ )ȦḂ S αAḂ ,22αAȦ
α A ȦA Ȧ
α{S , Q β B Ḃ } = iδβ δB δḂ D + δB δḂ (mµν ) β M µν − δβα δȦḂ (σ I )AB RCI − δβα δAB (σ İ )ȦḂ RHİ ,(3.1)

where (σµ )α β , (σ I )AB , (σ İ )ȦḂ are Pauli matrices. The indices are raised and lowered by εαβ ,εAB , εȦḂ and εαβ , εAB , εȦḂ in the convention that upper left indices contract with lower rightindices; for example,(σµ )αβ = εβγ (σµ )α γ ,(σµ )αβ = (σµ )γ β εγα .

(3.2)

The spacetime rotation matrices mµν are defined by(mµν )α β = − 1i (σµ )αγ (σν )γβ − (σν )αγ (σµ )γβ = εµνρ (σρ )α β .
4
2

(3.3)

The SU(2)C and SU(2)H are exchanged under a Z2 outer automorphism (see Section 3.2).
The unitary representations (superconformal multiplets) can be constructed by succesH
sively acting with Q αAȦ, M23 − i M31 , RC1 − iRC2 , and RH
1 − iR 2 on the highest weight states, whichH
are states annihilated by Kµ , S αAȦ, M23 + i M31 , RC1 + iRC2 , and RH
1 + iR 2 . The superconformal multiplets of OS p(4|4) are classified into long multiplets as well as A- and B -type shortmultiplets [50, 51, 78]. They are labeled by(2 j ;2 jH )X [2`]∆ C(3.4)

and satisfy the unitarity conditions

L:∆ > ` + jC + jH + 1,

A : ∆ = ` + jC + jH + 1,
B:

∆ = jC + jH ,(3.5)

` = 0,

where X = L, A, B , and ∆, ` ∈ Z2 , jC ∈ Z2 , jH ∈ Z2 are the dimension, spin and R-charges of the(2 j ;2 j )highest weight state of X [2`]∆ C H . The superconformal multiplets that will be important inthis paper are summarized in Table 4.
The quadratic Casimir operator C of OS p(4|4) is constructed out of bilinears of the generators,
1
C = C b + [S αAȦ, Q αAȦ] − RCI RCI − RHİ RHİ ,
2
(3.6)
1
1
C b = Mµν Mµν − D2 − {Pµ , Kµ },22and commutes with all the generators in the superconformal algebra. The Casimir operator
acting on a highest weight operator O∆,`, jC , jH gives[C, O∆,`, jC , jH ] = ρ(∆, `, jC , jH )O∆,`, jC , jH ,(3.7)

where the eigenvalue ρ(∆, `, jC , jH ) is given byρ(∆, `, jC , jH ) = ρ b (∆, `, jC , jH ) + 4∆ − jC ( jC + 1) − jH ( jH + 1),
ρ b (∆, `, jC , jH ) = ∆(∆ − 3) + `(` + 1).
19

(3.8)

SciPost Phys. 10, 097 (2021)

Table 4: Short N = 4 multiplets that contain conserved currents.

Name

Alias

Superconformal primary

Identity multiplet

Identity operator

Flavor current multiplet

Moment map operator

A[0]1

Stress tensor multiplet

Dimension-1 R-singlet scalar(0;0)
A[2`]`+1Higher spin multiplet

spin-` R-singlet current(0;0)
B [0]0(2;0)
B [0]1(0;2)
B [0]1(0;0)

3.2

Other important primaries

Flavor current
R-symmetry current
Stress tensor
Higher spin current

Z2 outer automorphism

The 3d N = 4 superconformal algebra has a Z2 outer automorphism that exchanges SU(2)Cwith SU(2)H . This outer automorphism has a close relation to mirror symmetry of 3d N = 4
theories. Mirror symmetry is a duality between two UV QFTs in the sense that they flow to
the same IR CFT. The Coulomb and the Higgs branch R-symmetries in the UV are embeddedin the opposite way into the IR R-symmetries. The outer automorphism is the action of mirror
symmetry that exchanges which SU(2) R-symmetry we call Coulomb, and the other Higgs.
Note that the Z2 outer automorphism is generally not a global symmetry of the theory.
The supercharges Q αAȦ are in the (2, 1) ⊕ (1, 2) = 4 representation ofSU(2)C × SU(2)H = SO(4)R . We will use the index M for the 4 of SO(4)R , and write thesupercharge as Q αM . The stress tensor Tµν is a conformal primary but a superconformal de(0;0)

scendant of the bottom component scalar O1,0,0,0 of the multiplet A[0]1
they are related by. More explicitly,Tµν = N εM1 M2 M3 M4 (σµ )α1 α2 (σν )β1 β2 Q α1 M1 Q α2 M2 Q β1 M3 Q β2 M4 O1,0,0,0 ,(3.9)

where N is a normalization constant. The Z2 outer automorphism exchanges SU(2)C with
SU(2)H , and hence is a reflection in O(4) = SO(4)R o Z2 . In particular, the Z2 outer automorphism flips the sign of εM1 M2 M3 M4 . This implies that in a self-mirror theory where this Z2 is atrue global symmetry, the superconformal primary O1,0,0,0 of the stress tensor multiplet is Z2
odd, while the stress tensor itself Tµν is of course Z2 even.21

3.3

Single-branch superconformal blocks(2;0)

The moment map operators in B [0]1
form a spin-1 representation of SU(2)C , and can be
written as a rank-2 symmetric traceless tensor in spinor indices, OCAB (x). It contracts with
auxiliary SU(2)C spinors Y A as OC (x, Y ) = OCAB (x)YA YB . The four-point function of OC (x, Y )is a homogeneous function of degree (−4, 8),

OC (x 1 , Y1 )OC (x 2 , Y2 )OC (x 3 , Y3 )OC (x 4 , Y4 ) =(Y1 · Y2 )2 (Y3 · Y4 )2
2 2
x 12
x 34

GC (u, v; w),(3.10)

where the cross ratios u, v and w are defined asu=21|x 12 |2 |x 34 |2,|x 13 |2 |x 24 |2v=|x 14 |2 |x 23 |2,|x 13 |2 |x 24 |2w=(Y1 · Y2 )(Y3 · Y4 ).
(Y1 · Y4 )(Y2 · Y3 )(3.11)

One can verify this in N = 8 theories where the Z2 self-mirror symmetry is embedded in the SO(8)R symmetry.

20

SciPost Phys. 10, 097 (2021)

The function G(u, v; w) can be expanded in terms of the single-branch superconformal blocks
as
X X
λ2
GC (u, v, w) =(2 jC ;0) AX [2`](2 jC ;0) (u, v, w).
(3.12)
X [2`]∆

X =L,A,B ∆,`, jC

∆

(0;2)
are denoted by OH (x, Ye ), where Ye Ȧ is an
Similarly, the moment map operators in B [0]1
auxiliary SU(2)H spinor. The four-point function of OH (x, Ye ) admits a similar single-branch
superconformal block expansion as〈OH (x 1 , Ye1 )OH (x 2 , Ye2 )OH (x 3 , Ye3 )OH (x 4 , Ye4 )〉 =e) =GH (u, v, w

X

X

λ2

X =L,A,B ∆,`, jH(0;2 j ;)X [2`]∆ H(Ye1 · Ye2 )2 (Ye3 · Ye4 )2
2 2
x 34
x 12

e ),
GH (u, v; w
(3.13)e ).
AX [2`](0;2 jH ) (u, v, w∆

By the Z2 outer automorphism, the single-branch superconformal blocks AX [2`](2 j;0) and∆AX [2`](0;2 j) are the same functions, i.e.
∆

AX [2`](2 j;0) (u, v, w) = AX [2`](0;2 j) (u, v, w) ≡ AX [2`]2 j (u, v, w),∆(3.14)∆∆and they were originally computed in [25, 44].

3.4

Mixed-branch superconformal blocks

Now, consider the four-point function of two OC and two OH operators,〈OC (x 1 , Y1 )OC (x 2 , Y2 )OH (x, Ye3 )OH (x, Ye4 )〉

2

2
(Y1 · Y2 )(Ye3 · Ye4 )(Y2 · Y3 )(Ye1 · Ye4 )=Gs (u, v) =G t (v, u),|x 12 ||x 34 ||x 23 ||x 14 |(3.15)

which can be expanded in terms of either the s-channel mixed-branch superconformal blocks
or the t-channel mixed-branch superconformal blocks
X
X
Gs (u, v) =λC,C,X [2`](2 jC ;2 jH ) λH,H,X [2`](0;0) As(0;0) (u, v),
G t (u, v) =X

X

X =L,A,B ∆,`, jC , jH

X [2`]∆∆∆

X =L,A,B ∆,`, jC , jH

λ2

(2 j ;2 j )C,H,X [2`]∆ C H

At(2 j ;2 j )X [2`]∆ C H(3.16)(u, v).

The s- and t-channel mixed-branch superconformal blocks can be computed by solving thesuper-Casimir equations. We first strip off the auxiliary variables Y and Ye in (3.15), and find
C
C
H
H
〈O++
(x 1 )O−−(x 2 )O+̇
(x )O−̇(x )〉 =
−̇ 4
+̇ 3

1
|x 12

|2 |x

34

|2

Gs (u, v) =

1
|x 23

|2 |x

2
14 |

G t (v, u).

(3.17)CCH
The commutators of the Casimir operator (3.6) with O++
O−−
and O+̇
OC give differential+̇ −−
operators acting on the functions Gs (u, v) and G t (u, v),

1
C s Gs (u, v),|x 12 |2 |x 34 |2
1
H
C
C
H
〈[C, O+̇(x )O−−(x 2 )]O++(x 3 )O−̇
(x )〉 =C t G t (u, v).
−̇ 4
+̇ 1
2
|x 12 | |x 34 |2C
C
H
H〈[C, O++(x 1 )O−−(x 2 )]O+̇
(x )O−̇(x )〉 =
−̇ 4
+̇ 3

21

(3.18)

SciPost Phys. 10, 097 (2021)

Expanding the above equations in terms of the mixed-branch superconformal blocks gives the
Casimir equations

C s Ass(2 j ;2 j ) (u, v) = ρ(∆, `, jC , jH )A(2 j ;2 j ) (u, v),
X [2`]∆ C H
X [2`]∆ C H
s
C t At(2 j ;2 j ) (u, v) = ρ(∆, `, jC , jH )A(2 j ;2 j ) (u, v).
X [2`]∆ C H
X [2`]∆ C H(3.19)The differential operators C s and C t are computed in Appendix B. We solve the above Casimir
equations by expanding the mixed-branch superconformal blocks in terms of the N = 2 superconformal blocks, i.e.

As

(2 j ;2 jH )X [2`]∆ Cs
N =2= f0,0G∆,`+ f 1s12 ,− 2

G N =2
1∆+ 2 ,`− 21+ f 1s 1 G N =2
1
2,2∆+ 2 ,`+ 21

s
N =2
s
N =2
s
N =2
+ f1,−1G∆+1,`−1+ f1,0G∆+1,`+ f1,1G∆+1,`+1+ f 3s12 ,− 2At(2 j ;2 j )X [2`]∆ C H

G N =23∆+ 2 ,`− 21+ f 3s 1 G N =2
3

t
N =2
= f0,0G∆,`+ f 1t12 ,− 22,2∆+ 2 ,`+ 12

G N =2
1∆+ 2 ,`− 21s
N =2+ f2,0G∆+2,`,+ f 1t 1 G N =212,2(3.20)∆+ 2 ,`+ 21

N =2
N =2
t
N =2
t
t
G∆+1,`+1G∆+1,`+ f1,1G∆+1,`−1+ f1,0+ f1,−1
N =2
t
N =2
t
+ f 3t 1 G N =2
3
1 + f3 1G
3
1 + f 2,0 G∆+2,` ,
2 ,− 2 ∆+ 2 ,`− 2
2 , 2 ∆+ 2 ,`+ 2
N =2
where the N = 2 superconformal blocks G∆,`
were computed in [79]. Their expansion interms of the bosonic conformal blocks is given by
N =2
N =2
N =2
N =2
G∆+2,` ,G∆+1,`−1 + f2,0G∆+1,`+1 + f1,−1G∆,`= G∆,` + f1,1

(3.21)

with the following coefficients22(` + 1)(∆ + `)2,2(2` + 1)(∆ + `)(∆ + ` + 1)`(∆ − ` − 1)2
N =2
=
,
f1,−1
2(2` + 1)(∆ − ` − 1)(∆ − `)∆2 (∆ + `)2 (∆ − ` − 1)2
N =2
=
f2,0
.
4(2∆ + 1)(2∆ − 1)(∆ + `)(∆ + ` + 1)(∆ − ` − 1)(∆ − `)
N =2
=
f1,1

(3.22)The bosonic conformal blocks are normalized asG∆,` (u, v) →

∆−`
(ε)ε
u 2 (1 − v)`(` + ε)ε

(3.23)

in the limit u → 0 then v → 1, where (x)n = Γ (x + n)/Γ (x) is the Pochhammer symbol andε = (d − 2)/2 = 1/2.
3.4.1 s-channel
It is straightforward to solve the s-channel Casimir equation. The long multiplet block is given
by
N =2
s
N =2
s
N =2
s
N =2
As
+ f1,−1G∆+1,`−1+ f1,1G∆+1,`+1+ f2,0G∆+2,`,(0;0) = G∆,`
(3.24)
L[2`]∆

22
Note that due to the difference in the normalization of the bosonic block, (3.23) versus (65) of [79], thecoefficients (3.22) are different from the coefficients given by (67) of [79].

