Motivation and Overview Overview Spaces of Superconformal field - - PowerPoint PPT Presentation

2D CFTs for a class of 4D N = 1 theories

Vladimir Mitev

PRISMA Cluster of Excellence, Institut f¨ ur Physik, THEP , Johannes Gutenberg Universit¨ at Mainz IGST Paris, July 18 2017

[V. Mitev, E. Pomoni, arXiv:1703.00736]

Motivation and Overview

Overview

Spaces of Superconformal field theories in 4D

• N = 4 SYM is unique up to the choice of a gauge group

Integrable in the planar limit

• N = 3: not known to exist until recently

[Garc´ ıa-Etxebarria, Regalado, 2015]

• The space of N = 2 SCFTs is rich

Theories of class S [Gaiotto, 2009] led to AGT

Overview

Spaces of Superconformal field theories in 4D

• N = 4 SYM is unique up to the choice of a gauge group

Integrable in the planar limit

• N = 3: not known to exist until recently

[Garc´ ıa-Etxebarria, Regalado, 2015]

• The space of N = 2 SCFTs is rich

Theories of class S [Gaiotto, 2009] led to AGT

• The space of N = 1 SCFTs much less understood

Let’s see which techniques carry over from N = 2!

Exact results for N = 2 SCFTs

• Seiberg-Witten: complete low-energy effective action (IR)

Exact results for N = 2 SCFTs

• Seiberg-Witten: complete low-energy effective action (IR)
• Nekrasov: instanton partition functions

Exact results for N = 2 SCFTs

• Seiberg-Witten: complete low-energy effective action (IR)
• Nekrasov: instanton partition functions
• Pestun: UV observables thanks to localization on S4

Exact results for N = 2 SCFTs

• Seiberg-Witten: complete low-energy effective action (IR)
• Nekrasov: instanton partition functions
• Pestun: UV observables thanks to localization on S4
• Gaiotto: 4D N = 2 class S, 6D (2,0) compactified on a

Riemann surface

Exact results for N = 2 SCFTs

• Seiberg-Witten: complete low-energy effective action (IR)
• Nekrasov: instanton partition functions
• Pestun: UV observables thanks to localization on S4
• Gaiotto: 4D N = 2 class S, 6D (2,0) compactified on a

Riemann surface

• 4D superconformal index = 2D correlation function of a TFT

Exact results for N = 2 SCFTs

• Seiberg-Witten: complete low-energy effective action (IR)
• Nekrasov: instanton partition functions
• Pestun: UV observables thanks to localization on S4
• Gaiotto: 4D N = 2 class S, 6D (2,0) compactified on a

Riemann surface

• 4D superconformal index = 2D correlation function of a TFT
• AGT: 4D partition functions = 2D CFT correlators

Exact results for N = 1 SCFTs

• Superconformal index
• Intriligator and Seiberg: generalized SW techniques
• Witten: IIA/M-theory approach to the SW curve

Beware!

• for N = 2 the SW curve fixes the prepotential (complete IR)
• for N = 1 the SW curve fixes the superpotential

BUT there are also K¨ ahler terms

Beware!

• for N = 2 the SW curve fixes the prepotential (complete IR)
• for N = 1 the SW curve fixes the superpotential

BUT there are also K¨ ahler terms

• No localization yet for N = 1
• S4 partition function Z plagued with scheme ambiguities

[Gerchkovitz, Gomis, Komargodski, 2014]

• BUT derivatives of log Z

can be scheme independent [Bobev, Elvang, Kol, Olson, Pufu, 2014]

A class of 4D N = 1 theories

The Sk theories

Nice Class of N = 1 theories

• N = 1 superconformal
• Constructed by orbifolding N = 2 theories (inheritance)
• Labeled by a punctured Riemann surface Σg,n
• Its index is a correlation function of a 2D TFT

[Gaiotto, Razamat, 2015]

