On 4d rank-one N=3 superconformal field theories

Takahiro Nishinaka( Yukawa Institute )

July 26, 2016 @ Osaka U.

arXiv: 1602.01503 w/ Yuji Tachikawa ( IPMU )

Introduction

4d SUSY QFTs

• N=1

• N=2

• N=4

SUSY standard model (MSSM)

exact computation (localization), SW curve

AdS/CFT, integrability

What about N=3 ???

Introduction4d SUSY QFTs

( No genuine N=3 Lagrangian theory )

• N=3 Lagrangian always preserves N=4

• But these days, we have seen many non-Lagrangian QFTs.

e.g.) M5-branes 4d QFTcompactify

’16 [TN - Tachikawa]’15 [García-Etxebarria - Regalado]

’16 [Aharony - Tachikawa]

’15 [Aharony - Evtikhiev]

• There could be Non-Lagrangian genuine N=3 theories.

’16 [Imamura - Yokoyama]

What do 4d genuin N=3 QFTs look like (if exist) ?Q:

Our answer

Focusing on 4d N=3 simplest conformal field theories ( SCFTs ),

• Moduli space of SUSY vacua

• conformal anomalies

• OPEs of BPS ops.

1/4 BPS operators described by 2d N=2 W-algebra

a = c =2` � 1

4

(` = 3, 4, 6)M = C3/Z`

Outline

1. N=3 Lagrangian preserves N=4

2. N=3 (non-Lagrangian) SCFTs

3. The simplest examples

4. OPEs of BPS operators

Outline

1. N=3 Lagrangian preserves N=4

2. N=3 (non-Lagrangian) SCFTs

3. The simplest examples

4. OPEs of BPS operators

Supersymmetry

superchargeQ↵

Q†↵

boson fermion

Aµ

�

N=3 Lagrangian preserves N=4

• massless N=1 SUSY multiplets

vector multiplet

1

1/2

0

-1

helicity

-1/2

1

1

�

chiral multiplet

1

2

1

vector chiral

CPT conjugate

1

1

N=3 Lagrangian preserves N=4

1

1/2

0

-1

-1/2

vector hyper

1

2

2

1

2

• massless N=2 SUSY multipletshelicity

1

2

1hyper multiplet

2

4

2

q q

N=1 chiral

N=1 vector

N=1 chiralN=1 chiral

�1 �2

�

vector multiplet

Aµ

N=3 Lagrangian preserves N=4

vector multiplet1

1/2

0

-1

-1/2

vector

1

4

4

1

6

• massless N=3 SUSY multiplets

This is equivalent to the N=4 vector multiplet!

N=2 vector N=2 hyper

helicity

3

1

1

3

The only N=3 massless multiplet is an N=4 vector multiplet!!

�1 �2

qq

Aµ

�

W = Tr �[Q, Q]

�

Q

Q

N=3 Lagrangian preserves N=4

• The only N=3 preserving renormalizable superpotential is

This is precisely the superpotential for N=4 SYM.

�1 �2

Aµ

What about interaction?

� q q

In summary,

• The only massless N=3 multiplet is equivalent to an N=4 vector multiplet.

• Moreover, every N=3 (renormalizable) Lagrangian preserves N=4.

No genuine N=3 renormalizable Lagrangian!!

• This particularly means that every N=3 free SCFT preserves N=4 SUSY.

�1 �2

qq

Aµ

�

Outline

1. N=3 Lagrangian preserves N=4

2. N=3 (non-Lagrangian) SCFTs

3. The simplest examples

4. OPEs of BPS operators

Let’s focus on 4d N=3 superconformal field theories.

Then, despite the lack of Lagrangian, we can learn about

( SCFT )( N=4 )

• Global symmetry

• Conformal anomalies

• Moduli space of SUSY vacua

N=3 (non-Lagrangian) SCFTs

Global symmetry

• Bosonic global sym.

SO(4,2) x U(3)R x GF

• No N=3 flavor symmetry is allowed.

Jµ : flavor current for GF

J↵µ =⇥Q3

↵, Jµ

⇤

J↵µ =hQ3↵, Jµ

i extra supercurrents leading to N=4 SUSY

conformal N=3 R-sym N=3 flavor sym

a =1

24, c =

1

12

N=3 (non-Lagrangian) SCFTs

Tµµ =

c

16⇡2(Weyl)2 +

a

16⇡2(Euler)

(Euler) = R2µ⌫⇢� � 4R2

µ⌫ + R2

(Weyl)2 = R2µ⌫⇢� � 2R2

µ⌫ +1

3R2

These coefficients capture the number of degrees of freedom in CFTs

e.g.) free hyper :

1

2

a

c

5

4• N=2 SUSY ’08 [ Hofman-Maldacena ]

Conformal anomalies

a

c= 1• N=3 SUSY ’15 [ Aharony, Evtikhiev ]

M

N=3 (non-Lagrangian) SCFTs

conformal point

SUSY vacua

Moduli space of vacua

flat directions preserving N=3 SUSY=

( but breaking conformal sym. )

dimC M = 3 x integer

�, q, q 2

=

parameterized by the VEV of a scalar in a massless N=3 multiplet

=

N=3 vector multiplet

N=3 vector multiplet

� q q�1 �2

Aµ

a = c

In summary, every genuine N=3 SCFT has

• Global symmetry

• Central charges

• Moduli space of SUSY vacua

conformal point

SUSY vacua3 x integerdimC M =

Tµµ =

c

16⇡2(Weyl)2 +

a

16⇡2(Euler)

SO(4,2) x U(3)R x GF

conformal N=3 R-sym

N=3 flavor sym

M

Outline

1. N=3 Lagrangian preserves N=4

2. N=3 (non-Lagrangian) SCFTs

3. The simplest examples

4. OPEs of BPS operators

conformal point

SUSY vacua

3 x integerdimC M =( rank )

The simplest examples are rank-one:

3dimC M =

We focus on this case and argue that

M = C3/Z`

(` = 3, 4, 6)

a = c =2` � 1

4

M

4d rank-one N=3 SCFTs conformal point

SUSY vacua

M

is parameterized by the VEVs of 3 scalars in an N=3 vector multiplet.M

Moduli space of vacua

3dimC M =

MH

Mh�i hqi, hqi

( N=2 Coulomb branch ) ( N=2 Higgs branch )⊂ ,

N=3 R-sym.