22SciPost Phys. 10, 097 (2021)

where the coefficients are(` + 1)(∆ + `)`(−∆ + ` + 1)s
, f1,−1=−,2(2` + 1)(∆ + ` + 1)
2(2` + 1)(` − ∆)(∆ + 1)2 (−∆ + ` + 1)(∆ + `)
s
f2,0
=
.
4(2∆ + 1)(2∆ + 3)(` − ∆)(∆ + ` + 1)
s
f1,1
=−

(3.25)The short multiplet blocks are
1 N =2N =2= G`+1,`− G`+2,`+1,4(0;0) = 1.

As

(0;0)

A[2`]`+1As(3.26)

B[0]0

3.4.2

t-channel

The t-channel Casimir equation admits the following classes of solutions,At(2 j ;2 jH )N =2
= G∆,`

with(2 j ;2 jH )= G N =2
1

with

(2 j ;2 jH )= G N =2
1

with

(2 j ;2 jH )N =2
= G∆+1,`withjC ( jC + 1) + jH ( jH + 1) = 0,

(2 j ;2 jH )N =2
= G∆+1,`−1withjC ( jC + 1) + jH ( jH + 1) = 2`.

X [2`]∆ CAt

X [2`]∆ CAt

X [2`]∆ CAt

X [2`]∆ CAt

X [2`]∆ C∆+ 2 ,`+ 21∆+ 2 ,`− 21

jC ( jC + 1) + jH ( jH + 1) = 2∆,
1
jC ( jC + 1) + jH ( jH + 1) = ∆ − ` − ,21
jC ( jC + 1) + jH ( jH + 1) = ∆ + ` + ,2(3.27)

There are several constraints on the possible superconformal blocks. First, the dimension,
spin and R-charges are constrained by the unitarity bounds (3.5). Second, by the Lorentz
H
C
and R-symmetry selection rules, the operators that appear in the O+̇
× O−−
OPE must have+̇
(2 j ;2 j )jC = − jH = 1, and ` ∈ Z. Hence, they must all be superconformal multiplets X [2`]∆ C H with`, jC and jH being all integers or all half integers. Third, the superconformal primaries that apH
Cpear in the O+̇
× O−−
OPE must be in the ( jC , jH ) = (1, 1) representation of SU(2)C ×SU(2)H ;+̇
hence, the first line of (3.27) can only take value ∆ = 2 and ` = 0.
The superconformal block belonging to the first line of (3.27) is

At

(2;2)

B[0]2

N =2
.
= G2,0(3.28)The superconformal blocks belonging to the second line of (3.27) are

At

(1;1)

= G N =2
5

= At

(1;3)

= G N =2
11

At

(3;3)

= G N =217A[2`]`+2

At

(3;1)

L[2`]`+4

L[2`]`+4

L[2`]`+8`+ 2 ,`+ 12`+ 2 ,`+ 12`+ 2 ,`+ 121
with ` ∈ Z≥0 + ,21with ` ∈ Z≥0 + ,21with ` ∈ Z≥0 + .
2(3.29)The superconformal blocks belonging to the third line of (3.27) are

At

(3;1)

A[1] 7
2

= At

(1;3)

A[1] 7

N =2= G4,0
,

2

At

(3;3)

A[3] 11

N =2= G6,1
,

2

At

(3;3)

L[1] 132

23

N =2= G7,0.

(3.30)

SciPost Phys. 10, 097 (2021)

Finally, the superconformal blocks belonging to the fourth line of (3.27) are
(0;0)

N =2
= G`+2,`
,

= At

(0;2)

N =2
= G4,0
,

At

(2;2)

N =2
= G6,1,AtA[2`]`+1

At

(2;0)

A[2]3

A[2]3
A[4]5

At

(4;0)

= At

(0;4)

N =2
= G7,2
,

At

(4;2)

= At

(2;4)

N =2
= G9,3
,

At

(4;4)

N =2
= G12,5
,

A[6]6
A[8]8

A[6]6
A[8]8

A[12]11At(0;0)
L[2`]∆

At

(2;0)

L[2]∆

with ∆ > ` + 1, ` ∈ Z≥0 ,

= At

(0;2)

N =2
= G∆+1,0
with ∆ > 3,

At

(2;2)

N =2
= G∆+1,1
with ∆ > 5,

L[2]∆

L[4]∆

At

(4;0)

= At

(0;4)

N =2
= G∆+1,2
with ∆ > 6,

At

(4;2)

= At

(2;4)

N =2
= G∆+1,3
with ∆ > 8,

At

(4;4)

N =2
with ∆ > 11.
= G∆+1,5

L[6]∆

L[8]∆

L[6]∆

L[8]∆

L[12]∆

4

N =2= G∆+1,`(3.31)The protected sector

In [22, 31, 32], the authors showed that every three-dimensional N = 4 SCFT admits a protected sector as a topological quantum mechanics (TQM), which lives on a straight line in
the three dimensional Euclidean space R3 . The operators in the TQM are either the Coulomb
or Higgs branch chiral ring operators with suitable twisting, where the translation along thestraight line is accompanied with certain SU(2)C or SU(2)H rotations. The correlation functions of these operators depend only on the ordering of the operators on the straight line but
not their positions. This implies that the OPE along the straight line forms an associative algebra, which is called the protected associative algebra. Crossing symmetry of the four-point
function amounts to associativity of the protected associative algebra.
The multiplication of the Coulomb or Higgs branch chiral ring operators in the protectedassociative algebra is a non-commutative deformation of the commutative chiral ring multiplication. It was observed in [31] that the leading order deformation is determined by the
Poisson bracket of the chiral ring; hence, the protected associative algebra is a deformation
quantization of the chiral ring.
When the Coulomb or Higgs branch of a given theory coincides with the minimal nilpotent
orbit of a complex simple Lie algebra g (except g = sl2 ), the protected associative algebra isunique [31]. We will study these theories by two equivalent approaches in Sections 4.1 and 4.2.
In Section 4.1, we consider the crossing equations of the four-point function of the momentmap operators in the TQM, and derive analytical bounds on the flavor central charges andother OPE coefficients. We dub this approach the “mini-bootstrap”. We find that the minimal
nilpotent theories sit at the kinks of the allowed regions, where the values of the charges nicely
agree with the ones computed in Section 2 and Appendix A by localization. In Section 4.2, westudy the deformation quantization of the chiral ring of the minimal nilpotent theories. Using
associativity, we fix the coefficients in the protected associative algebra up to quadratic order.
We also compute the four-point function of the moment map operators using the protected
24

SciPost Phys. 10, 097 (2021)

associative algebra, and find agreement with the four-point function computed using minibootstrap.

4.1

Mini-bootstrap

Consider a three dimensional N = 4 SCFT with a simple flavor group G, which is generated
by the flavor currents in the flavor current multiplets. Without loss of generality, we assume
(2;0)
(0;2)
these flavor current multiplets are of type B [0]1
(as opposed to B [0]1 ). The momentmap operators OaC (x, Y ) are in the adjoint representation of the flavor group G, and have thefour-point function

OaC (x 1 , Y1 )OCb (x 2 , Y2 )OcC (x 3 , Y3 )OdC (x 4 , Y4 ) =(Y1 · Y2 )2 (Y3 · Y4 )2
2 2
x 34
x 12

Gabcd (u, v; w).

(4.1)

The four-point function simplifies when all the four-points are along a straight line and witha suitable twisting by the R-symmetry rotation. More precisely, we consider

OaC (s) ≡ OaC (x = (0, 0, s), Y = (1, s/2r)),(4.2)

so that the four-point function becomesOaC (s1 )OCb (s2 )OcC (s3 )OdC (s4 ) = Gabcd (uw , vw ; w).

(4.3)

In the above, r is a dimensionful parameter to make the combination 2rs dimensionless; wedefineYi · Y j =1(si − s j ),2rw=

s12 s34,s14 s23u = uw ≡w2,(1 + w)2v = vw ≡1.
(1 + w)2(4.4)

The function Ga bcd (uw , vw ; w) has a very simple superconformal block expansion. In general,
(2;0)
gives
the fusion of two flavor current multiplets B [0]1(2;0)

B [0]1(2;0)× B [0]1(0;0)= B [0]0(2;0)+ B [0]1(4;0)+ B [0]2

+

∞
X
`=0

+

∞
X
`=0

(0;0)

A[2`]`+1

(4.5)

(2;0)
(0;0)
A[2`]`+2 + L[2`]∆ .

However, in the particular configuration (4.3), the long multiplet blocks and the A-type shortmultiplet blocks vanish identically, and the B -type short multiplet blocks only depend on theordering of the si . More explicitly, we have
Ga bcd (uw , vw ; w)
abcd 2= P1a bcd AB[0](0;0) (uw , vw ; w) + Padj
λA (2;0) (uw , vw ; w)(2;0)
B[0]1 ,adj B[0]10
abcd 2+ P1a bcd λ2 (4;0) + P2adj
λ(4;0)
B[0]2 ,1
B[0]2 ,2adj
Xabcd 2+PR
λAB[0](4;0) (uw , vw ; w)
(4;0)
(S,i)

ab

cd

B[0]2

,R(S,i)ia bcd 2
− 2Padj λsgn(s12 s34 s13 s24 )
(2;0)
B[0]
,adj

=δ δ

+ 6 P1a bcd λ2

(4.6)

2

1

(4;0)

B[0]2

,1

abcd 2+ P2adj
λ

(4;0)

B[0]2

25

,2adj

+

X
i

abcd 2
PR
λ
(S,i)


(4;0)

B[0]2

,R(S,i),SciPost Phys. 10, 097 (2021)

a bcd
a bcd
a bcd
where the Padj
, P2adjand PRare projection matrices defined in Appendix C, and we have(S,i)

assumed that in the OaC × OCb OPE, the identity multiplet must be in the trivial representationof the flavor group, and the flavor current multiplets must be in the adjoint representation.
Defining
1
P dc ba Gabcd (u, v; w),
GRi (u, v; w) =(4.7)
dim(Ri ) Ri
the crossing equation of the four-point function (4.1) can be written asu
GRi (v, u; w−1 ),vw2

Fi j GR j (u, v; w) =(4.8)

where Fi j is the crossing matrix defined in Appendix C. After the specialization (4.2), thebootstrap equation reduces to
F i j GR j = GR i ,(4.9)
where the GRi can be computed by use of (4.6) and (4.7),
G1 = dim(G) + 6λ2

(4;0)

B[0]2

GR(S,i) = 6λ2(4;0)
B[0]2 ,R(S,i),1

Gadj = −2λ2

,

(2;0)

B[0]1

,adj

(2;0)

B[0]1

,adj

λ2

(4;0)

B[0]2

,2adj

(4;0)

, λ2

B[0]2

(2;0)

B[0]1

,adj

(4;0)

B[0]2

and λ2the positivity conditions

which give bounds on λ2

(4;0)

B[0]2

,2adj

,
(4.10)

GR(A,i) = 0.

,

For simple Lie groups, we find that the coefficients λ2
by the coefficients λ2

G2adj = 6λ2

,

,1

(4;0)

B[0]2

and λ2

(4;0)

B[0]2

,R(S,i)are determined

via the crossing equation (4.9). Unitarity gives
(4;0)

B[0]2

and λ2,1,2adj,R(S,i)≥ 0,(4.11). The bounds for flavor groups G = SU(2),SU(3), G2 , and E6 are given in Figure 2, where the shaded region is ruled out.
The maximal nilpotent orbit condition is
λ2

(4;0)

B[0]2

,1

= 0,

(4.12)

which combines with the positivity conditions (4.11) gives a line segment in the λ2
λ2

(2;0)

B[0]1

(4;0)

B[0]2

,2adj

,adj

-

plane. As discussed in Section 2, the T [G] theories are examples of theories on

this line segment.
The minimal nilpotent orbit condition is
λ2

(4;0)

B[0]2

,1

= λ2

(4;0)

B[0]2

,R(S,i)= 0.

(4.13)For G = SU(2), the equation (4.9) and the conditions (4.13) give a linear relation


1
2
2
1+λ
.
λ
=
(2;0)
(4;0)
B[0]1 ,adj
B[0]2 ,2adj
5(4.14)As discussed in Section 2, the examples of theories on this line are the Z2 gauge field coupled
to a free hypermultiplet, the SQED with N f = 2, and the U(2)2 × U(1)−2 ABJ theory. For othersimple Lie groups, the equation (4.9) and the conditions (4.13) uniquely determine λ2 (2;0)
and λ2

(4;0)

B[0]2

,2adj

. The results are given in Table 5. The OPE coefficient λ2

to the flavor central charge CJ by a general formula
λ2

(2;0)
B[0]1 ,adj

26

=

8h∨
.
CJ

B[0]1

(2;0)

B[0]1

,adj

,adj

is related(4.15)

SciPost Phys. 10, 097 (2021)

Table 5: Values of the flavor central charge CJ and the OPE coefficient λ2

(4;0)

B[0]2

,2adj

fixed by the minimal nilpotent orbit condition and solved by mini-bootstrap.

λ2

GAn−1 , n ≥ 3

Bn

Cn

Dn

E6

E7

E8

F4

G2

CJ

8n
n+1
2(n+1)(n+2)
3n(n+3)8(2n−3)nn(2n−1)2(n+1)(2n−3)432(n−2)2n−1
2n2 −3n+1
4n2 −6n−4

192
13
13
27

384
19
95
216

960
31
217
540

12

64
9
2
3

(4;0)

B[0]2

4.2

,2adj

1

14
27

Deformation quantization

Let v a be a vector in the adjoint representation of a simple Lie algebra g. The coordinate ring
A of the minimal nilpotent orbit of g is
I

(S,i)
A = C[v a ]/{v a v a = 0, v a v b (ΓR(S,i) )ab= 0},(4.16)

where R(S,i) are the representations appeared in the symmetric tensor product (adj ⊗ adj)SI

that are not 2adj, and (ΓR(S,i) )a(S,i)
are the associated Clebsch-Gordan coefficients. The Poissonbbracket on the minimal nilpotent orbit is{v a , v b }P.B = f abc v c .

