Takahiro Nishinaka ( Ritsumeikan U. ) 1. Review of [ Beem, Lemos, - - PowerPoint PPT Presentation

Takahiro Nishinaka

( Ritsumeikan U. )

Matt Buican, Jaewang Choi, Kazuki Kiyoshige, Zoltan Laczko, Hironori Mori, Sanefumi Moriyama, Yuji Tachikawa, Seiji Terashima, Ruidong Zhu

thanks to :

Chiral algebras for 4d superconformal field theories N ≥ 2

arXiv: 1706.03797

• 2. The last part is based on

w/ Matt Buican, Zoltan Laczko

( Queen Mary )

• 1. Review of

[ Beem, Lemos, Liendo, Peelaers, Rastelli, van Rees ]

arXiv: 1312.5344

What do we know about the OPEs when we have SUSY.

Q:

O1(x) O2(0) = X

k

c12k(x) Ok(0)

OPEs in 4d CFTs

Let us focus on chiral primary operators such that

O

Introduction

The OPE of two chiral primaries is non-singular chiral

+

O1(x) O2(0) = O3(0) Φ = O + θψ + θ2F ⇥ Q ˙

α, O

⇤ = 0

• r

We can safely set to get

x = 0

( chiral ring )

O1(0) O2(0) = O3(0)

[Sα, O] = ⇥ S ˙

α, O

⇤ = 0 ( )

4d N=1 SCFT

( 0d OPEs )

⊃

∀

Super Conformal Field Theory

This is useful to study SUSY vacua of the theory.

O1(0) O2(0) = O3(0)

2d chiral algebra 4d N=2 SCFT

( Vertex Operator Algebra )

⊃

∀

∃

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

x3,4

z Since here is a coordinate dependence, this algebra captures more than the SUSY vacua of the theory.

Goal of this talk

• 1. I will review…
• 4d N=2 SCFTs 2d chiral algebras

Virasoro algebra ( )

c < 0

• 2. I will also talk about our recent work.

⊃

• Its character 4d superconformal index

=

’13 [ Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees ] ’17 [ Buican - Laczko - TN ]

( exotic 4d N=2 SCFT whose detail is totally unclear )

Outline

• 1. 2d chiral algebra
• 2. Examples
• 3. What’s still to be understood
• 4. Our recent work

3 slides 6 slides 7 slides 7 slides

Outline

• 1. 2d chiral algebra
• 2. Examples
• 3. What’s still to be understood
• 4. Our recent work

3 slides 6 slides 7 slides 7 slides

O

I = 1, 2 N=2 supercharges

2d chiral algebra

su(2)R charge scaling dim. so(4) spins 4d N=2 SCFT Schur operators : annihilated by

Q1

−, Q 2 ˙ −, S1−, S 2 ˙ −

QI

±, QI ˙ ±, SI ±, S I ˙ ±

= ⇒

Q ≡ Q1− + S

2 ˙ −

{Q, Q†} = ∆ − (j1 + j2) − 2R ∆ − (j1 + j2) − 2R = 0

Q

Ker / Im Q

'

=

Schur ops.

• {Q, Q†}

Ker cohomology

2d chiral algebra

Schur operators

∆ = 2R, j1 = j2 = 0

∆ = 3, j1 = j2 = 1 2, R = 1

Jµσµ

+ ˙ +

O

SU(2)R current derivative

e.g. )

Higgs branch op.

σµ

+ ˙ +∂µ

∆ = 1, j1 = j2 = 1 2

The spectrum of Schur operators is generally highly non-trivial. su(2)R charge scaling dim. so(4) spins

∆ − (j1 + j2) − 2R = 0

2d chiral algebra

4d 2d

x3,4 z = x1 + ix2

O2(0)

x3 = x4 = 0

O1(z, ¯ z)

SU(2)R lowering op.