• SW curves known

[Coman, Pomoni, Taki, Yagi, 2015]

The quiver construction

• Circles are SU(N) gauge groups (N = 2 vector multiplets)
• Squares are SU(N) flavor symmetry
• Line segments are N = 2 hypermultiplets

The quiver construction

• N = 2 vector ⇒ N = 1 vector and N = 1 chiral (blue)
• N = 2 hyper ⇒ N = 1 chiral (red) and N = 1 chiral (green)

The quiver construction

The SW curves

Seiberg-Witten theory

SU(N)M −→ U(1)M(N−1)

The Coulomb branch low energy effective action is given by

• an auxiliary algebraic curve given by G(x, t) = 0
• the meromorphic SW differential λSW = x dt

Seiberg-Witten theory

SU(N)M −→ U(1)M(N−1)

The Coulomb branch low energy effective action is given by

• an auxiliary algebraic curve given by G(x, t) = 0
• the meromorphic SW differential λSW = x dt

a =

• α

λSW

aD =

• β

λSW = ∂F ∂a τIR = ∂aD ∂a

Brane contruction

Gauge theory as the world-volume theory on D4 branes ending on NS5 branes

Brane contruction

Gauge theory as the world-volume theory on D4 branes ending on NS5 branes

Brane contruction

Gauge theory as the world-volume theory on D4 branes ending on NS5 branes

Uplift to M-theory

Uplift to M-theory

The brane configuration for Sk

Orbifold identification: v ∼ e

2πi k v

for k = 1, 2, 3, . . .

The brane configuration for Sk

Orbifold identification: v ∼ e

2πi k v

for k = 1, 2, 3, . . .

The brane configuration for Sk

Orbifold identification: v ∼ e

2πi k v

for k = 1, 2, 3, . . .

The Sk SW curve

The SW curve for one SU(N) gauge group

t2

N

• i=1
• vk − mk

L, i

• + t
• − (1 + q)vNk

+

N

• ℓ=1

ukℓv(N−ℓ)k

+ q

N

• i=1
• vk − mk

R, i

• = 0

[Coman, Pomoni, Taki, Yagi, 2015] q = e2πiτ = coupling us = Coulomb moduli

Set v = xt and rewrite:

N

• ℓ=0

φ(4)

kℓ (t)xk(N−ℓ) = 0

φ(4)

kℓ (t) =

(−1)ℓc(ℓ,k)

L

t2 + ukℓt + (−1)ℓc(ℓ,k)

R

q tkℓ(t − 1)(t − q) for ℓ = 1, . . . , N

• Casimirs c(ℓ,k) =

i1<···<iℓ mk i1 · · · mk iℓ

• φ(4) because there are four poles in t:

two full ⊙ at t = 0, t = ∞ and two simple • at t = 1, t = q

Set v = xt and rewrite:

N

• ℓ=0

φ(4)

kℓ (t)xk(N−ℓ) = 0

φ(4)

kℓ (t) =

(−1)ℓc(ℓ,k)

L

t2 + ukℓt + (−1)ℓc(ℓ,k)

R

q tkℓ(t − 1)(t − q) for ℓ = 1, . . . , N

• Casimirs c(ℓ,k) =

i1<···<iℓ mk i1 · · · mk iℓ

• φ(4) because there are four poles in t:

two full ⊙ at t = 0, t = ∞ and two simple • at t = 1, t = q The 2D CFT duals of φ(n) are blocks with insertions of currents

Trinion curve

Weak coupling limit

q → 0 gives the trinion curve It describes a bunch of free chiral hypermultiplets

Observations

SW curve for N = 1 Sk and SU(N) is the SW curve of N = 2 S with SU(kN) with mL, j+Ns −→ mL, j e

2πi k s

mR, j+Ns −→ mR, j e

2πi k s

aj+Ns −→ aj e

2πi k s

us −→

• us if s mod k = 0

0 otherwise

The AGT correspondence

• Zinst = exp
• −

1 ǫ1ǫ2

F

• SW prepotential

from SW curve

+ · · ·

• Zinst = exp
• −

1 ǫ1ǫ2

F

• SW prepotential

from SW curve

+ · · ·

• limǫi→0

Ws(t) 2D CFT Blocks with currents = SW curve coefficients φs(t).

