Counting instantons in N=1 theories of class S k Elli Pomoni - - PowerPoint PPT Presentation

Counting instantons in N=1 theories of class Sk

Elli Pomoni

• [1512.06079 Coman,EP,Taki,Yagi]

[1703.00736 Mitev,EP] [1712.01288 Bourton,EP]

Motivation: N=2 exact results

Gaiotto: 4D N=2 class S: 6D (2,0) on Riemann surface AGT: 4D partition functions = 2D CFT correlators 4D SC Index = 2D correlation function of a TFT Seiberg-Witten theory: effective theory in the IR Nekrasov: instanton partition function Pestun: observables in the UV (path integral on the sphere localizes)

String/M-/F-theory realizations 2D/ 4D relations

What can we do for N=1 theories?

Superconformal Index Intriligator and Seiberg: generalized SW technology Witten: IIA/M-theory approach to curves

Holomorphy fixes: N=2 theories: prepotential (that’ s all in the IR) N=1 theories: superpotential (there are also Kähler terms)

No Localization (No Nekrasov, no Pestun) An S4 partition function plagued with scheme ambiguities. Derivatives of the free energy scheme independent.

[Romelsberge 2005] [Kinney,Maldacena,Minwalla,Raju 2005] [Gerchkovitz, Gomis, Komargodski 2014] [Bobev, Elvang, Kol, Olson, Pufu 2014]

What can we do for N=1 theories?

Conformal Obtained by orbifolding N=2 (inheritance) Labeled by punctured Riemann Surface Index = 2D correlation function of a TFT

[Gaiotto,Razamat 2015]

Class Sk (SΓ):

Can construct conformal N=1 theories. AdS/CFT natural route to several examples. 6D (1,0) on a Riemann Surface.

[Gaiotto,Razamat 2015] [Leigh,Strassler 1995]

[Kachru,Silverstein 1998] [Lawrence,Nekrasov,Vafa1998]

[Heckman,Vafa….]

2D/ 4D relation

Plan

Introduce N=1 theories in class Sk Spectral curves for N=1 theories in class Sk From the curves: 2D symmetry algebra and representations Conformal Blocks Instanton partition function Instanton partition function from Dp/D(p-4) branes on orbifold Free trinion partition functions on S4 3pt functions

Is there AGTk ? 4D partition functions = 2D CFT correlators

Class Sk

Class S and Sk

6D (2,0) SCFT on Riemann surface: 4D N=2 theories of class S 6D (1,0) SCFT on Riemann surface: 4D N=1 theories of class Sk

• [Gaiotto 2009]

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 N M5-branes − − − − . . − . . . − Ak−1 orbifold . . . . − − . − − . .

N M5 branes on X4 x Cg,n SU(N) theory on X4 2D theory on Cg,n

4D/2D relation

[Gaiotto,Razamat 2015]

Class Sk

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 (x10) M NS5 branes − − − − − − . . . . . N D4-branes − − − − . . − . . . − Ak−1 orbifold . . . . − − . − − . .

m1 m2 m4 m3 a1 a2 D 4 D 4 NS 5 NS5 R

4/Z2

−m4 −m3 −m1 −m2 −a2 −a1

[Gaiotto,Razamat 2015]

m1 m2 m4 m3 a1 a2 D4 D4 NS5 NS5

Type IIA _____ _____ ✏ = Γ0Γ1Γ2Γ3Γ4Γ5✏ = Γ0Γ1Γ2Γ3Γ6✏ = Γ4Γ5Γ7Γ8✏ ______ U(1)r SU(2)R

• f N=2

x6

x4, x5

U(1)R

1/g2

• 1/g2

Class Sk

Type IIB N=1 orbifold daughter of N=4 SYM Useful for AdS/CFT (orbifold inheritance) String theory technics to calculate instantons

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 AM−1 orbifold . . . . . . − − − − N D3-branes − − − − . . . . . . Ak−1 orbifold . . . . − − . − − .