(4.17)

We consider the deformation quantization algebra of the chiral ring A. The algebra is acommutative ring A[r −1 ] equipped with an associative star product? : A[r −1 ] ⊗ A[r −1 ] → A[r −1 ].

(4.18)The star product satisfies the conditions
1{ f (v), g(v)}P.B ,2r
1
f (v) ? g(v) = f (v)g(v) + { f (v), g(v)}P.B + O(r −2 ).
4r[ f (v), g(v)]? =(4.19)The star product of two v a takes the general form as
1 abc c1
f
v + 2 λ2 δ ab ,
4r
4r
1 e
1
a
α
aα A
bc abd d c
ab bv ? v = (Γ3adj )A v + (Γ2adj )α f
v v + 2λ2 (Γ2adj )α v ,2r4rv a ? v b = (Γ2adj )αa b v α +(4.20)

e 2 are coefficients to be determined, v α and v A are defined bywhere λ2 and λv α ≡ (Γ2adj )αab v a v b ,v A ≡ (Γ3adj )Aaα v a v α ,(4.21)and (Γ3adj )Aaα for A = 1, · · · dim(3adj) are the Clebsch-Gordan coefficients of the decomposition
adj ⊗ 2adj ⊃ 3adj with the normalization(Γ3adj )Aaα (Γ3adj )Baα = δAB .

(4.22)

Contracting the first equation of (4.20) with (Γ2adj )αab , we findv α = (Γ2adj )αab v a ? v b .
27(4.23)

SciPost Phys. 10, 097 (2021)

SU(2)λ2B[0](4;0) ,2adj

SU(3)λ2B[0](4;0) ,2adj

2

2

0.9

0.9

free hyper/Z2

0.6

SQED with Nf =3

0.6

U(2)2 xU(1)-2 ABJ andSQED with Nf =2

T[SU(3)]

0.3

0.3

3

6

9

λ2B[0](2;0)

3

1

G2

λ2B[0](4;0) ,2adj

9

λ2B[0](2;0)
1E6

λ2B[0](4;0) ,2adj

2

6

2

0.9

0.9

0.6

0.6

T3

T[G2 ]
0.3

0.3

3

6

9

λ2B[0](2;0)

T[E6 ]

31

6

9

λ2B[0](2;0)
1Figure 2: Analytic bounds from the mini-bootstrap equations (4.10) and the positivity conditions (4.11). The shaded regions are ruled out. The solid red lines are
the maximal nilpotent condition (4.12). The dashed lines show the values of theOPE coefficient λ2 (2;0)
determined via (4.15) by the central charges computed
B[0]1

,adj

in Section 2 for various theories.
e 2 by imposing associativity. Consider
Let us try to fix λ2 and λv a ? (v b ? v c ) = (v a ? v b ) ? v c


1
1 e
cβ
cd d
= (Γ2adj )βa b (Γ3adj )A v A − (Γ2adj )βde f cd f v f v e + 2 λ(Γ)
v
2 2adj β
2r
4r
1 a bd1
1+f(Γ2adj )βd c v β +f abd f dce v e + 2 λ2 δab v c ,
2
4r
16r
4r

(4.24)

where we have used the commutators
1 a α
1{v , v } =(Γ2adj )αbc {v a , v b v c }
2r
2r
1
1
= (Γ2adj )αbc f abd v d v c = (Γ2adj )αbc f abd (Γ2adj )βdc v β .
rr[v a , v α ]? =(4.25)

Contracting (4.24) with the Clebsch-Gordan coefficients (C.4), the O(r −2 ) order of the equa-

28

SciPost Phys. 10, 097 (2021)

tion givese 2 (Γ1 ) bc (Γ2adj )a b (Γ2adj )ce + 1 (Γ1 ) bc f abd f dce + λ2 (Γ1 )ae = λ2 dim(G)(Γ1 )ae ,
λ
β
β
4
α
a
b
ce
e 2 (Γ2adj )ae ,e 2 (Γ2adj ) (Γ2adj ) (Γ2adj ) + 1 (Γ2adj )α f abd f dce + λ2 (Γ2adj )α = λ
λ
ae
α
bc
bc
β
β
4
I(S,i)e 2 (ΓR ) I(S,i) (Γ2adj )a b (Γ2adj )ce + 1 (ΓR ) I(S,i) f abd f dce + λ2 (ΓR )ae
= 0,
λ
β
β
(S,i)
(S,i) bc(S,i) bc4
e 2 (Γadj ) f (Γ2adj )a b (Γ2adj )ce + 1 (Γadj ) f f abd f dce + λ2 (Γadj ) f = 1 (Γadj ) f f bcd f ade ,
λ
ae
β
β
bc
bc
bc
4
4
1
I(A,i)e 2 (ΓR ) I(A,i) (Γ2adj )a b (Γ2adj )ce + (ΓR ) I(A,i) f abd f dce + λ2 (ΓR )ae
λ
= 0,
β
β
(A,i) bc
(A,i) bc
(A,i)
4

(4.26)

which can be rewritten in terms of the crossing matrices defined in (C.6) as∨

e 2 F1 2adj + h F1 adj + λ2 = λ2 dim(G),
λ
2
∨
h
2adj
e2,
e 2 F2adj
F2adj adj + λ2 = λλ+2∨e 2 FR 2adj + h FR adj + λ2 = 0,
λ
(S,i)
(S,i)
2
∨
∨
e 2 Fadj 2adj + h Fadj adj + λ2 = h ,λ22∨e 2 FR 2adj + h FR adj + λ2 = 0.
λ
(A,i)
(A,i)
2

(4.27)For SU(2), these equations reduce toe 2 = 0.
3 − 6λ2 + 5λ(4.28)

e 2 are uniquely determined. The results are summaFor the other simply Lie groups, λ2 and λrized in Table 6.
Table 6: The leading coefficients in the deformation quantization algebra of the chiralring of the minimal nilpotent theories.

G

An−1 , n ≥ 3

Bn

Cn

Dn

E6

E7

E8

F4

G2

λ2
e2
λ

n
− 4(n+1)− 2n−3
4n

− 18

n−2
− 2n−1

6
− 13

− 12
19

− 30
31

− 38

− 92

− n+2
n+3

− 4(n+1)3(2n−1)− 34− 3n−3
2n+1

− 34

− 53

− 73

− 76

− 98

For each homogeneous polynomial p(v), there is an associated chiral ring operator O f (v) .
The correlation functions of these operators in the topological quantum mechanics can becomputed by taking the constant term of the start product as1

O p1 (v) (s1 ) · · · O pn (v) (sn ) = (2r) 2 (d1 +···+dn ) C.T.(p1 (v) ? · · · ? pn (v)) with s1 ≥ · · · ≥ sn ,(4.29)
where di is the degree of the polynomial pi (v), and C.T.(· · · ) denotes the constant term of(· · · ). The moment map operators are given by
Oa = Oev a ,1eva = p va.
λ2
29(4.30)

SciPost Phys. 10, 097 (2021)

Using the star products (4.20), we compute the four-point function of the moment map operator Oa ,Oa (s1 )O b (s2 )Oc (s3 )Od (s4 ) = (2r)4 C.T.(eva ? evb ? evc ? evd )= δ a b δ cd +(4.31)h∨ abcd(2r)4
abcd
Padj +
C.T.(v α ? v α )P2adj.
2λ2
dim(2adj)Comparing (4.31) with (4.6) and using (4.15), we find the relation
CJ = −32λ2 ,(4.32)

which perfectly agrees with Table 5 and Table 6.

5

Superconformal bootstrap

5.1

Coulomb-Higgs mixed correlators and Z2 outer automorphism

This subsection sets up the bootstrap equations for the four-point function of flavor current
multiplets. We begin by considering the mixed correlator system that involves one flavor current in each of SU(2)C and SU(2)H , with some focus on the effect of the Z2 outer automorphismthat exchanges SU(2)C and SU(2)H on the bootstrap, and then set up the bootstrap system forthe most general flavor symmetry group.
5.1.1

U(1) × U(1) flavor symmetry(2;0)

Let us consider the four-point functions involving U(1) flavor current multiplets B [0]1and

〈OC (x 1 , Y1 )OC (x 2 , Y2 )OH (x, Ye3 )OH (x, Ye4 )〉
2

2

(Y2 · Y3 )(Ye1 · Ye4 )(Y1 · Y2 )(Ye3 · Ye4 )Gs (u, v) =G t (v, u),=|x 12 ||x 34 ||x 23 ||x 14 |2
(Y1 · Y2 )(Y3 · Y4 )CCCCGC (u, v, w),〈O (x 1 , Y1 )O (x 2 , Y2 )O (x, Y3 )O (x, Y4 )〉 =|x 12 ||x 34 |
2
(Ye1 · Ye2 )(Ye3 · Ye4 )
H
H
H
H
e ).
〈O (x 1 , Ye1 )O (x 2 , Ye2 )O (x, Ye3 )O (x, Ye4 )〉 =GH (u, v, w
|x 12 ||x 34 |

(5.1)

(0;2)
B [0]1 (0, 2, or 4 of each):

The four-point functions can be expanded in terms of the various superconformal blocks as
X X
Gs (u, v) =λC,C,X [2`](0;0) λH,H,X [2`](0;0) As(0;0) (u, v),∆

X =L,A,B ∆,`G t (u, v) =X

λ2∆,`, jC , jH

GC (u, v, w) =X

X

X =L,A,B ∆,`, jCe) =GH (u, v, w

X

(2 j ;2 jH )C,H,L[2`]∆ CX

X =L,A,B ∆,`, jH

λ2∆

X [2`]∆tA∆,`,
j , j (u, v),
C H(2 jC ;0) AX [2`](2 jC ;0) (u, v, w),

C,C,X [2`]∆λ2∆(0;2 jH )H,H,X [2`]∆

(5.2)

e ),
AX [2`](0;2 jH ) (u, v, w∆

where by the Z2 outer automorphism,

AX [2`](0;2 j) (u, v, w) = AX [2`](2 j;0) (u, v, w).
∆

∆

30

(5.3)

SciPost Phys. 10, 097 (2021)

Putting the above together gives the crossing equations
X X
v
λC,C,X [2`](0;0) λH,H,X [2`](0;0) As
X

=u

λ2

∆,`, jC , jH

X

X

λ2

X

λ2

X =L,A,B ∆,`, jH(u, v)tA∆,`,
j , j (v, u),
C H(2 jC ;0) FX [2`](2 jC ;0) (u, v, w) = 0,

C,C,X [2`]∆X =L,A,B ∆,`, jC

X(2 j ;2 jH )C,H,L[2`]∆ C(0;0)

X [2`]∆∆∆

X =L,A,B ∆,`

(5.4)

∆

(0;2 jH )H,H,X [2`]∆e ) = 0,FX [2`](0;2 jH ) (u, v, w∆e ) arewhere the functions FX [2`](2 jC ;0) (u, v, w) and FX [2`](0;2 jH ) (u, v, w∆∆

FX [2`](2 j;0) (u, v, w) = FX [2`](0;2 j) (u, v, w)
∆
∆
(5.5)
u
≡ vwAX [2`](0;2 j) (u, v, w) − AX [2`](0;2 j) (v, u, w−1 ).
∆∆w
These equations comprise a mixed system by the fact that λC,C,X [2`](0;0) and λH,H,X [2`](0;0) are∆∆common coefficients. A more explicit form of (5.4) is given in (D.1).
5.1.2

Mirror symmetry and the Z2 outer automorphism

We are particularly interested in theories where the Z2 outer automorphism that exchanges theSU(2)C with SU(2)H is a true global symmetry of the theory. For j ≡ jC = jH , let us denote a Z2
( j, j)even/odd intermediate multiplet as X [2`]∆,± , respectively. When the Z2 outer automorphism
is a global symmetry, the OPE coefficients are related as
λC,C,X [2`]( j, j) = ±λH,H,X [2`]( j, j) ,
∆,±

∆,±

λC,C,X [2`](2 jC ,2 jH ) = λH,H,X [2`](2 jH ,2 jC ) ,
∆

for

∆

jH 6= jC .