= ⇒

⇥ Q, L−1 ⇤ = 0

b L−1 =

• Q, Q

1 ˙ −

✓ Q ≡ Q1

− + S 2 ˙ −◆

( -exact )

Q

Schur Twisted translation Schur

O1(z, ¯ z) ≡ ezL−1+¯

z b L−1 O1(0) e−zL−1−¯ z b L−1

L−1 ≡ P1 − iP2 b L−1 ≡ P1 + iP2 + R−

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

2d chiral algebra

O1(z, ¯ z) ≡ ezL−1+¯

z b L−1 O1(0) e−zL−1−¯ z b L−1

Then the 4d OPE implies the following “2d OPE”

O1(z, ¯ z)O2(0, 0) = X

k

c12k zh1+h2−hk Ok(0, 0)

• exact

Q

+

h = ∆ − R

( ) In the sense of the -cohomology,

Q

O1(z)O2(0) = X

k

c12k zh1+h2−hk Ok(0)

2d chiral algebra

• exact

Q

4d 2d

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

J+ ˙

+ ≡ Jµσµ + ˙ +

T (z) ≡ ezL−1+¯

z b L−1J+ ˙ +(0) e−zL−1−¯ z b L−1

SU(2)R current Virasoro sub-algebra x3,4

z

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

2d chiral algebra

J I

µ(x)J J ν (0) ∼ −3c4d

4⇡4 IJ x2gµν − 2xµxν x8 + 2i ⇡2 ✏IJK xµxνxρ · J ρ

K(0)

x6 + · · ·

4d Virasoro algebra w/ 2d

c2d = −12c4d < 0 T (z)T (0) ∼ −6c4d z4 + 2T (0) z2 + · · ·

O(z)

Ok

twisted translation

4d Schur op. 2d chiral algebra

h = ∆ − R

∆ = j1 + j2 + 2R

cohomology

J+ ˙

+

T (z) M A

O

O(z)

JA(z)

stress tensor affine GF current SU(2)R current flavor GF current Higgs branch

• perators

Virasoro primary

4d 2d

J A

µ

SUSY

Outline

• 1. 2d chiral algebra
• 2. Examples
• 3. What’s still to be understood
• 4. Our recent work

3 slides 7 slides 7 slides

Examples

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

free hypermultiplet

ψ

q

4d

q†(x) q(0) ∼ 1 x2 , φ†(x) φ(0) ∼ 1 x2

φ

χ

SU(2)R doublet

q φ† !

Schur

Examples

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

free hypermultiplet

ψ

q

4d

x3,4

z

2d ( 2d symplectic boson )

qtwisted(z) = q(z, ¯ z) + ¯ z φ†(z, ¯ z)

qtwisted(z) φ(0) ∼ 1 z

T = 1 2(q∂φ − φ∂q)

φ

χ

SU(2)R doublet

q φ† !

Schur

q†(x) q(0) ∼ 1 z¯ z , φ†(x) φ(0) ∼ 1 z¯ z

The same analysis for a free vector mult. is easy. But, for interacting theories, it is not straightforward to identify the corresponding 2d chiral algebra. Many guessworks have been done by using the equivalence:

Trchiral alg.(−1)F qL0

Tr4d local ops.(−1)F q∆−R

=

4d 2d ( superconformal index ) ( character of chiral alg. )

Examples

= q− 27

120

Q∞

n=1(1 − qn)

X

`∈Z

✓ q

(20`−3)2 40

− q

(20`+7)2 40

◆ identical to the character of Virasoro algebra w/ c2d = −22 5 ( = −12c4d ) N=2 pure SU(3) deep IR

H0 Argyres-Douglas massless monopole, dyon

’95 [ Argyres - Douglas ] ’95 [ Argyres - Plesser - Seiberg -Witten ] ’96 [ Eguchi - Hori - Ito - Yang ]

’15 [ Cordova - Shao ]

= 1 + q2 + q3 + q4 + q5 + 2q6 + 2q7 + 3q8 + 3q9 + 4q10 + 4q11 + 6q12 + · · ·

Tr4d local ops.(−1)F q∆−R

H0 Argyres-Douglas theory ( N=2 SCFT )

Examples

This result strongly suggests that, for this AD theory, 2d chiral algebra

=

Virasoro algebra w/ c2d = −22 5

H0 Argyres-Douglas theory

’15 [ Cordova - Shao ]

This immediately implies the absence of fermionic Schur ops.