Γ = Zk × ZM

AdS5 × S5/ (Zk × ZM)

[Bershadsky, Kakushadze,Vafa 1998]

_ _____ _____ __________ _

[Dorey, Hollowood, Khoze, Mattis,…] [Lerda,…]

Class Sk

[Gaiotto,Razamat 2015]

4D field theory point of view

Large global symmetry group

• U(1)t

U(1)αc U(1)βi+1−c U(1)γi V(i,c) Φ(i,c) 1 1 +1 Q(i,c−1) +1/2 1 +1 e Q(i,c−1) +1/2 +1 1

WS =

M−1

X

c=1

⇣ Q(c−1)Φ(c) ˜ Q(c−1) − ˜ Q(c)Φ(c)Q(c) ⌘ WSk =

k

X

i=1 M−1

X

c=1

⇣ Q(i,c−1)Φ(i,c) ˜ Q(i,c−1) ˜ Q(i,c)Φ(i,c)Q(i+1,c) ⌘

N Φ(c) = B B B B B B @ Φ(1,c) Φ(2,c) ... Φ(k−1,c) Φ(k,c) 1 C C C C C C A

Class Sk

Q(c) = B B B B @ Q(1,c) Q(2,c) ... Q(k,c) 1 C C C C A e Q(c) = B B B B @ e Q(k,c) e Q(1,c) ... e Q(k−1,c) 1 C C C C A

• kNxkN

NxN

Begin with N=2 class S with SU(kN) gauge groups: Orbifold projection:

[Douglas,Moore 1996]

Coulomb and Higgs branch

• Coulomb Branch: with and

Higgs Branch: with operators e.g.

mi = 0

hφi = a = 0 hQi = 0

hi = diag (a1, . . . , aN)

u` = htrφ`i

µIJ = htr

• Q{I ¯

QJ }

• i

u`k = htr

• Φ(1) · · · Φ(k)

`i ,

• hU −1ΦUi = diag (a1, a2, · · · , aN) ⌦ diag

⇣ 1, e

2⇡i k , e 4⇡i k · · · e 2⇡i(k−1) k

⌘

CB and HB do not mix (no relations): charged under different charges!

Coulomb Branch: parameterised by Higgs Branch: similar w/ mother theory, operators charged under new beta and gamma symmetries.

hQi = 0

[Bourton,Pini,EP to appear]

E = r E = 2R E = r

Curves

[1512.06079 Coman,EP,Taki,Yagi]

Generic N=1 Curves

The spectral curve computes the effective YM coupling constants. M-theory on CY3 locally two holomorphic line bundles on the curve Cg,n N=1 spectral curve is an overdetermined algebraic system of eqns. For class Sk on the Coulomb Branch ( ) only one equation exactly like for N=2 theories.

R3,1 × CY3 × R1,

[Bah, Beem, Bobev, Wecht] [Bonelli,Giacomelli,Maruyoshi,Tanzini] [Xie… ]

hQi = 0

[Coman,EP,Taki,Yagi] [Intriligator,Seiberg]

m1 m2 m4 m3 a1 a2 D 4 D 4 NS 5 NS5 R

−m4 −m3 −m1 −m2 −a2 −a1

Zero vevs for Higgs branch

• perators!

[Coman,EP,Taki,Yagi]

Sk curves

• (v , w) ∼

⇣ e

2πi k v , e− 2πi k w

⌘ v = x4 + ix5

w = x7 + ix8 t = e− x6+ix10

R10

• hQi = 0

[Lykken,Poppitz,Trivedi 97]

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 (x10) M NS5 branes − − − − − − . . . . . N D4-branes − − − − . . − . . . − Ak−1 orbifold . . . . − − . − − . .

_____ _____ U(1)r SU(2)R

• f N=2

U(1)R

Curves from M-theory

• M-theory: a single M5 brane with non trivial topology

[Witten 1997] 2D surface F(t,v)=0 in the 4D space {x4, x5, x6, x10}={v,t}.

m1 m2 m4 m3 a1 a2 D4 D4 NS5 NS5

(t − 1)(t − q)v2 − P1(t)v + P2(t) = 0

coupling constant q=e2πi

• M=m1+m2+m3+m4

u = tr2 _

• _

(v − m1)(v − m2)t2 +

• −(1 + q)v2 + qMv + u
• t + q(v − m3)(v − m4) = 0
• q
• v = x4 + ix5

t = e− x6+ix10

R10

Class S Curve

(t − 1)(t − q)v2 − P1(t)v + P2(t) = 0

SU(2) with 4 flavors

q=e2πi

• M=m1+m2+m3+m4

u = tr2

_

• _

(v − m1)(v − m2)t2 +

• −(1 + q)v2 + qMv + u
• t + q(v − m3)(v − m4) = 0

m1 m2 m4 m3 a1 a2 D4 D4 NS5 NS5

• m1

m2 m4 m3 a1 a2 D 4 D 4 NS 5 NS5 R

−m4 −m3 −m1 −m2 −a2 −a1

(vk − mk

1)(vk − mk 2)t2 + P(v)t + q(vk − mk 3)(vk − mk 4) = 0

P(v) = −(1 + q)v2k + ukvk + u2k

_ _

• htr
• Φ(1) · · · Φ(k)