(5.6)

Assuming the Z2 symmetry, the crossing equation (5.4) becomes

XX
2
As
v
λ2
−
λ
(0;0)
(0;0)
(0;0) (u, v)
C,C,X [2`]∆,+

X ∆,`

X

=u

∆,`, jC , jH

XX X
X ∆,` jC 6=0

λ2(2 j ;2 jH )C,H,L[2`]∆ C

λ2(2 j ;0)
C,C,X [2`]∆ C

XX
+
λ2
X ∆,`

C,C,X [2`]∆,−

X [2`]∆tA∆,`,
j , j (v, u),
C H

(5.7)

FX [2`]2 jC (u, v, w)∆

2(0;0) + λC,C,X [2`]∆,+


(0;0)

C,C,X [2`]∆,−FX [2`]2 jC (u, v, w) = 0 .
∆A more explicit form of (5.7) is given in (D.2).
Every solution to (5.7) can clearly be lifted to a solution to the more general crossing
equation (5.4) (not assuming Z2 symmetry), so the bootstrap constraints imposed by the Z2
symmetric (5.7) are no weaker than those imposed by the general (5.4). On the other hand,
every solution λ to the more general crossing equation (5.4) induces a Z2 symmetric solution
λ̃ to (5.7), by1
λ̃C,C,X [2`](0;0) =λC,C,X [2`](0;0) ± λH,H,X [2`](0;0) ,
∆,±
∆
∆
2

1
2
2
(5.8)
λ̃2
=
λ
+
λ
,
for j 6= 0 ,(2 j;0)(2 j;0)(0;2 j)
C,C,X [2`]∆C,C,X [2`]∆
H,H,X [2`]∆2λ̃C,H,L = λC,H,L .
31

SciPost Phys. 10, 097 (2021)

The lesson here is that we can always construct a Z2 symmetric correlator that solves the
crossing equations (5.7) from a non-symmetric one. Therefore, the crossing equations (5.7)
can be used to constrain the quantities on the right hand sides of (5.8) even for theories without
the Z2 symmetry.23
But then, how do we test whether a theory we want to bootstrap truly has this additional
Z2 global symmetry?
The answer is as follows. If the Z2 outer automorphism is a true symmetry, then in aninteracting theory without further global symmetry (in addition to the Z2 ), we do not expect
the Z2 even spectrum to coincide with the Z2 odd spectrum. In other words, we expect acollection of the Z2 even states to not have a Z2 odd “partner” with the same ∆, `, r, and viceversa. By contrast, the induced solution (5.8) to the crossing equations (5.7) from a genericsolution to the crossing equations (5.4) has no relation between λCC and λHH , so the Z2 evenand odd contributions appear together for most if not all ∆, `, r.
In conclusion, the hallmark of a genuinely Z2 symmetric four-point function is that a collection of the Z2 even states do not have Z2 odd “partners” with the same ∆, `, r, or vice versa.
As will be described in the next subsection, the spectrum of multiplets contributing to thefour-point function can be fully determined by the bootstrap if we extremize an OPE coefficient.
Testing the extremal spectrum against the above criterion allows us to verify whether theZ2 outer automorphism is a genuine symmetry of the extremal theory. In the field theory
context, the Z2 outer automorphism becomes a genuine symmetry when the theory has a UV
construction that is self-mirror.
5.1.3

GC × GH flavor symmetry

Let us consider the four-point functions analogous to (5.1), but for GC flavor current multiplets
(0;2)
(2;0)
B [0]1 and GH flavor current multiplets B [0]1 . The crossing equations are
1 XX
s(u, v)
λ(0;0) λ(0;0) Au X ∆,` C,C,X [2`]∆ ,1 H,H,X [2`]∆ ,1 X [2`](0;0)
∆
1 X
t
=
λ2
(v, u),(j ;j ) Av ∆,`, j , j C,H,L[2`]∆C H ∆,`, jC , jH
X C H
X X
r0C
λ2
(u, v, w) = 0,
(2 jC ;0) 0 FX [2`](2 jC ;0) ,r
X ∆,`, j r0 ∈adjC ⊗adjC

X XC

X

X ∆,`, jH r0H ∈adjH ⊗adjH

C,C,X [2`]∆,rC∆(5.10)

C0

λ2(0;2 j )H,H,X [2`]∆ H ,r0H

FX [2`](0;2 jH ) ,r rH (u, v, w) = 0,∆

H

where rC , rH denote representations of GC , GH , and we have defined00FX [2`](2 j;0) ,r r (u, v, w) = FX [2`](0;2 j) ,r r (u, v, w)
∆

∆

0 1
w
1
≡ Fr r AX [2`](0;2 j) (u, v, w) − δrr
A(0;2 j) (v, u,
w ),∆u
vw X [2`]∆0(5.11)

0with Fr r the crossing matrix (6j symbol) defined in appendix C. A more explicit form of (5.10)
is given in (D.3).
23

However, for theories without the Z2 symmetry, the crossing equation (5.7) cannot constrain the linear combination of the OPE coefficients
λ2(2 j;0)

C,C,X [2`]∆− λ2(0;2 j)

H,H,X [2`]∆

32

,

for

j 6= 0.

(5.9)

SciPost Phys. 10, 097 (2021)

When GC = GH , and when the Z2 outer automorphism that exchanges GC and GH is a
global symmetry, the crossing equations (5.10) reduce to

XX
2
2
As
−λ
v
λ
(0;0)
(0;0)
(0;0) (u, v)
=u

X

λ2

∆,`, jC , jH

XX X

XX

X

X

X ∆,` r0C ∈adjC ⊗adjC(2 j ;2 jH )C,H,L[2`]∆ C

X [2`]∆tA∆,`,
j , j (v, u),C H(5.12)

0

λ2

X ∆,` jC 6=0 r0C ∈adjC ⊗adjC+

C,C,X [2`]∆,− ,1

C,C,X [2`]∆,+ ,1

X ∆,`(2 j ;0)
C,C,X [2`]∆ C ,r0C

FX [2`]2 jC ,r rC (u, v, w)
∆

C


λ2

+ λ2
(0;0)
(0;0)
C,C,X [2`]∆,− ,r0C
C,C,X [2`]∆,+ ,r0C0

FX [2`]2 jC ,r rC (u, v, w) = 0 .
∆CA more explicit form of (5.12) is given in (D.4).
Nonetheless, these crossing equations can also constrain theories without the Z2 symmetry,as the discussion in Section 5.1.2 for the U(1) × U(1) case also applies to the general case:any solution to the Z2 symmetric (5.12) can be lifted to a solution to the general (5.10).
Conversely, a solution to (5.10) induces a solution to (5.12), via (5.8). Thus, (5.12) constrains
the combinations of OPE coefficients appearing on the right hand sides of (5.8).

5.2

The linear functional method

The linear functional method is a powerful tool for constraining and solving unitary conformal
field theories. We give a schematic explanation of the method here, and refer the reader toearlier papers by some of the authors for more details.
We act a vector valued linear functional α on the bootstrap equations (5.12), and the result
can be written schematically as
X
r
].
0=
λ2X ,r α[KX
(5.13)X ,rHere, K involves A or F , X denotes the multiplet, and r denotes the flavor representation. A
linear functional that satisfies
α[K1

(0;0)

B[0]0

] = −1,rα[KX] ≥ 0 for all X , r(5.14)

implies a bound on the OPE coefficients
λ2X ,r ≤

1
r .
α[KX
]

(5.15)

If we maximize α[KX ,J ] while satisfying (5.14), we obtain the optimal upper bound on λ2X ,J .
r
An extremal functional is one that maximizes α[KX], which we denote by αX ,r . If there
exists a four-point function that saturates the bound (5.15), then the OPE coefficients satisfyX
r00=
λ2X 0 ,r0 αX ,J [KX
0 ].
(5.16)(X 0 ,r0 )6=(X ,r)0r24
In light of (5.14), this means that the spectrum can be read off from the zeros of αX ,r [KX
0 ].
In practice, we perform the bootstrap in the following basis of linear functionals. Define z
and z̄ byu = zz̄, v = (1 − z)(1 − z̄),(5.17)
24
While it is often assumed that the extremal correlator is unique, there are situations where we explicitly know
this to be false. Nonetheless, we will assume that the spectrum is unique when the extremal theory is believed to
be interacting.

33

SciPost Phys. 10, 097 (2021)

so that crossing u ↔ v is equivalent to (z, z̄) ↔ (1 − z, 1 − z̄), and consider the space of linear
functionals at derivative order Λ:ΛXα=αm,n ∂zm ∂z̄n |z=z̄= 1 ,
2

m,n=0

αm,n ∈ R.

(5.18)The optimal bound is obtained by extrapolation to infinite Λ.25 The semidefinite programming
computations are performed using the SDPB solver [80, 81].

5.3

Numerical bounds

We are interested in the conformal central charge C T , the flavor central charge CJC of the
Coulomb branch flavor group GC , and the flavor central charge CJH of the Higgs branch flavor
group GH . They are related to the OPE coefficients by26
λ2

(0;0)

= λ2

(2;0)

=

C,C,A[0]1,− ,1

λ2

C,C,B[0]1

,adjC

(0;0)

H,H,A[0]1,− ,1

8h∨G
G

C

CJ C

,

=

24
,
CT

λ2

(0;2)

H,H,B[0]1

,adjH

=

8h∨G
G

(5.19)H

CJ H.

For numerical efficiency, we focus on the bootstrap equations with Z2 symmetry (5.7) and(5.12). One drawback is that we can only give bounds on the “average" flavor central charge
avg
CJ , given by


1
1
1
1
.
+
(5.20)avg =2 C GC C GH
CJ
J

5.3.1

J

Single branch

Let us first investigate the bootstrap bounds obtained from the crossing equation of flavor current multiplets with symmetry group G in a single branch. Indeed, there are many interesting
theories that only have Higgs branch flavor currents (charged under SU(2)H ), but no Coulomb
branch ones (charged under SU(2)C ). The left side of Table 7 lists the universal lower bounds
on CJ and C T for various choices of G. The G = An bounds on CJ is saturated by (Z2 -gauged)n free hypers, as described in Section 2.3. The authors are unaware of candidate theories orcorrelators that saturate the other bounds.
A special class of theories satisfy the minimal nilpotent orbit condition, under which theflavor central charge CJ is completely fixed by the minibootstrap program of Section 4.1. The
right side of Table 7 provides a list of the minimal (smallest C T ) known such theories forvarious choices of G, and compares them with the corresponding bootstrap bounds on C T
with the minimal nilpotent orbit condition imposed. We observe the following.
• Free theories with n hypermultiplets saturate the minimal nilpotent C T bounds withflavor symmetry Cn .
• Interacting minimal theories are consistent with but do not saturate the nilpotent C T
bounds. In fact, they have values of C T that lie very close to the bound, especially the T3
theory in the case of G = E6 . Although these theories have no Coulomb branch moment25

We estimate the error of such an extrapolation by the the variation of results using different ansatze. We markthe error of the last significant digit in parentheses following standard convention, in Table 7, (5.21) and (5.22).
26
As explained in Section 3.2, the N = 4 superconformal primary of the stress tensor block is odd under the Z2
outer automorphism, though the stress tensor itself is even as is required by conformal Ward identities.

34

SciPost Phys. 10, 097 (2021)

map operators (N = 4 primaries of flavor current multiplets charged under SU(2)C ),they do have nontrivial Coulomb branches parameterized by the chiral operators of other(4;0)
B -type multiplets, such as B [0]3 . Therefore, in order to bootstrap these theories, it
may be necessary to consider the mixed correlator bootstrap with these higher B -type
multiplets included as external operators.
• It would be interesting to see if the SQED and SQCD theories saturate the bootstrap
bounds with U(1) × GH flavor symmetry. We leave this for future work.
Table 7: Single correlator bootstrap lower bounds on the conformal and flavor central
charges, both assuming and not assuming the minimal nilpotent orbit condition. Also
listed are theories that saturate or are close to saturating the bounds on (C T )min-nil .
(CJ )min

G
U(1)
C ∼
=A

(C T )min(CJ )min-nil

6.02(3)(C T )min-nil
minReference theoryCT

6.02(3)1

4.02(5)9.0(1)≥ 3.99(3)11.9(2)

Free hyper

12

C2 ∼
= B2

4.09(3)17.8(1)

4

23.93(4)Free hyper

24

C3

4.04(4)28.3(3)435.7(2)

Free hyper

36

A2

4.47(5)14.93(6)629.68(5)

SQED with 3 hypers

34.5

44.2(3)

SQED with 4 hypers

46.3

57.9(1)

SU(2) SQCD with

72.1

1

A3

4.58(2)

20.1(2)32
5 = 6.4

B3

6.19(5)29.7(5)8
192
13 = 14.77
384
19 = 20.21
960
31 = 30.97

155.6(3)

7 fund half-hypers
E6

13.66(4)102.0(9)E719.32(2)167.6(6)E830.30(5)304(5)

F4

10.68(5)70.1(9)12113.9(2)19.3(1)64
9 = 7.11

43.95(9)

G2

5.3.2

4.89(2)

T3

160.2

239.1(4)406.0(6)GC = GH = U(1)For the mixed branch bootstrap, the simplest flavor symmetry to consider is GC = GH = U(1).
In this case, we cannot access CJ (in the absence of a preferred normalization of the abeliancurrent), but can only bound C T . The result after extrapolating to infinite derivative order is(C T )min = 12.0(2).

(5.21)

This value comes close to that of a single free hypermultiplet. However, a single free hypermultiplet has only GH = SU(2) and no Coulomb branch, GC = empty, so it is not a candidate
for a Z2 symmetric four-point function; we cannot even apply the construction of adding Z2
images described in Section 5.1.2.
5.3.3

GC = GH = SU(2)For GC = GH = SU(2) flavor symmetry, the allowed region in the C T − CJ plane is shown in
Figure 3, with and without imposing the minimal nilpotent orbit condition. For the former,
the self-mirror T [SU(2)] theory appears to sit at a (soft) corner. Certain BLG, ABJ, and ABJM
theories with this flavor symmetry are also included in the figure.