O

Hall-Littlewood

• perators :

annihilated by

Q1

−, Q 2 ˙ −, S1−, S 2 ˙ −

Q

2 ˙ +

When mapped to 2d, they cannot be generated by T (z) Moreover,

S

2 ˙ +

Examples

H0 Argyres-Douglas theory

Absence of such ops in this thy!

This result strongly suggests that, for this AD theory, 2d chiral algebra

=

Virasoro algebra w/ c2d = −22 5

O

Hall-Littlewood

• perators :

annihilated by

Q1

−, Q 2 ˙ −, S1−, S 2 ˙ −

Q

2 ˙ +

When mapped to 2d, they cannot be generated by T (z) Moreover, This immediately implies the absence of fermionic Schur ops.

S

2 ˙ +

’15 [ Cordova - Shao ]

We can learn much about the spectrum of 4d Schur ops. from the 2d chiral algebra. It is more powerful than the superconformal index. Combining it w/ other 4d data will give us more information.

’15 [ Liendo - Ramirez - Seo ]

c4d ≥ 11 30

c2d ≤ −22 5

( for interacting N=2 SCFTs )

2d stress tensor correlator 4d fusion rules e.g.)

+

= ⇒

new 4d unitarity bound

⇐ ⇒

’16 [ TN - Tachikawa ]

N > 2 SCFT

’16 [ Lemos, Liendo, Meneghelli, Mitev ] ’13 [ Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees ]

Examples

=

Schur ops.

• {Q, Q†}

Ker

✓ Q ≡ Q1

− + S 2 ˙ −◆

• If you have N>2 SCA, there are such supercharges :

4d N=3 SCFT 4d N=4 SCFT 2d N=2 2d N=4

• 4d N=2 superconformal algebra (SCA) has no supercharge that

commutes w/ . 2d N=0 chiral alg.

= ⇒

{Q, Q†}

In summary,

• For interacting theories, the chiral algebra can be guessed w/ help of

Trchiral alg.(−1)F qL0

Tr4d local ops.(−1)F q∆−R

=

( superconformal index ) ( character of chiral alg. ) e.g.) SU(2) Nf = 4, MN’s E6, E7, E8, AD theories, …

• 2d chiral algebra tells much about the spectrum of Schur ops.

and more.

Outline

• 1. 2d chiral algebra
• 2. Examples
• 3. What’s still to be understood
• 4. Our recent work

3 slides 7 slides

The full set of relevant chiral algebras?

• Not all 2d chiral algebras are related to 4d

2d chiral algebra 4d N=2 SCFT

∀

∃

( The converse is not true. )

• Which class of 2d chiral algebras is related to 4d N=2 SCFTs?

c2d ≤ −22 5

c2d < 0

( if the 4d is interacting ) There are perhaps more constraints, which are NOT fully understood.

The full set of relevant chiral algebras?

Recent conjecture

’17 [ Beem - Rastelli ]

( 2 weeks ago )

chiral algebra V

⊃ C2(V )

• perators involving

a derivative

{ {

≡

nilpotent elements

⇣ V/C2(V ) ⌘ .

{ {

4d Higgs branch chiral ring

'

conjecture

[ Arakawa ]

This immediately implies that

is null.

T k + ϕ

k > 0, ϕ ∈ C2(V )

∃ ( )

[ Zhu ]

4d N=2 SCFT

∀

2d chiral algebra

∃

4d N=2 SCFT

∀

We might be able to classify 4d N=2 SCFT in terms of 2d chiral algebras.

2d chiral algebra in a class

∀

not established yet

Outline

• 1. 2d chiral algebra
• 2. Examples
• 3. What’s still to be understood
• 4. Our recent work

7 slides

Our recent work

exotic AD

SU(2) exotic AD AD

S-dual

We want to know the chiral algebra for this exotic AD theory.

’14 [ Buidan - Giacomelli - TN - Papageorgakis ]

AD SU(3) AD 3

( β = 0 )

H2 Argyres-Douglas theory w/ flavor SU(3)

’17 [ Buican - Laczko - TN ]

Our recent work

Motivation

• A similar S-duality

SU(3)

6

S-dual

SU(2)

SCFT w/ flavor E6

1 led to the discovery of an infinite number of N=2 S-dualities, and the AGT conjecture.