• Zinst = exp
• −

1 ǫ1ǫ2

F

• SW prepotential

from SW curve

+ · · ·

• limǫi→0

Ws(t) 2D CFT Blocks with currents = SW curve coefficients φs(t).

The SW curve knows a lot about the 2D CFT: type of algebra and nature of the primary fields

Repeat

• The standard AGT correspondence is well established and

has given many new results

• It can be generalized in many directions

(parafermionic Toda, etc...)

• We want to generalize to the N = 1 SCFTs
• The class Sk SCFTs are obtained by orbifolding the usual

N = 2 class S SCFTs with a Zk

• Their SW curves are known.

What do they say about the 2D CFT?

The 2D CFT side

WN Toda overview

N − 1 free bosons with an exponential potential One coupling b2 = ǫ1

ǫ2

WN Toda overview

N − 1 free bosons with an exponential potential One coupling b2 = ǫ1

ǫ2

• N = 2 ⇒ Liouville CFT is the simplest non-rational 2D CFT

stress-tensor: W2(z) = T(z) =

∞

• n=−∞

z−n−2Ln

[ Lm , Ln ] =

c 12m(m2 − 1)δm,−n + (m − n)Lm+n

WN Toda overview

N − 1 free bosons with an exponential potential One coupling b2 = ǫ1

ǫ2

• N = 2 ⇒ Liouville CFT is the simplest non-rational 2D CFT

stress-tensor: W2(z) = T(z) =

∞

• n=−∞

z−n−2Ln

[ Lm , Ln ] =

c 12m(m2 − 1)δm,−n + (m − n)Lm+n

• Toda: Virasoro → WN with higher spin currents W2, . . . , WN

The algebra becomes non-linear

Decomposition of the correlation function

• V1(∞)V2(1) V3(q)V4(0)
• OPE
• =

Decomposition of the correlation function

• V1(∞)V2(1) V3(q)V4(0)
• OPE
• =

=

• dα
• primary
• Young diagram Y
• descendants
• C(L−Yα)34

q∆3+∆4−∆(L−Yα)

V1(∞)V2(1)(L−YVα)(0)

Decomposition of the correlation function

• V1(∞)V2(1) V3(q)V4(0)
• OPE
• =

=

• dα
• primary
• Young diagram Y
• descendants
• C(L−Yα)34

q∆3+∆4−∆(L−Yα)

V1(∞)V2(1)(L−YVα)(0) =

• dαC12αCα34q∆α−∆3−∆4

Y,Y′

q|Y|

V1(∞)V2(1)(L−YVα)(0) V1(∞)V2(1)Vα(0)

• =γ12α(Y)

× Q−1

∆α(Y, Y′)

• Shapovalov form

L−YVα | V3(1)V4(0) Vα | V3(1)V4(0)

Decomposition of the correlation function

• V1(∞)V2(1) V3(q)V4(0)
• OPE
• =

=

• dα
• primary
• Young diagram Y
• descendants
• C(L−Yα)34

q∆3+∆4−∆(L−Yα)

V1(∞)V2(1)(L−YVα)(0) =

• dα C12αCα34 Bα(∆1, . . . , ∆4)

3 and 4-point blocks

3-point block γ123(Y) = V1(∞)V2(1)(L−YV3)(0) V1(∞)V2(1)V3(0)

is fixed by symmetry if one of the Vi is a simple puncture •

The 4-pt block B =

• Y,Y′

γ12∆(Y) Q−1

∆ (Y, Y′) ¯

γ∆;34(Y′) =1 + q(∆ − ∆1 + ∆2)(∆ + ∆3 − ∆4)