35

SciPost Phys. 10, 097 (2021)

GC = GH = SU(2)

1/CT
0.07
0.06
●

0.05

●

●

●
●
●
●
●

0.04

●

●

●

●

●

●

●

●

●

U(2)2 xU(1)-2 ●ABJ ●●0.02

SU(2)k xSU(2)-k BLG ●

●

●

●
●●
●●●
●
●
●
●
●
●
●

0.01

●

U(1)1 xU(1)-1
andU(1)2 xU(1)-2
ABJM

●

●
●

●

● ●
●●●
●●
●●
●
●
●
●
●
●●
●●
●●
●
T[SU(2)] ●●
●
●●
●
●

●

0.03●

●

●

●

●

●

0.00
0.00

0.05

0.10

●

0.15

avg

0.20

0.25

1/CJ
0.30

Figure 3: Allowed region in the C T − CJ plane for GC = GH = SU(2) flavor symmetry,at derivative order Λ = 32. The stronger (red) bounds are with the minimal nilpotent
orbit condition imposed, while the weaker (blue) bounds are without.
In Figure 4, we read off the extremal spectrum (in the OPE of moment map operators) fromthe extremal functional at derivative order Λ = 32. The Z2 even and odd sectors appear to havedifferent spectra, providing evidence that the extremal four-point function has a genuine Z2
symmetry, per the discussion of Section 5.1.2. Moreover, it appears that the lightest multiplet
is Z2 odd, the second lightest is a Z2 doublet.27
5.3.4

GC = GH = SU(3)For GC = GH = SU(3) flavor symmetry, the allowed regions in the C T − CJ plane are shownin Figure 5, with and without imposing the maximal nilpotent orbit condition. The minimal
candidate SCFT with GC = GH = SU(3) flavor symmetry is the T [SU(3)] theory, whose C T and
CJ values are inside the allowed region, but do not appear to sit at the boundary. In Figure 6,
we extrapolate our lower bounds on C T with CJ = 8 fixed to infinite derivative order, and find
(C T )min = 51.7(4),(5.22)

which is far from the value C T = 75.5 of T [SU(3)].
If we instead impose the minimal nilpotent orbit condition, then we learn from the minibootstrap of Section 4.1 that the flavor central charge is fixed to be CJ = 6. Fixing this valueof CJ , we find that the universal lower bound on the conformal central charge at derivative
order Λ = 28 is28
(C T )min-nil= 36.9.
(5.23)
min
27

Since our data is insufficient for reliable precision study, we do not provide numerical values for the scaling
dimensions of multiplets appearing in the OPE of moment map operators. Moreover, the lightest Z2 odd multiplet
may not actually exist, if the corresponding zero of the extremal functional converges towards the unitarity boundas Λ → ∞. If existent, then it would be interesting to study the deformation by this Z2 -odd relevant operator.
28
Since we do not have a reference theory/value in mind, we choose to present the rigorous bound at the highest
derivative order that we have achieved in the bootstrap analysis, instead of the extrapolated result. Thus there is
no extrapolation error here, and we keep three significant digits because the next digit changes between Λ = 24
and Λ = 28, and would likely change further for Λ > 28 giving stronger bounds.

36

SciPost Phys. 10, 097 (2021)

T[SU(2)]: flavor singlet
1
log α[KL[0]
]
Δ,0
10

Z2 even

50

Z2 odd
-5
Δ
4

6

8

10

12

-15
1
Figure 4: Extremal functional α[KL
[0]

∆,0

] for GC = GH = SU(2) flavor symmetry

with CJ = 16
3 fixed and C T minimized at derivative order Λ = 32. Shown here are
the sectors that are singlet under the flavor symmetry, and even or odd under the Z2
outer automorphism. The zeros correspond to the scaling dimensions appearing in
(0;2)
(2;0)
in the T [SU(2)] theory. The mismatch between
and B [0]0
the OPE of B [0]0
the Z2 even and odd sectors indicates that the Z2 is a true global symmetry in theextremal theory.
GC = GH = SU(3)

1/CT
0.04
0.03

●

●

●

●

●

●

●

●●
●●
●●
●
●
●
●
●
● ●
T[SU(3)]
●
●
●
●

●

●

0.02

●
●
●
●

●

0.01

●

0.00
0.00

0.05

●

0.10

avg

0.15

0.20

1/CJ
0.25

Figure 5: Allowed region in the C T − CJ plane for GC = GH = SU(3) flavor symmetry,at derivative order Λ = 28. The stronger (red) bounds are with the maximal nilpotent
orbit condition imposed, while the weaker (blue) bounds are without.

6

Conclusion

In this work we utilized the non-perturbative methods of the conformal bootstrap and supersymmetric localization, together with special properties of 3d N = 4 such as mirror symmetry
and a protected subsector described by topological quantum mechanics (TQM), to obtain universal constraints on 3d N = 4 superconformal field theories. A key ingredient in the bootstrap
analysis was the determination of the mixed-branch superconformal blocks. The main results
are summarized as follows:• We studied the single branch bootstrap with flavor symmetry G = U(1), G = ABC of
low ranks, and G = E F G. Free theories with n hypermultiplets appeared to saturate theuniversal lower bounds on C T with G = Cn and the minimal nilpotent orbit condition
imposed. Other theories that came relatively close to saturating the bounds were the
SQED with n + 1 hypers to G = An and the T3 theory to G = E6 . We highlight thesevalues below:

37

SciPost Phys. 10, 097 (2021)

GC = GH = SU(3): CJ = 8

min CT

T[SU(3)]

80
60
●

●

●

●

●

0.06

0.08

40
20
0
0.00

0.02

0.04

1/Λ
0.10

Figure 6: Lower bounds on C T with CJ = 8 fixed to be the value for the T [SU(3)]theory, for derivative order 12 ≤ Λ ≤ 28, and extrapolated to infinite Λ. Also shownis the actual value of C T = 75.5 for T [SU(3)].
G
∼
C =A

(C T )min-nil
minReference theoryCT1

11.9(2)

Free hyper

12

C2 ∼
= B2

23.93(4)Free hyper

24

C335.7(2)

Free hyper

36

A2

29.68(5)

SQED with 3 hypers

34.5

A3

44.2(3)

SQED with 4 hypers

46.3

E6

155.6(3)

T3

160.2

1

• For the mixed branch bootstrap with flavor symmetry GC = GH = SU(2), we found that
the T [SU(2)] theory sits at a corner in the allowed region in the C T − CJ plane. The
spectrum is read off from the extremal functional, and the mismatch between the evenand odd sectors indicated that the Z2 outer automorphism is a true global symmetry in
T [SU(2)] (it cannot be reproduced by certain spurious Z2 symmetric solutions to thecrossing equations).
• For the mixed branch bootstrap with flavor symmetry GC = GH = SU(3), we found that
the T [SU(3)] theory sits in the interior of the allowed region in the C T − CJ plane.
The framework developed here can readily be applied to a wider range of flavor symmetries
and assumptions, including self-mirror theories beyond SU(2) and SU(3) flavor factors, andnon-self-mirrors situations with GC 6= GH or GC = GH but different flavor central charges.
In particular, many of the known theories that came close to (but not quite) saturating oursingle-branch bootstrap bounds have at least U(1) flavor symmetry in the other branch, whose
incorporation into bootstrap may bring the theories (closer) to actual saturation.
Another promising direction is to explore bounds on certain protected OPE coefficients(2;2)
beyond the TQM sectors.29 For example, the OPE coefficient for the B [0]2
operator thatappears in the OPE of one Higgs branch and one Coulomb branch moment map operator is
not captured by the Higgs branch or Coulomb branch TQM sector, but is part of the chiral ringdata of the theory viewed as an N = 2 SCFT. In particular, when the SCFT has a UV gauge
theory description, this OPE coefficient may accessed from taking multiple derivatives of the(appropriately deformed) S 3 partition function with respect to mass and FI parameters at thesame time. This provides additional input to refine our exploration of mirror symmetry.
29

We thank Silviu Pufu for an interesting comment on this point.

38

SciPost Phys. 10, 097 (2021)

Futhermore, the application of the analytic bootstrap [82–85] or the OPE inversion formula[86–88] can shed light on interesting limits of 3d N = 4 superconformal field theory, that may
be relevant for AdS/CFT.
One can also consider the four-point functions of chiral operators beyond moment map
operators, to study theories without continuous flavor symmetry. For non-self-mirror theories
such as the 3d Minahan-Nemeschansky theories, the incorporation of the Coulomb branchprotected algebra (which no longer involve moment map operators) will be important to improve the bootstrap bounds. Another obvious avenue for future exploration is the bootstrap
analysis of the four-point function of the stress-tensor multiplet. In particular, the most minimal known theories [89] have no chiral operators, and are inaccessible by the study of such
four-point functions. Some superconformal blocks that are necessary in the bootstrap of thesemore general four-point functions have been computed in [45, 90].

Acknowledgments
We are grateful to Silviu Pufu for interesting comments on the draft. The computations inthis paper are performed on the Helios computing cluster supported by the School of Natural Sciences Computing Staff at the Institute for Advanced Study. C.C. is supported in partby the U.S. Department of Energy grant DE-SC0009999. The work of M.F. is supported by
an SNS fellowship P400P2-180740, the Princeton physics department, the JSPS Grant-In-Aid
for Scientific Research Wakate(A) 17H04837, the WPI Initiative, MEXT, Japan at IPMU, theUniversity of Tokyo, and the David and Ellen Lee Postdoctoral Scholarship. Y.L. is supportedby the Sherman Fairchild Foundation, and both M.F. and Y.L. by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632.
The work of S.H.S. is supported by the National Science Foundation grant PHY-1606531, theRoger Dashen Membership, and a grant from the Simons Foundation/SFARI (651444, NS).
The work of Y.W. is supported in part by the US NSF under Grant No. PHY-1620059 and by
the Simons Foundation Grant No. 488653. This research was supported in part by Perimeter
Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development
and by the Province of Ontario through the Ministry of Research and Innovation. C.C., M.F.,
S.H.S. and Y.W. thank the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1607611, for hospitality during the finishing stages of this paper.

A

Localization computation of the central charges

In this appendix, we provide detailed formulae for computing the conformal and flavor central
charges, C T and CJ , of 3d N = 4 superconformal field theories, derived from the matrix models
obtained by supersymmetric localization [21,52–57]. Using these formulae, we obtain explicit
analytic expressions for these central charges in certain classes of theories.

A.1

Double-sine functions and derivatives

We first start by introducing the main players, the double sine functions, and their various
properties. The double sine functions were introduced in [91, 92], and are closely related toBarnes’ double-gamma functions [93] (see also [94]). They are most straightforwardly defined

39

SciPost Phys. 10, 097 (2021)

using their infinite product form,
S2 (z|ω1 , ω2 ) =∞Yz + n1 ω 1 + n2 ω 2.
−z + (n1 + 1)ω1 + (n2 + 1)ω2
n ,n =0
1

(A.1)

2

However, for our purposes in the following, the integral expression is more useful: for 0 < Re ω j
and 0 < Re z < Re |ω1 + ω2 |,

Z

πi
d`
ez`
S2 (z|ω1 , ω2 ) = exp
B2,2 (z|ω1 , ω2 ) +.
(A.2)
ω1 ` − 1)(e ω2 ` − 1)2C ` (e
The integration contour C is along the real axis except near the (essential) singularity at theorigin where it runs along an infinitesimal half-circle in the upper half plane, and the multiple
Bernoulli function B2,2 (z|ωi ) is given byω1 + ω2 + 3ω1 ω2
ω1 + ω 2z2
B2,2 (z|ω1 , ω2 ) =
−
z+
.
ω1 ω2
ω1 ω2
6ω1 ω22

2(A.3)

Some useful identities are
S2 (cz|cω1 , cω2 ) = S2 (z|ωi ),S2 (x|ω1 , ω2 ) = S2 (ω1 + ω2 − x|ω1 , ω2 )−1 ,
πx
πx
sin
.
S2 (x|ω1 , ω2 )S2 (−x|ω1 , ω2 ) = −4 sin
ω1
ω2

(A.4)

In the (round) limit ωi → 1, we have
S2 (±i x + 1/2|1, 1) ≡ S2 (i x + 1/2|1, 1)S2 (−i x + 1/2|1, 1) = 2coshπx.

(A.5)

Throughout this paper, S2 (±z) is understood as the product S2 (z)S2 (−z).
As we are interested in computing the conformal central charges from the squashed S 3
partition function, we shall define
S2 (z|b) ≡ S2 (z|b, b−1 ),Q ≡ b + b−1 .

and

(A.6)

Furthermore, we are required to take derivatives of the double sine functions with respect tothe squashing parameter b. This is most conveniently done using the explicit integral expression (A.2), and one can explicitly compute∂ b | b=1 S2 (z|b) = 0,

∂ b | b=1 S2 (z + Q/4|b) = 0.

(A.7)

The non-trivial contributions will come from ∂ b2 acting on S2 (z|b), namely,∂ b2 b=1 S2 (z|b)
S2 (z|1)∂ b2 b=1 S2 (z + Q/4|b)
S2 (z + Q/4|1)==π (2π(z − 2)(z − 1)z + (3z − 2) sin 2πz),
6 sin2 πz

π2 8z 3 − 12z 2 − 2z + 3 − π(6z + 1) sin 2πz
24 cos 2πz

(A.8)
.

