SCFTs in 6d and IR symmetry enhancement in 4dShlomo S. Razamat
Technion

Kim, SSR, Vafa, Zafrir – 1709.02496, 1802.00620, and 1806.06720
SSR, Sela, Zafrir – 1711.02789

June 28, 2018 – Strings 18
Okinawa, Japan.

Lagrangians/non Lagrangian models
This talk is about 4d QFTS with N = 1 supersymmetry
We believe that certain models exist for which one cannot write a
Lagrangian with all the (super)symmetries manifest

Many N = 2 examples; Argyres-Douglas models,
Minahan-Nemeschansky models, TN models

Insisting on manifestly N = 2 supersymmetric Lagrangians nodescription can be found which reproduces some of the propertiesof these models
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 20182 / 26

Compactifications from six dimensions – no LagrangiansF

N = (1, 0)
CF T 6d

N =1
CF T 4d

6D

4D
2D

Evidence for existence of non-Lagrangian models studying
compactifications of 6d conformal field theories to 4d
Choices involved – 6d model, surface, flux for G6d
General choices lead to models with no known Lagrangian
There is a tention between having a Lagrangian description andhaving all the symmetries of the models manifest
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 20183 / 26

How should we think about such models?

Two options:
Give up Lagrangians:
One can attempt to develop new tools to avoid the need of defining
Lagrangians (see Bootstrap)

Alternatively:···
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 2018

4 / 26Give up symmetry!!

Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 2018

5 / 26

UV symmetry and IR symmetry

Global (super)symmetries can be enhanced in IR, GIR might be
not equal to GU V
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 2018

6 / 26

Compactifications from six dimensions – LagrangiansF

N = (1, 0)

N =1

CF T 6d

CF T 4d

6D

4D
2D

What will be an organizing principle?
A strong hint comes by studying same type of compactifications
Many compactifications do have Lagrangian descriptions
In many cases the UV symmetry is enhanced in the IR
We will discuss how it comes about in the case of a torus with flux
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 20187 / 26

Flux and symmetry
The 6d (1, 0) theories have some symmetry G6dUpon compactification one can turn on flux for abelian subgroupsof this symmetry preserving N = 1 supersymmetry in 4d
Flux is specified by r = Rank G6d integers
F = (F1 , F2 · · · , Fr−1 , Fr )The 4d symmetry G4d is the subgroup of G6d which commuteswith the flux
For example if G6d is SU (r + 1),
F = (r − 1, −1, −1, · · · , −1)

preserves SU (r) × U (1)Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 20188 / 26

Torus built from tubes
FmYE

IAIft
IEIF

We can think of the theories
obtained by compactification on
tori as combined from simple
duality building blocks

IFErinThe blocks correspond toa tubes with some value of flux Fj
Each tube is a simple4d theory, the IR symmetry visible in UV
PHowever, combining tubes the flux nj=1 Fj might be indicating
enhanced symmetry, bigger than the symmetry of the blocks
Example, U (1)2 → U (1) × SU (2),(4, −1, −3) + (4, −3, −1) → (8, −4, −4)
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 2018

9 / 26

What are the tube models?
FTo try and understand what are the tube models one cancompactify the 6d models first on a circle to 5d
Many 6d theories have effective description as gauge theory in 5d
which then can be used to understand the 4d models
Example: ADE conformal matter (N M5 branes on ADEsingularity), 5d description as quivers in the shape of affine ADE
Dynkin diagram
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201810 / 26

Five dimensions, domain walls, and flux
m1

m! 1x

Upon compactification to 5d we have a choice of holonomy whichtranslates to a choice of mass parameters
Different holonomies might lead to different effective theories in 5d(Hayashi, Kim, Lee, Taki, Yagi 15)
The 5d manifestation of the flux is in terms of domain walls
interpolating between different values of the mass parameters, ordifferent five dimensional descriptions
The four dimensional theories can be constructed byunderstanding the theories on the domain walls (see N. Paquette’s talk)
Punctures: make the cylinder finite, choose bc for 5d fields
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201811 / 26

The basic reduction toolbox
How do we know theories in four dimensions correspond to somecompactification?
Anomaly,ZΣg,06dT 4d (T 6d ,Σg,0 ,F )I8T (F) = I6(Benini, Tachikawa, Wecht 9)
Indices,
1+X

relevantsni (qp)Ri /2 + (M arg − Currents)qp + . . .