3 (q, q)

4d rank-one N=3 SCFTs

Higgs branch

conformal point

SUSY vacua

M

MH

U(3)R U(1)R x SU(2)R x U(1)F⊂

N=2 R-sym. N=2 flavor sym.

N=3 R-sym :

MH = C2/Z` ` = 3, 4, 5, 6, · · ·

MH is acted on by SU(2)R and U(1)F• 　

MH : hyper Kahler cone w/ U(1) isometry=)

charged

C3/Z`

= C2/Z`C/Z`

4d rank-one N=3 SCFTsCoulomb branch

` = 3, 4, 5, 6, · · ·

N=3 R-sym.

Mh�i hqi, hqi

( N=2 Coulomb branch ) ( N=2 Higgs branch )⊂ +

=) ` = 3, 4, 6

e2⇡i`

h�i

h�i = 0

conformal point

The Seiberg-Witten curve is a singular -fiber over C/Z`T 2

conformal point

SUSY vacua

M

(` = 3, 4, 6)M = C3/Z`

2a � c =1

4

X

i

(2[�i] � 1)

4d rank-one N=3 SCFTsCentral charges

’08 [Shapere - Tachikawa]

Formula widely applicable to N=2 SCFTs

a = c

a = c =2` � 1

4

Combined w/ , we find

Scaling dim. of Coulomb branch operators

2a � c =1

4(2[�] � 1) =

2` � 1

4

In our case, read off from the SW curve

In summary,

4d rank-one N=3 SCFTs should have...

• Moduli space of vacua

• Central charges

a = c =2` � 1

4

(` = 3, 4, 6)M = C3/Z`

` = 1

` = 2

c.f.) N=4 U(1) SYM

N=4 SU(2) SYM

Outline

1. N=3 Lagrangian preserves N=4

2. N=3 (non-Lagrangian) SCFTs

3. The simplest examples

4. OPEs of BPS operators

O1(x)O2(0) =X

k

c12k(x)Ok(0)

OPEs of 4d CFTs

• OPEs relate n-pt functions to (n-1)-pt functions.

Operator Product Expansion

• How to compute this without Lagrangian...?

O4d(x) O2d(z)

’13 [Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees]

For any 4d N=2 SCFT

1/4 BPS operators 2d local operator

x

3,4

zz = x

1 + ix

2

R4 = R2 ⇥ R2

2

4d 2d map

O4d1 (x)O4d

2 (0) =X

k

c12k(x)O4dk (0) O2d

1 (z)O2d2 (0) =

X

k

O2dk (0)

zh1+h2�hk

4d OPE 2d chiral algebra

The detail of the 2d chiral algebra depends on the 4d SCFT you are studying.

What are the 2d chiral algebras corresponding to the 4d N=3 SCFTs we are studying?

Q:

J

OPEs of BPS operators

SU(2)R current N=2 super Virasoro

4d 2d

WfW

�C2/Z`

�Higgs branch operators

W fW ⇠ J `Higgs branch relation :

chiral primary

anti-chiral primary

• 4d N=3 SUSY implies 2d N=2 SUSY

c2d = �12c4d = �3(2` � 1)w/

(` = 3, 4, 6)

The label of 4d N=3 SCFT

OPEs of BPS operators

4d

J

↵ ↵

�+↵ ��

↵

W+ W�

J i↵↵

stress tensor multiplet Higgs branch operator multiplet

J` = W+W�

G GW W

H H

N=2 super Virasoro

T

J

chiral primary anti-chiral primary

2d

stress tensor

h =`

2

Our conjecture

• We assume all the 2d operators are generated by these.

• The 2d chiral algebra associated w/ the 4d rank-one N=3 SCFT for is an N=2 W-algebra generated by

J

W

fW

c2d = �3(2` � 1)N=2 super Virasoro

+chiral primary

anti-chiral primary+ w/

w/

[Odake]

[Inami-Matsuo-Yamanaka]

[Blumenhagen]

` = 3, 4, 6

W(Z)W(0) fW(Z) fW(0)

W(Z) fW(0)

Ansatz for OPEs

~ regular ~ regular

~ composites of and its descendantsJ

Requirement from Jacobi identities

The above ansatz should be consistent with Jacobi identities

c2dconstraint on the value of

null operator relations

OPEs of BPS operators

chiral primary anti-chiral primary

Requirement from Jacobi

`

�15

�21

�3318, �15,

12, �9,

c2dallowed values of

3

4

6

c2d = �12c4d

= �3(2` � 1)

expected value from 4d

W fW ⇠ J `

4d Higgs branch relation is realized as a null relation

( Only for these values, )

OPEs of BPS operators

M = C3/Z` a = c =2` � 1

4

Summary• We have studied 4d rank-one N=3 SCFTs.

` = 3, 4, 6• The theories are labeled by .

• The corresponding 2d chiral algebras are conjectured to be N=2 W-algebras.

• There is no Lagrangian genuine N=3 theory. Therefore they have to be non-Lagrangian.

moduli space of SUSY vacua central charge

Open problemThe superconformal index of these N=3 SCFTs

The character of the 2d W-algebra we identified

=