For instance, we can now immediately conclude2∂ b2 b=1 S2 (Q/4|b)
S2 (Q/4|1)
40

=

π2
,
4

(A.9)

SciPost Phys. 10, 097 (2021)

leading to C T = 12 for the free hypermultiple using the formula (2.15). The generalization ton free hypermultiplets follows immediately, C T = 12n.
Finally, we remark that the matrix model integrals possess a z → −z symmetry, so thefollowing combinations are useful:∂ b2 b=1 S2 (z|b)
S2 (z|1)∂ b2 b=1 S2 (z + Q/4|b)
S2 (z + Q/4|1)

A.2

+

+

∂ b2 b=1 S2 (−z|b)S2 (−z|1)∂ b2 b=1 S2 (−z + Q/4|b)S2 (−z + Q/4|1)=πz(−2πz + sin 2πz)sin2 πzπ(π − 4πz 2 − 2z sin 2πz)=.
4 cos2 πz(A.10)

Squashed S 3 partition function

Before we proceed with the evaluation of the conformal and flavor central charges, let us
briefly recall the explicit localization results for N = 4 (Lagrangian) theories on the squashed
S 3 background (2.28) [21,52–57] (see also [58]). The contribution to the matrix model arisingfrom N = 4 vector multiplets associated to a gauge group H is given by
Z
Z
Y
Y

1
1
vector=Zb
S2 (iα(σ)|b) ≡4 sinh (πbα(σ)) sinh πb−1 α(σ) ,dσdσ|W | t|W | tα∈∆+ ∪∆−
α∈∆+(A.11)
where we integrate over the Cartan subalgebra t ⊂ h = Lie H, ∆± is the set of all positive/negative roots of H, W is the Weyl group of H, and in the second equality we used thethird identity in (A.4).
Additionally, the contribution from N = 4 hypermultiplets in a representation (R GUV , RH )of the maximal subgroup GUV × H ⊂ US p(2n) to the localized path integral is given by
Y Y
1
hyper
Zb
=
,
(A.12)
S (±iρ(m) ± i ρ̂(σ) + Q/4|b)
ρ∈R GUV ρ̂∈RH

2

where the products are over the weights. Here GUV is the flavor symmetry realized by thegauged hypermultiplets and the dependence on m encodes CJ (GUV ).
As will be used in the following, we remark that in the round limit b → 1, the hypermultiplet contribution reduces to
Y Y
1
hyper
Z b=1 =,
(A.13)
2 cosh π(ρ(m) ± ρ̂(σ))ρ∈R GUV ρ̂∈RH

where we used the identity in (A.5).
Finally, we may add Fayet-Iliopoulos parameters to our theory which serve as mass parameters for topological U(1) symmetries. Namely, for a theory of gauge group G, we may
introduceP(A.14)Z bF I = e−2πi a ηa σa ,where the sum is taken to run over each (UV) abelian factor in h.
To get the full partition function of a Lagrangian N = 4 theory with masses for flavor symmetries, we put the above expressions together and integrate over the gauge group according
to (A.11).

A.3

Computation of C T

We now proceed to provide some details on the explicit evaluation of the conformal central
charge C T for various N = 4 SCFTs by using its relation to derivatives with respect to thesquashing parameter of the squashed S 3 partition function as expressed in the formula (2.15).
41

SciPost Phys. 10, 097 (2021)

SQED with k unit charge hypermultiplets
The squashed S 3 partition function of the N = 4 SQED theory, with quiver
U(1)

k

Z

1

(A.15)

is given by
SQEDk
Zb
=

dσ
R

[S2 (±iσ + Q/4|b)]k.

(A.16)

Thus, using the central charge formula (2.15) as well as the relation (A.8), we get
SQED

2
k
48
1 ∂ Zb
C T (SQEDk ) = − 2 SQED
k
π Z
∂ b2
b
b=1
Z
48 k
dσ
π(π + 4πσ2 + 2σ sinh 2πσ)
= 2 SQED.
kk
π Z
4 cosh2 πσR (2 cosh πσ)(A.17)

b=1

SQED

Now, we briefly recall the result for Z b=1 k [68], which can be computed for example bysumming over poles in the upper half-plane,nX
e2πiησ1e2πimα ηQ
dσ Q= n−1 πη,i (e − (−1)n e−πη ) α=1 β6=α 2 sinh π(mαβ )α 2 cosh π(σ − mα )R
(A.18)
where we turned on mass parameters mα for when we compute CJ and an FI parameter ηwhich acts as a regulator. Then, together with following integration identities,ZZ
2πσ sinh πσdσ
dσ
=
,
k+1
k
R (2 cosh πσ)
R k(2 cosh πσ)

Z
Γ 2k
dσ
=p
,
(A.19)kπ2k Γ k+1
R (2 cosh πσ)2ZΓ (2 + 2k )2 Γ (1, k+2(2πσ)2 dσ
4
2 )
=
,
k+2
π(k + 2)Γ (3 + k)
R (2 cosh πσ)
SQEDZ b=1 k (mα ) =Z

we findC T (SQEDk ) =12 2k2 ψ(1)k2
2 +1 + π k+4
π2 (k + 1)





k+4


,

(A.20)

Γ 0 (z)dwhere ψ(1) (z) ≡ dz
Γ (z) is the z-derivative of the digamma function.

SU(2) SYM with k fundamental half-hypermultiplets
We now compute the conformal central charge for the 3d N = 4 SU(2) theory with k > 4
fundamental half-hypermultiplets, i.e.

SU(2)

k

42

(A.21)

SciPost Phys. 10, 097 (2021)

This theory has an SO(k) global symmetry. Given the general rules outlined in the previous
subsection, the localized squashed sphere partition function of that theory readsZdσ
4 sinh2 (2πσ)
SQCDk
Zb
=
.
(A.22)k
R 2 [S2 (±iσ + Q/4|b)]Using the central charge fomula (2.15), and recalling the general insertion rule in (A.10) forcomputing derivatives, we end up with the following expression
Z
48 1
dσ 4 sinh2 (2πσ) π(π + 4πσ2 + 2σ sinh 2πσ)k
C T (SQCDk ) = 2 SQCDkk
π Z
4 cosh2 πσ
R 2 (2 cosh πσ)b=1(A.23)
πσ(2πσ − sinh 2πσ)+.
sinh2 πσ
Evaluating the round sphere partition function yields
22−k Γ
SQCD
Z b=1 k = p

πΓ

k
2 −2

k−1
2


,

(A.24)

and repeatedly applying the integral identities in equation (A.19), we end up with
12
C T (SQCDk ) = 2
n π2 (k − 4)(k − 2) + 12k − 20 (k − 4)
π (k − 4)(k − 2)(k − 1)

2 (1) k − 2+ 2(k − 2)(3k − 2)(k − 4) ψ
+ 48 .
2(A.25)T [SU(3)] from gauging T [SU(2)]As discussed in Section 2.4, the T [SU(3)] theory can be described by the quiver [41, 43, 65]
U(1)

U(2)

3

(A.26)

Thus, it is obtained from gauging the SU(2) × U(1) flavor symmetry of the T [SU(2)] theory(i.e. SQEDk=2 ) and three additional hypermultiplets in the doublet representation of U(2).
The localized squashed S 3 partition function is then given byZ
T [SU(2)]Zb(u12 /2)
du1 du2 Y
T [SU(3)]S2 (±iu j` |b) Q2,
Zb=(A.27)
2!
S2 (±iu j + Q/4|b)3
R
j<`
j=1

where
T [SU(2)]
Zb
(u) =

Z
dσ
R

1
,
S2 (±iσ + iu + Q/4|b)S2 (±iσ − iu + Q/4|b)(A.28)

and where we used the shorthand u12 = u1 − u2 . Then, using the C T formula (2.15) and theidentities in (A.10), we get the following expression for the central chargeZ
du1 du2 4 sinh2 (πu12 ) T [SU(2)]48
1Q
C T (T [SU(3)]) = 2 T [SU(3)]Z(u12 /2)π Z
2!
(2 cosh πu j )3 b=1
j
R
b=1
 X
π(π + 4πu2j + 2u j sinh 2πu j ) πu (−2πu + sinh 2πu )
12
12
12
× 3
−
(A.29)
2
2
4 cosh πu jsinh πu12jT [SU(2)]∂ 2 Z b=1
1
− T [SU(2)]
,
∂ b2
Z
b=1

b=1

43

SciPost Phys. 10, 097 (2021)

where we can by the same methods evaluate
T [SU(2)]−∂ 2 Zb

(u)

∂ b2

=
b=1

π (1 + 6u2 ) sinh 4πu − 4πu(1 + 4u2 ),6
sinh3 2πu

(A.30)

and we further have the solutions for the round sphere (see e.g. [68, 69])
T [SU(2)]

Z b=1

(u) =

u
,
sinh 2πu

T [SU(3)]

Z b=1

=

1
.
16π3

(A.31)

Putting the pieces together, we can numerically evaluate the the pieces in (A.29) to get30
C T (T [SU(3)]) = 75.5329.

(A.32)

T3 from gauging three T [SU(3)] theories
The (mirror of the) 3d T3 theory can be obtained by gauging three T [SU(3)] theories [95](see also [96]), i.e.31
U (1)

U (2)

SU (3)

U (2)

U (1)

U (2)

(A.33)U (1)Thus, the localized squashed S 3 partition function of the T3 (mirror) theory is given by
Z Q3

3
X Y
j=1 d v jT3
T [SU(3)](A.34)
4 sinh2 (πvk` ) Z b
(~
v) ,
δ
vj
Zb =
3!
R
j
k<`
T [SU(3)]with notation as above and where Z b
S 3 T [SU(3)] partition function, i.e.
T [SU(3)]
Zb
(~
v) =

(v j ) is the (Higgs branch) mass deformed squashed
T [SU(2)]Z

Zb

du1 du2 Y
S2 (±iu`k |b) Q3
2! `<kR(u12 /2)Q`,k=1 S2 (±i(u` + vk ) + 4 |b).

(A.35)In the round limit, this is evaluates to [68, 69]
T [SU(3)]

Z b=1

(~
v) =

v12 v13 v23.
16 sinh (πv12 ) sinh (πv13 ) sinh (πv23 )(A.36)

Again, using our trusty C T formula (2.15) and the identities in (A.10), we find
Z Q3
X Y

3
48 1
j=1 d v j
T [SU(3)]
C T (T3 ) = − 2 Tδvj
4 sinh2 (πvk` ) Z b=1
(~
v)
π Z 3
3!
R
j
k<`
b=1

×

X πv (sinh(2πv ) − 2πv )
k`
k`
k`
k<`

sinh2 πvk`+

30

T [SU(3)]∂ 2 Zb3
T [SU(3)]

Z b=1

(~
v)

∂ b2

(A.37)

!

(~
v)

,
b=1

All the numerical integrals throughout this paper are evaluated using Mathematica with 6 significant digits.
There are alternative constructions; for example the proposals in [97, 98] suggest N = 1 Lagrangian descriptions.
31

44

SciPost Phys. 10, 097 (2021)

where (similar to before) we can write
T [SU(3)]−×−∂ 2 Zb

(~
v)

∂ b2

T [SU(2)]2(u12 /2)
du1 du2 4 sinh (πu12 )Z b=1
Q=2!
k,` 2 cosh π(uk + v` )
R

Z

b=1

X

π(π + 4π(uk + v` )2 + 2(uk + v` ) sinh 2π(uk + v` ))k,`

4 cosh2 π(uk + v` )πu12 (sinh 2πu12 − 2πu12 )sinh2 πu12+

π(A.38) 
3u212 /2 + 1 sinh(2πu12 ) − 2πu12 u212 + 1.
3u12 sinh2 πu12

We can now numerically evaluate the required integrals. However, we can do better; namely,
we can explicitly evaluate the second and third term in (A.38). To do so, let us first define thefollowing quantityZ
e2πiη y12F (η) =
d y1 d y2,
Q2 Q n(A.39)
2 sinh π y12 i=1 α=1 2 cosh π( yi − mα )Rwhich is convergent for generic η 6= 0 and (mildly) divergent at η = 0 (F 0 (η) is well-defined
everywhere). Thus, we may use the Fourier transform of the distributionZ1= i d y e−2πi x a 2 tanh πa,(A.40)
sinh πxand writei
F (η) =2Z

e2πi(η−a) y12 2 tanh πa
.
Qn
2
cosh
π(
y
−
m
)
i
α
i=1
α=1

d y1 d y2 d a Q2

(A.41)

This can be thought of as an analytic continuation (analogous to Principal Value regularization)of (A.39) which is now well-defined for arbitrary η. We remark that the integrand contains
two factors of the SQEDn integral in equation (A.18), and thus (after a change of variables)
we may explicitly evaluate those contributions to findF (η) =

mαβ sinh(2πη) − sin(2πmαβ η)QQ.
8 sinh2 πη α,β sinh πmαβ i6=α 2 sinh πmαi j6=β 2 sinh πmβ jiX(A.42)

With the quantity F (η) in hand, the integral of the second and third terms in equation(A.38) can be expressed in terms of the function (A.39) as follows
π
1  00i 000
F (−i) − F (i) + 2i F 0 (0) +F (−i) − F 00 (i) +F (0).
38π
3π(A.43)

Then, the explicit numerical evaluation of (A.37) gives
C T (T3 ) = 160.246.

(A.44)

Type II k (n + 1) Chern-Simons matter theories
So far we focused on 3d SCFTs which have UV descriptions that involve no Chern-Simons couplings. Now, we consider the U(1)k × U(1)−k Chern-Simons matter theories of type II k (n + 1)[59], which in N = 4 language are described by one vector multiplet and n unit-charge hypermultiplets, one twisted vector multiplet and one unit-charge twisted hypermultiplet, witha BF coupling (between the vector and twisted vector multiplets) of even level k.

45

SciPost Phys. 10, 097 (2021)

For illustration, we focus on the II k (3) theories, i.e. we fix n = 2. The squashed S 3 partition
function isZ
e−kiπστII k (3)
Zb
(m) =
dσdτ
S2 (±iσ + Q/4|b)2 S2 (±iτ + Q/4|b)
ZR(A.45)1=,dσS2 (±iσ + Q/4|b)2 S2 (±ikσ/2 + Q/4|b)R
where in the second equality above we have used the Fourier transformation identity for the
double-sine functions in [99]. Using the central charge formula (2.15) and the relation (A.8),
we find
2 CS
48 1 ∂ Z b
π2 Z bCS ∂ b2
b=1
Z

48 1
dσ
2π(π + 4πσ2 + 2σ sinh 2πσ)
= 2 CS+(A.46)
π Z b=1 R (2 cosh πσ)2 (2 cosh πkσ )4 cosh2 πσ2π(π + k2 πσ2 + kσ sinh πkσ)+
4 cosh2 πkσ
2

C T (II k (3)) = −= 34.5463, 34.7619, 35.0577, 35.2887, . . .
for k = 2, 4, 6, 8, . . . .