2∆

+ · · ·

3 and 4 point blocks with insertions

J(t)

n = n-point block with insertion of J(t)

n-point block

3 and 4 point blocks with insertions

J(t)

n = n-point block with insertion of J(t)

n-point block

3-points J(t)

3 =

V1(∞)V2(1)J(t)V3(0) V1(∞)V2(1)V3(0)

3 and 4 point blocks with insertions

J(t)

n = n-point block with insertion of J(t)

n-point block

3-points J(t)

3 =

V1(∞)V2(1)J(t)V3(0) V1(∞)V2(1)V3(0) 4-points J(t)

4 =

• Y,Y′ q|Y|γ12w(J(t); Y)Q−1

w (Y, Y′)¯

γw;34(Y′)

• Y,Y′ q|Y|γ12w(Y)Q−1

w (Y, Y′)¯

γw;34(Y′)

AGT for the curves

lim

ǫi→0

Ws(t)

n = φ(n) s (t)

for the currents W2, W3, . . . n = 3 trinion n = 4 one gauge group

In words

ratio of blocks with insertions of the spin s current

= sth differential of the SW curve

Results

• The Sk SU(N) SW curves are reproduced by WkN blocks

Results

• The Sk SU(N) SW curves are reproduced by WkN blocks
• All the Ws with s kℓ vanish because

mj+Np −→ mj e

2πi k p

sends the Ws charges to zero

• limǫi→0

Ws(t)

n = φ(n) s (t)

tells us the structure of the CFT representations

Structure of the simple punctures

The norm of L−1V• for k > 1 is zero

|| | L−1V• ||2 = L−1V• | L−1V• = 2∆• V• | V• = 0 ⇒ We are dealing with a non-unitary theory!

Predictions

• Since B = Zinst, we get a prediction for the Sk instanton

partition functions

Predictions

• Since B = Zinst, we get a prediction for the Sk instanton

partition functions

• We need c to complete the prediction

We conjecture that it remains the same c = (kN − 1)

• 1 + kN(kN + 1)Q2

Q = ǫ1 + ǫ2

√ǫ1ǫ2

for Sk SU(N)

• Hence

class Sk SU(N) Zinst =

• class S SU(kN) Zinst
• mj+Ns−→mj e

2πi k s

How to build the CFT dual

• The blocks/instantons do not give the full CFT correlation

function, we need the 3-pt functions

• If we had localization for the N = 1 SCFTs,

we could see if they are given by a 2d CFT

How to build the CFT dual

• The blocks/instantons do not give the full CFT correlation

function, we need the 3-pt functions

• If we had localization for the N = 1 SCFTs,

we could see if they are given by a 2d CFT

• Since we don’t, we have to bootstrap our way to victory!
• Simplest case N = 1, k = 2: Liouville conformal blocks and
• ne extra non-unitary representation

• The simplest 3-pt correlation function

V⊙(∞)V•(1)V⊙(0)

is the S4 partition function of the trinion theory

• The simplest 3-pt correlation function

V⊙(∞)V•(1)V⊙(0)

is the S4 partition function of the trinion theory

• That theory is just a bunch of free N = 1 hypermultiplets
• Hence, its partition function is just det(∆)

The correlation functions have to obey crossing [J.P . Carstensen, VM, E. Pomoni, work in progress]

Summary and Outlook

So far...

• The SW curves have been constructed
• From the curves we got the 2D symmetry algebra WkN
• The curve/block comparisons give the structure of the

representations

• The conformal blocks give a prediction for the instanton

partition functions

Is there AGTk?

• Check the instanton partition functions directly
• Free trinion partition functions on S4 = 3-point functions
• Verify the crossing symmetry?
• Get localization for these N = 1 theories
• Get away from the orbifold point

Thank you