(Beem, Gadde 12)
If at order qp you see adjoint of some group it has to be thesymmetry of the theory (unless there is an accidental U (1) )Consistency checks
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201812 / 26

Examples: ADE conformal matter
DA

E7

E6

E8

We will give examples of G = ADE conformal matter

Five dimensional description is in terms of G affine quiver

Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201813 / 26

Flip is the trick

W =✏i1···iN ✏j1···jN Qji11 · · · QjiNNQ.
The theories are usual quiver theories
In addition to gauge charged matter one also has gauge singlets
The gauge singlet fields flip some of the gauge invariant baryons
The flips are very important, without these the anomalies do notmatch with 6d and in some cases the symmetries do not enhance
Generally these are just free fields

Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201814 / 26

Example of A
FkTypical building block is a WZ model
The UV symmetry is two copies of SU (N )k associated to theboundary and U (1)2k−1 which is the Cartan subgroup of
G6d = SU (k) × SU (k) × U (1)The pattern of the bifundamental fields is related to flux F
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201815 / 26

Example of Akk1Gluing k − 1 blocks (triangulation of torus) one obtains theory
corresponding to flux
F = (k, −1, −1, · · · − 1, 0, 0, ...)
Symmetry enhanced from U (1)2k−1 to SU (k) × SU (k − 1) × U (1)2
Anomalies and indices work
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201816 / 26

Example of D1

122
2k+2 / 1

2

2

2

2

2

1/ 12

1

1

1
1 2

1
1/( 122k+2 )

1
1 3

1
1 4

1
1 k

1

1
1 k

1
1 k+1

1 k+2
1 k+2

2

2

1

2k+3

1 k+2
1 k+2

2
2

2

22kk

3
3

1
12kk

2
2k+3 / 11k+1 k+3k+1 k+3k+1 k+3k+1 k+31k

The building block becomes more complicated
Two copies of affine Dynkin diagram connected by bifundamental
fields

Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201817 / 26

Example of D5 minimal

On the left flux is such that SU (4)2 × U (1)2 enhances toSO(15) × U (1)On the right flux is such that SU (4)2 × U (1)2 enhances toSO(12) × SU (2) × U (1)Anomalies and indices are consistent with thisShlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201818 / 26

Dualities, oldenEmFr
I1ME4din

FIE fHI HItoHean
I1IE
Fm En HIHIHeMEFrinI

E

gu

Changing the order in which we combine the blocks one obtains
En IE
different
quiver theories
Fm lookingIHI HIan
I

Six 1dimensional interpretation tells all these have to be the same
CFTs, dual to each otherEuTheseFrreduce
to Seiberg (Intriligator-Pouliot, etc) dualities

HegShlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201819 / 26

Dualities, newa 1 s2

a 1 s1

2a1a2

N +1

42222a22a 3 s2

a 3 s1a32a 2 s2

a 2 s12a3a442a4a1
a 4 s12a 4 s2

22

NDifferent 5d descriptions of a 6d model can give different blocks
but equivalent 4d models for closed surfaces
Minimal DN +3 conformal matter has three description, SU (2)N ,U Sp(2N ), SU (N + 1)
The models above are dual (r = N + 1) with different manifest
symmetry which enhances to SO(2N + 10) × SU (N + 1) × U (1)Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201820 / 26

Complicated to simple
66226222222

Theories obtained by compactification can be simplified bydeformations, still exhibit enhanced symmetry
Example, reductions with D4 minimal conformal matter deformed
The 6d logic implies should have E6 × U (1) symmetry
The theory is SU (2) SQCD with four flavors and a superpotential
42

1 + 27h−1 (qp) 9 + h3 (qp) 3 + ... + (−78 − 1)qp + ... .
Assumptions of no accidental abelian symmetry, CFT, proof
symmetry is E6 × U (1)Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201821 / 26

22

6Symmetry and duality
2AQ

2B

2g

4Ae
Q

The symmetry is related to Seiberg duality
The duality mixes SU (2) with SU (6) enhancing it to E6
The conformal manifold is a point
Generalizations: Self-dualities of Spin(4 + n) gauge theories withn vectors and spinors with 32 components (Csaki, Schmaltz, Skiba, Terning
97, Karch 97) lead to models with enhanced symmetry, for example
symmetry rotating the spinors is commutant of SU (2) in E9−n .
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201822 / 26

Summary
Can construct a lot of examples of theories with IR symmetry
being much bigger than UV symmetry through compactifications
The construction is (almost) algorithmic
Simple compactifications lead to involved models
Simple enhancements often related to deformations ofcompactifications
Symmetry emerges but is not completely accidental

Shlomo S. Razamat (Technion)6d to 4d and symmetry enhancement

June 28, 201823 / 26

Open questions
FN = (1, 0)
CF T

6dN =1
CF T 4d6D4D
2DThe key is to identify a set of building blocks
Understand more systematically the domain walls
Higher genus known for A0 N = 2 and N = 3, A1 and N = 2, D4
and N = 1, pure glue SU (3) and SO(8) 6d SCFTS (SSR, Zafrir 18)
Develop the general dictionary between six and four dimensions
Three dimensions
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201824 / 26

Is everything Lagrangian?
Derived Lagrangians for many models with flavor symmetryenhancing

Have Lagrangian for Argyres-Douglas models (Maruyoshi, Song 2016)

Have Lagrangian constructions for E6 (and many others, E7 , ...)

In such examples also supersymmetry enhances

Can we write a Lagrangian for any model?
Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201825 / 26

Thank You

Shlomo S. Razamat (Technion)

6d to 4d and symmetry enhancement

June 28, 201826 / 26