A.4

Computation of CJ

Now, let us turn to the evaluation of the flavor central charges CJ for a variety of examples.
We shall use the formula (2.46), which relates mass deformations of the round S 3 partition
function to the flavor central charge.
Free hypermultiplets
Let us start by considering k free hypermultiplets, for which we turn on the following mass matrix M = Diag (m1 , m2 , . . . , mk , −mk , . . . , −m2 , −m1 ) ∈ US p(2k). Then, it follows immediatelyfrom (A.13), thatk
X
(A.47)
F (M) =
log 2 cosh(πmi ),i=1

and therefore, using (2.46), we conclude
USp(2k)CJ(hyperk ) = 4.

(A.48)

In case we are interested in their SU(k) ⊂ US p(2k) flavor symmetry, then the minimal achievable CJ for free hypermultiplets is¨ USp(4)CJ(hyper2 ) = 4,
k = 2,SU(k)CJ (hyperk ) =(A.49)
USp(2k)ISU(k),→USp(2k) CJ(hyperk ) = 8, k > 2,
where as before we denote by I the embedding index.
Z2 gauged k hypermultiplets
The US p(2k) 1-instanton moduli space is C2k /Z2 , and can be realized by k Z2 gauged hypermultiplets. However, since discrete gauging does not affect local correlators of Z2 invariant
operators, we get
USp(2k)CJ(hyperk /Z2 ) = 4,
(A.50)
i.e., the free field value.
46

SciPost Phys. 10, 097 (2021)

SQED with k unit charge hypermultiplets
Let us now turn to the next example: the 3d N = 4 SQED with k unit charge hypermultiplets.
The mass-deformed S 3 partition function is already computed in (A.18), where now the massPkmatrix is given by M = Diag (m1 , . . . , mk , −mk , . . . , −m1 ) ∈ US p(2k) subject to i=1 mi = 0(note that k ≥ 2 to satisfy this constraint). The formula (2.46) then gives
CJSU(k) (SQEDk ) =

8k
.
k+1

(A.51)

Notice that in the large k limit, this formula reproduces (A.48). Moreover, for given flavor
symmetry SU(k), the interacting theory obtained in the infrared of the SQED has smaller CJ
than the one realizable by free theories. This means that the bootstrap gives us access tointeracting theories despite the fact that we cannot exclude higher spin conserved currents
from the onset.
SU(2) SYM with k fundamental hypermultiplets
Now, let us consider 3d N = 4 SU(2) gauge theories with k fundamental hypermultiplets; theirHiggs branches have SO(2k) flavor symmetry and are described by the 1-instanton moduli
space of SO(2k). The mass deformed S 3 partition function is then given byZ
sinh2 (2πσ)
SQCD2kdσ Qk,Z b=1 (m) =(A.52)
R
i=1 4 cosh π(±σ + mi )where MSO(2k) = Diag (m1 , . . . , mk , −mk , . . . , −m1 ). The formula (2.46) then gives
CJSO(2k) (SQCD2k ) =32(k − 2).
2k − 1(A.53)

Notice that here we take k ≥ 3, otherwise the quiver is not good (ugly or bad in the senseof [43]) and the naive S 3 partition function diverges.
SU(2) SYM with k fundamental hypermultiplets and an additional half-hypermultiplet
If we include an additional half-hypermultiplet in the doublet representation to the gauge
theory described in the previous example, we obtain a Higgs branch with SO(2k + 1) flavor
symmetry, described by the 1-instanton moduli space of SO(2k + 1).
Then, the SO(2k + 1) mass deformed S 3 partition function for this theory readsZ
sinh2 (2πσ)
SQCD2k+1
Z b=1
=
dσ
,
(A.54)Qk
2 cosh(πσ) i=1 4 cosh π(±σ + mi )R
where MSO(2k+1) = Diag (m1 , . . . , mk , 0, −mk , . . . , −m1 ). Thus, the formula (2.46) gives
CJSO(2k+1) (SQCD2k+1 ) =16(2k − 3).
2k

(A.55)

Notice that here we take k ≥ 2, since otherwise the naive S 3 partition function diverges. Thus,
together we find that for 3d N = 4 SU(2) SYM with n half-hypermultiplets in the doublet
representation, the flavor central charge is given by32
CJSO(n) (SQCDn ) =32

16(n − 4).
n−1(A.56)

It is also easy to see that the central charge CJ of the SU(k) ⊂ SO(2k) subgroup is bigger than the ones in(2.52).

47

SciPost Phys. 10, 097 (2021)

Affine quiver gauge theories
Finally, let us turn to the T3 theory; The mass-deformed partition function was computedin [68, 69], and for convenience we shall recall it here,1T3m~ (1) , m
~ (2) , m
~ (3) = Q3 Q
Z b=1
(α)
3! α=1 `< j 2 sinh πm` j(A.57)
X

X

X

P (α)  X
(α)
(α)
(α)
×
(−1) α ρmρ(α) (1) cothmρ(α) (1) cothmρ(α) (3) ,αρ (α) ∈S3ααwhere S3 is the symmetric group and the mass parameters are turned on for SU(3)3 ⊂ E6 .
Then, we immediately find3
12 2 X  (α) 2πm.
F M2 =(A.58)
13 i,α=1 iThen, using the formula (2.46), we conclude that the SU(3) central charge is
192.
13
The E6 representation, 27, decomposes into SU(3) representations as follows
CJSU(3) (T [SU(3)]) =27 → (3, 1, 3) ⊕ (3, 3, 1) ⊕ (1, 3, 3).

(A.59)(A.60)

Furthermore, we have the following quadratic indices of the representation (see e.g. [100])
TE6 (27) = 3,

TSU(3) (3) =1,
2

TSU(3) (1) = 0.

(A.61)

Then, the embedding index is given by
ISU(3),→E6 =6TSU(3) (3) + 9TSU(3) (1)TE6 (27)

and we conclude thatE

CJ 6 (T3 ) = CJSU(3) (T [SU(3)]) =

= 1,

192
.
13

(A.62)(A.63)

Type II k (n + 1) Chern-Simons matter theories
The U(1)k × U(1)−k Chern-Simons matter theory of type II k (n + 1) [59] with even k has a
Higgs branch of quaternionic dimension n from the hypermultiplets, and a Coulomb branch
given by C2 /Zk/2+n from the twisted hypermultiplets. The Higgs and Coulomb branch flavor
symmetries are SU(n) × U(1) and U(1), respectively. For k = 2, the Higgs branch symmetry
is enhanced to SU(n + 1) and the theory is known to flow to the same IR SCFT as the SQEDwith n + 1 flavors.
Here, we consider the type II k (3) Chern-Simons matter theories with GH = SU(2) × U(1)and focus on the SU(2) factor. The SU(2)-mass deformed S 3 partition function isZ
e−kiπστII k (3)Z b=1 (m) =dσdτ
8 cosh π(±σ + m) cosh(πτ)
ZR(A.64)
1=dσ,
8 cosh π(±σ + m) cosh(πkσ/2)
Rgiving rise toR8d x sech4 (πx) sech(πk x/2)
CJSU(2) (II k (3)) = R= 6, 6.65633, 7.06667, 7.3258, . . .
d x sech2 (πx) sech(πk x/2)
for k = 2, 4, 6, 8 . . . .
48(A.65)

SciPost Phys. 10, 097 (2021)

B

Derivation of the superconformal Casimir equations

In this appendix we detail the derivation of the superconformal Casimir equations for the sand t-channels of the mixed Coulomb and Higgs branch four-point functions.

B.1

s-channel

We first consider the four-point functionCCH〈O (x 1 , Y1 )O (x 2 , Y2 )O (x, Ye3 )OH (x, Ye4 )〉 =



(Y1 · Y2 )(Ye3 · Ye4 )|x 12 ||x 34 |

2Gmixed (u, v),(B.1)

where the operators OC (x, Y ) and OH (x, Ye ) are defined as

OC (x, Y ) = Y A1 · · · Y Ak OAC1 ···Ak (x),OH (x, Ye ) = Ye Ȧ1 · · · Ye Ȧk OȦH ···Ȧ (x),
1

k

(B.2)

where OAC ···A (x) and OȦH ···Ȧ (x) are Coulomb branch and Higgs branch operators, respectively.
1k1k
For k = 2, we haveH
H
CC
(x 2 )O+̇
(x )O−̇(x )〉 =
(x 1 )O−−
〈O++
−̇ 4
+̇ 3

1
|x 12

|2 |x

34 |

2

Gmixed (u, v),(B.3)

with the following cross ratiosu=|x 12 |2 |x 34 |2,|x 13 |2 |x 24 |2v=|x 14 |2 |x 23 |2.
|x 13 |2 |x 24 |2(B.4)

Now, the Casimir operator acts on the four-point function as a differential operator, i.e.
HHCC(x )O−̇(x )〉 =
(x 2 )]O+̇
(x 1 )O−−〈[C, O++
−̇ 4
+̇ 3

1
C s Gmixed (u, v),|x 12 |2 |x 34 |2(B.5)

and we may solve for the superconformal blocks via the (eigenvalue) equation

C s As∆,`, j , j (u, v) = λC As∆,`, j , j (u, v) .
C H

C H

(B.6)

We can generically write the differential operator C s ass
C s = C bs + CSQ+ CRs ,

(B.7)

where by the results by Dolan and Osborn [101], we can read off the piece from the bosonicsubalgebra,

C bs = 2z 2 (1 − z)∂ 2 + 2z̄ 2 (1 − z̄)∂¯ 2 − 2(z 2 ∂ + z̄ 2 ∂¯ ) + 2zz̄
[(1 − z)∂ − (1 − z̄)∂¯ ].
z − z̄

(B.8)

Furthermore, for our purposes, the intermediate operators should be R-symmetry singlets, andthus we have
CRs = 0.
(B.9)
Thus, it remains to compute CSQ . The BPS conditions for the moment map operators areC[S αAȦ, OAB(0)]= [S αAȦ, OȦHḂ (0)]

= 0,

C
C
[Q α+Ȧ, O++(x)] = [Q α−Ȧ, O−−(x)] = 0,H[Q αA+̇ , O+̇
(x)]
+̇

=

H
[Q αA−̇ , O−̇
(x)]
−̇

49

= 0,

(B.10)

SciPost Phys. 10, 097 (2021)

and the superconformal S-supercharge acts on a superconformal primary at position x as follows[S αAȦ, O(x)] = −i x µ εAB εȦḂ σµαβ [Q β B Ḃ , O(x)],(B.11)
where we have used the commutators in (3.1) as well as

O(x) = e i x·P O(0)e−i x·P .

(B.12)

To compute the differential operator CSQ , we consider
1 αAȦCC[S , Q αAȦ]O++(x 1 )O−−(x 2 )|0〉
2 ¦

©
C
C
C
C
= −i x 12,µ εȦḂ σµαβ [Q β−Ḃ , O++(x 1 )][Q α+Ȧ, O−−(x 2 )] + 8O++(x 1 )O−−(x 2 ) |0〉.

(B.13)

Now, given the following Ward identitiesCC
HH(x )}〉,
0 = 〈{Q α++̇ , [Q β−−̇ , O++(x 1 )]O−−(x 2 )O+̇
(x )O−̇
−̇ 4
+̇ 3
C
C
H
H
(x )}〉,
0 = 〈{Q β−+̇ , O++(x 1 )[Q α+−̇ , O−−(x 2 )]O+̇
(x )O−̇−̇ 4+̇ 3

we find(B.14)H
HCC(x )O−̇(x )〉(x 2 )]O+̇
(x 1 )][Q α++̇ , O−−〈[Q β−−̇ , O++
−̇ 4
+̇ 3
H
H
C
C
(x )[Q α++̇ , O−̇(x )]〉(x 2 )O+̇
(x 1 )]O−−+ 〈[Q β−−̇ , O++
−̇ 4
+̇ 3

= −i(σas well asµ(B.15)H
HCC(x )O−̇(x )〉,(x 2 )O+̇
(x 1 )O−−)αβ ∂ x 1,µ 〈O++
−̇ 4
+̇ 3

H
H
C
C
(x )O−̇(x )〉(x 2 )]O+̇
(x 1 )][Q α+−̇ , O−−〈[Q β−+̇ , O++
−̇ 4
+̇ 3
H
H
C
C
(x )[Q β−+̇ , O−̇(x )]〉(x 2 )]O+̇
(x 1 )[Q α+−̇ , O−−− 〈O++−̇ 4+̇ 3(B.16)H
HCC(x )O−̇(x )〉.
(x 2 )O+̇
(x 1 )O−−= −i(σµ )αβ ∂ x 2,µ 〈O++
−̇ 4
+̇ 3

Therefore, taking the x 4 → ∞ limit we obtainH
H
C
C(x 2 )]O+̇
(x )O−̇(x )〉(x 1 )][Q α++̇ , O−−− i x 12,µ σµαβ 〈[Q β−−̇ , O++
−̇ 4
+̇ 3
H
H
C
C
(x 1 )O−−(x 2 )O+̇
(x )O−̇(x )〉,∼ 2x 12,µ ∂ x 1,µ 〈O++
−̇ 4
+̇ 3

and

C
H
H
C
(x 1 )][Q α+−̇ , O−−(x 2 )]O+̇
(x )O−̇(x )〉i x 12,µ σµαβ 〈[Q β−+̇ , O++
−̇ 4
+̇ 3
H
C
C
H
(x )O−̇(x )〉.
∼ 2x 21,µ ∂ x 2,µ 〈O++(x 1 )O−−(x 2 )O+̇
−̇ 4
+̇ 3