’96 [ Minahan - Nemeschansky] ’07 [ Argyres - Seiberg ]

No known UV Lagrangian constructed by M5 wrapping S2

AD SU(3) AD 3

We want to check this S-duality.

SU(2) exotic AD AD

S-dual Quite mysterious

AD SU(3) AD 3 SU(2)

exotic AD

AD

S-dual

Can we construct the chiral algebra for this ??

Q :

Let us compute the superconformal index.

Our recent work

Superconformal index

AD SU(3) AD 3 SU(2) exotic AD AD

S-dual

An inversion formula to extract this piece!

∃

flavor U(3) flavor U(1) flavor SU(3)

Tr(−1)F q∆−Raf1

1 af2 2 af3 3

: flavor charges

fi

Z dµSU(3) Ivect(µ) h IAD(µ) i2 I3fund(µ, a1, a2, a3)

=

Z dµSU(2) Ivect(µ) IAD(µ, a1) Iexotic(µ, a2, a3)

Our recent work

sign of a single FREE hyper

exotic AD

???

=

A FREE hyper

+

Let us construct a chiral algebra whose character coincides w/ this.

I??? = Iexotic AD . Ifree hyp

+ ⇣ 2 + χsu(2)

3

+ χsu(2)

5

+ 2χsu(3)

8

+ χsu(2)

3

χsu(3)

8

+ χsu(3)

27

⌘ q2 + O(q

5 2 )

= 1 + ⇣ χsu(2)

3

+ χsu(3)

8

⌘ q + χsu(2)

2

χsu(3)

8

q

3 2

NEW exotic AD theory

Iexotic(µ, a2, a3) = 1 + (µ + µ−1)q

1 2 + O(q)

Our recent work

Chiral algebra

???

exotic AD free hyp.

=

stress tensor w/

T (z)

JA

SU(2)(z)

JI

SU(3)(z)

affine SU(2) x SU(3) current

at critical levels

bosonic primary of dim 3/2

WaI(z)

( 2 x 8 rep of su(2) x su(3) )

generators

c2d = −26

The OPE is uniquely fixed by Jacobi identities. Wa

I(z) Wb J(0)

The character of this algebra reproduces the 4d index

I???

AD SU(3) AD 3 SU(2)

exotic AD

AD

S-dual

We have managed to construct a consistent chiral alg. for this!

AD SU(3) AD 3

S-dual

We have managed to construct a consistent chiral alg. for this!

SU(2)

???

AD Single hyper

???

is a NEW exotic Arygres-Douglas SCFT.

SW curve? UV Lagrangian? 3d reduction? BPS spectrum?

Summary 1

• This 4d/2d relation is available for ANY 4d N=2 SCFT.
• The 2d chiral algebra tells us a lot about the spectrum of Schur
• perators. It is more powerful than the superconformal index.

Lagrangian / non-Lagrangian class S / others do not matter !!

• Sometimes, we can get more information. e.g.) c4d ≥ 11

30

2d chiral algebra 4d N=2 SCFT ∃

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

Schur ops.

∀

’15 [ Liendo - Ramirez - Seo ]

Summary 2

• We constructed the chiral algebra for a NEW exotic AD.

AD SU(3) AD 3

S-dual

SU(2)

???

AD Single hyper

???

SW curve? UV Lagrangian? 3d reduction? BPS spectrum?

• flavor SU(2) x SU(3)
• We know the chiral algebra.

SU(2)

???

4 hypers

Witten’s anomaly

( β = 0 )

’17 [ Buican - Laczko - TN ]

Open problem

Marginal gauging w/ Witten anomaly This gauging is conformal but Witten anomalous…!!!

SU(2)

???

4 hypers

NO

( β = 0 )

YES

• It seems possible to “compute” the superconformal index…

index ~ ZS1×S3 6= ZS4 …

• 2d chiral algebra interpretation of the 4d Witten anomaly???

∃

’17 [ Buican - Laczko - TN ]