(B.17)(B.18)On the one hand, for the case in equation (B.17), we consider the parametrization


 
 
1
z
z̄
1
z
z̄
µ
x1 = 1 ++,−,0 ,
2 1 − z 1 − z̄ 2 1 − z 1 − z̄
(B.19)µ
x4µµ
x 2 = (1, 0, 0) ,x3 == (0, 0, 0) ,|x 4 |2
and thus, we haveu = zz̄,µx 12

v = (1 − z)(1 − z̄),

∂

∂

∂

(B.20)+ z̄(1 − z̄) .
µ = z(1 − z)∂z∂ z̄∂ x 12On the other hand, for the case in (B.18), we consider the parametrizationµx 1 = (1, 0, 0) ,

µ
x2 =



1
1
1 − (z + z̄), (z − z̄), 0 ,
2
2
50

µ
x3 =

µ

x4

|x 4 |2= (0, 0, 0) ,(B.21)

SciPost Phys. 10, 097 (2021)

and thus, we haveu = zz̄,µx 21

v = (1 − z)(1 − z̄),

∂

∂

∂

+ z̄ .
µ =z
∂z
∂ z̄
∂ x2

(B.22)s
Putting all the pieces together, we end up with the following expression for CSQ
,
s
CSQ
=2

B.2






∂
∂
∂
∂
+ z̄(1 − z̄)
+2 z
+ z̄
.
z(1 − z)
∂z
∂ z̄
∂z
∂ z̄

(B.23)

t-channel

We now proceed by computing the t-channel Casimir operator, i.e. we are deriving C t inHCCH(x )〉 =〈[C, O+̇(x )O−−(x 2 )]O++(x 3 )O−̇
−̇ 4
+̇ 1

h
i
1
t u
C
G
(v,
u)
.
mixed
|x 12 |2 |x 34 |2v(B.24)

Again, we can write the differential operator C t as followst+ CRt ,
C t = C bt + CSQ(B.25)

and as before, given [101], we may read offzz̄
C bt = 2z 2 (1 − z)∂ 2 + 2z̄ 2 (1 − z̄)∂¯ 2 − 2(z 2 ∂ + z̄ 2 ∂¯ ) + 2[(1 − z)∂ − (1 − z̄)∂¯ ].
z − z̄

(B.26)

However, now the intermediate operator are in the representation (2 jC , 2 jH ) = (2, 2) of theSU(2)C × SU(2)H R-symmetry, and so we have

CRt = −2 − 2 = −4.

(B.27)

t
Finally, we compute the remnant piece, CSQ. To do so, we consider

1 αAȦ
C
H(x 2 )|0〉(x )O−−[S , Q αAȦ]O+̇
+̇ 1
2 ¦

©
H
C
H
C
(x
)][Q
,
O
(x
)]
+
8
O
(x
)
O
(x
)
|0〉.
= −i x 12,µ σµαβ [Q β−−̇ , O+̇
α++̇
−− 2
−− 2
+̇+̇ 1+̇ 1(B.28)

Now, the Ward identity
HCC
H
0 = 〈{Q α++̇ , [Q β−−̇ , O+̇(x )]O−−(x 2 )O++(x 3 )O−̇
(x )}〉,−̇ 4+̇ 1(B.29)

gives
H
C
C
H
〈[Q β−−̇ , O+̇(x )][Q α++̇ , O−−(x 2 )]O++(x 3 )O−̇
(x )〉
−̇ 4
+̇ 1
H
C
C
H
+ 〈[Q β−−̇ , O+̇(x )]O−−(x 2 )O++(x 3 )[Q α++̇ , O−̇(x )]〉−̇ 4+̇ 1(B.30)H
CC
H= −i(σµ )αβ ∂ x 1,µ 〈O+̇(x )O−−(x 2 )O++(x 3 )O−̇
(x )〉.
−̇ 4+̇ 1

Therefore, in the x 4 → ∞ limit we obtainH
CCH− i x 12,µ σµαβ 〈[Q β−−̇ , O+̇(x )][Q α++̇ , O−−(x 2 )]O++(x 3 )O−̇
(x )〉
−̇ 4
+̇ 1
H
C
C
H
∼ 2x 12,µ ∂ x 1,µ 〈O+̇(x )O−−(x 2 )O++(x 3 )O−̇
(x )〉.
−̇ 4+̇ 1

Putting everything together, we end up with


∂
∂
t
CSQ = 2 z(1 − z)+ z̄(1 − z̄)
+ 4.
∂z
∂ z̄
51

(B.31)(B.32)

SciPost Phys. 10, 097 (2021)

C

Crossing matrices (6j symbols)

Consider a simple Lie group G, the tensor product of two adjoint representations decomposesasM(adj ⊗ adj)S = 1 ⊕ 2adj ⊕R(S,i) ,i(adj ⊗ adj)A = adj ⊕

M

(C.1)

R(A,i) ,i

where 2adj denotes the representation whose Dynkin label is twice the Dynkin label of theadjoint representation, and R(S,i) (R(A,i) ) are representations that appear in the symmetric(antisymmetric) tensor product and which are not 1, adj and 2adj.
Let us denote the Clebsch-Gordan coefficients by1
f abc ,
(Γadj )ca b = p
2h∨1(Γ1 )a b = p δa b ,|G|(Γ2adj )αab ,

I

(S,i)
(ΓR(S,i) )ab
,

I

(A,i)
(ΓR(A,i) )ab
,

(C.2)

where a = 1, · · · , dim(G), I(S,i) = 1, · · · , dim(R(S,i) ), I(A,i) = 1, · · · , dim(R(A,i) ), and f abc is the
structure constant of G, which is normalized byf aed f bde = 2h∨ δ ab .

(C.3)

The other Clebsch-Gordan coefficients are normalized byβ(Γ2adj )αa b (Γ2adj )a b = δαβ ,
I

J

I

(S,i)
(S,i)
= δ I(S,i) J(S,i) ,(ΓR(S,i) )ab(ΓR(S,i) )abJ= δ I(A,i) J(A,i) .
(ΓR(A,i) )a(A,i)(ΓR(A,i) )a(A,i)
b
b

(C.4)

The projection matrices are defined as
cd IR JRa bcd,
= (ΓR )abPR
I (ΓR )J δ
R

R

(C.5)

and the crossing matrix (6j symbol) is defined as follows
Fi j =

D1
abcd.
P dabc PRjdim(R j ) Ri

(C.6)

Crossing equations in matrix form

In this appendix, we rewrite the crossing equations (5.4), (5.7), (5.10), and (5.12) in forms
that are more explicit and readily usable for semidefinite programming.
The crossing equations for theories with U(1)×U(1) flavor symmetry (5.4) can be rewritten

52

SciPost Phys. 10, 097 (2021)

as




2v As (0;0) (u, v)
t
2u
A
(v,
u)
B[0]0
∆,`,0,0
 X



−
λ2C,H,(L,∆,`) 0=
F0(0;0) (u, v, w) B[0]0∆,`0e)FB[0](0;0) (u, v, w
0


0
v As
(u,
v)
(0;0)
L[2`]∆

 v As0(0;0) (u, v)


L
[2`]
 λi

∆

XX i


(0;0)
i
FL[2`](0;0) (u, v, w) 0  C,C,L[2`]∆ λ+(0;0) λ
(0;0)
∆
 λi

C,C,L[2`]∆
H,H,L[2`]∆


(0;0)
∆,` i
0
0
H,H,L[2`]∆






0
0


e)
0 FL[2`](0;0) (u, v, w
∆





0
0
X




2
2
F
0
(2;0) (u, v, w) + λ+λ(2;0)  A[2`](0;2) `+2
C,C,A[2`]`+2
H,H,A[2`]`+2
`
e
F
(u,
v,
w
)
(0;2)
0
A[2`]`+2




0
0




F (4;0) (u, v, w) + λ2
0
+ λ2
.
(4;0)  B [0](0;4) 2
C,C,B[0]2
H,H,B[0]2e)FB[0](0;4) (u, v, w
0
2

(D.1)
(0;0)
Here, the sum over A[2`]`+1 is omitted because these blocks are smooth ∆ → 1 limits of the(0;0)

L[2`]∆ blocks. In the second line, we explicitly included a sum over degenerate multipletslabeled by i with the same ∆, `. By contrast, in the other lines, we simply defined λ2 to be thesum of degenerate contributions.
For the (U(1) × U(1)) o Z2 flavor symmetry, the crossing equations (5.7) can be rewritten
as




t
X
2v As (0;0) (u, v)
2uA∆,`,0,0(v, u)
B
[0]
2
−
0
0=
λC,H,(L,∆,`)
0
FB[0](0;0) (u, v, w)
∆,`
0




s
X
X
2v As
(u,
v)
−2v
A
(u,
v)
(0;0)
(0;0)
L[2`]∆
L[2`]∆
+ (D.2)


+
λ2
λ2
(0;0)
(0;0)
C,C,L[2`]∆ ,+ F
C,C,
L
[2`]
,−
∆
(u,
v,
w)
F
(u,
v,
w)
(0;0)
(0;0)
∆,`
∆,`
L[2`]∆
L[2`]∆
!


X
0
0
+
λ2
+ λ2
.
(2;0)
(4;0)
C,C,B[0]2
C,C,A[2`]`+2FA[2`](2;0) (u, v, w)
FB[0](4;0) (u, v, w)
`

`+2

2

For the most general GC × GH flavor symmetry, the crossing equations (5.10) can be rewrit-

53

SciPost Phys. 10, 097 (2021)

ten as


2v As

(0;0)
B[0]0

(u, v)





(0;0)
A[0]1



1
 + λ2
0=
(u,
v,
w)
F
(0;0)

 B[0] ,∗(0;0)
C,C,A[0]1 ,10e)
FB[0](0;0) ,∗ 1 (u, v, w

−2v As(u, v)





1

F
(u,
v,
w)
(0;0)

 A[0]1 ,∗e)
FA[0](0;0) ,∗ 1 (u, v, w
1

0




(u,
v)
(0;0)
L[2`]∆

 v As0(0;0) (u, v)


!
L
[2`]



 ∆
X

λC,C,L[2`](0;0) ,1

1FL[2`](0;0) ,∗ (u, v, w) 0∆
λC,C,L[2`](0;0) ,1 λH,H,L[2`](0;0) ,1 
+
∆
 λ

∆
∆
(0;0)


H,H,L[2`]∆ ,1
∆,`
0
0






0
0


e)
0 FL[2`](0;0) ,∗ 1 (u, v, w
∆




0
0




adj
2
0∗
+ λ2
FB[0]1(2;0) ,∗ C (u, v, w) + λ


(2;0)
(0;2)
C,C,B[0]1 ,adjC
H,H,B[0]1 ,adjH
adjH
e
F
(u,
v,
w
)
(0;2)
0∗
B[0]1 ,∗





0
0


X 2

 X 2
rC
0∗
F
(u,
v,
w)
(4;0)
λ
+
λ



+
(0;4)
(4;0)
B[0]2 ,∗
H,H,B[0]2 ,rH
C,C,B[0]2 ,rC
rH
rH
rC
e)
FB[0](0;4) ,∗ (u, v, w
0∗
v As

0

2




+

X X
λ2






0

 X 2

r
λ
FA[2`](2;0) ,∗ C (u, v, w) +

0





0∗


`+2
rH
rH
rC
`
e)
FA[2`](0;2) ,∗ (u, v, w
0∗
`+2





0
0
XX

 X 2


r
0∗
+
λ2
λ

FL[2`]∆(0;0) ,∗ C (u, v, w) +


(0;0)
(0;0)
C,C,L[2`]∆ ,rC
H,H,L[2`]∆ ,rH
rH∆,` rC 6=1
rH 6=1e)FL[2`](0;0) ,∗ (u, v, w
0∗
∆


t
2uA∆,`,0,0(v, u)X
2−
λC,H,(L,∆,`) 
0∗
,
∆,`
0∗
(D.3)
where * represents a column vector over representations in adjC ⊗ adjH for the respective GC
or GH . When GC (resp. GH ) is empty, one simply omits the crossing equations involving thefirst and second (resp. third) entries.
Consider GC = GH , for the (GC × GH ) o Z2 flavor symmetry, the crossing equations (5.12)(2;0)
C,C,A[2`]`+2 ,rC(0;2)
H,H,A[2`]`+2 ,rH

54

SciPost Phys. 10, 097 (2021)

can be rewritten as




s
t
X
vA
(0;0) (u, v)
uA∆,`,0,0(v, u)
B
[0]
2
−
0
0=
λC,H,(L,∆,`)0(FB[0](0;0) )∗1 (u, v, w)
∆,`
0




s
X
X
−v
A
(u,
v)
v As
(u,
v)
(0;0)
(0;0)
L[2`]∆
L[2`]∆


+

λ2
+
λ2
(0;0)
(0;0)
1
1
C,C,L[2`]∆ ,1,− (FC,C,L[2`]∆ ,1,+ (F(0;0) ) (u, v, w)(0;0) ) (u, v, w)
∆,`
∆,`
∗
∗
L[2`]∆
L[2`]∆


X
0
+
λ2
(0;0)
r
C,C,L[2`]∆ ,r (F(0;0) ) (u, v, w)∗
L[2`]∆∆,`,r6=1




X
X
0
0
2
2
+
λ
+
λ
(2;0)
(4;0)
r
r
C,C,A[2`]`+2 ,r (FC,C,B[0]2 ,r (FA[2`]2`+2 )∗ (u, v, w)
B[0]42 )∗ (u, v, w)
r
`,r


0
2
+λ
,
adj
(2;0)
C,C,B[0]1 ,adjC (FB[0]21 )∗ (u, v, w)(D.4)
where * represents a column vector over representations in adj ⊗ adj.

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