2i ⇠ u2k

vevs of gauge invariant operators: parameterize the Coulomb branch

• htr
• Φ(1) · · · Φ(k)
• i ⇠ uk

Class Sk Curve

t = e v ∼ e

2πi k v

q

_ _

(4)

k` (t) = (−1)` c(`,k) L

t2 + uk`t + (−1)` c(`,k)

R

q tk`(t − 1)(t − q)

c(s,k) =

N

X

i1<···<is=1

mk

i1 · · · mk is

Gaiotto or UV curve a sphere with n punctures SW or IR curve

• f g=kN-1

Sk curves

(3)

k` (t) = (1)` c(`,k) L

t c(`,k)

R

tk`(t 1)

X xkN = −

N

X

`=1

(4)

k` (t)xk(N−`)

Full/maximal puncture: k images of N mass parameters Simple puncture: No further parameters

Curve decomposition and S-duality

Different S-duality frames (as for N=2 class S) Weak coupling: free trinions Strong coupling limit: new type of punctures!

[Coman,EP,Taki,Yagi]

∞ q 1 A B C D ∞ 1 A C B D q'=1/q ∞ 1 A D C B q'=1−q V S C

q

• >

q

• >

1 q

• >

q

• >
• igure
• The
• q
• >

1

S-duality = 2D crossing equation

Instantons from the 2D Blocks

Zinst = Bw(w1, w2, w3, w4|q)

[1703.00736 Mitev,EP]

The AGT-W correspondence

4D N=2 theories of class S with SU(2)/SU(N) factors 2D Liouville/Toda CFT

A relation between:

4D gauge theory 2D CFT instanton partition function conformal block perturbative part 3-point function coupling constants cross ratios masses external momenta Coulomb moduli internal momenta generalized S-duality crossing symmetry Omega background Coupling constant/central charge

[Alday,Gaiotto,Tachikawa] [Wyllard]

ZS4 [Tg,n] = Z da Zpert |Zinst|2 = Z dα C . . . C |Bαi

α |2 = h n

Y

i=1

VαiiCg,n

er b2 = ✏1

✏2

The AGT relation from the curve

Close to the punctures:

• hi = m2

i

Recall 2D Ward Identities:

Example: N=2 SU(2) Free trinion N=2 SU(N) Free trinion

hT(z)V1(z1)V2(z2)V3(z3)i =

3

X

j=1

 hj (z zj)2 + ∂j z zj

• hV1(z1)V2(z2)V3(z3)i

h

φ(3)

2 (z) ⇠ m2

j

(z−zj)2

φ(3)

2 (z) = h T(z)V1(z1)V2(z2)V3(z3) i

h V1(z1)V2(z2)V3(z3) i = ⌦ ↵ h i φ(3)

` (z) =

⌦ W`(z)V(z1)V•(z2)V(z3) ↵ ⌦ V(z1)V•(z2)V(z3) ↵

A B C

Dynkin diag

From the curves to the 2D CFT

The symmetry algebra that underlies the 2D CFT = WkN algebra The reps are standard reps of the WkN algebra Obtain them from the N=2 SU(kN) after replacing: mSU(Nk)

j+Ns

7 ! mj e

2πi k s

aSU(Nk)

j+Ns

7 ! aj e

2πi k s

ÈÈ J(t) ÍÍn

def

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

[1703.00736 Mitev,EP]

lim

✏1,2→0 hh J`(t) iin = (n) `

(t)

2D Conformal Blocks = Instanton P.F.

We have the reps of the WkN algebra for ε1,2 = 0 (from the curve) Demand: the structure of the multiplet (null states) not change ε1,2 ≠ 0 The blocks for ε1,2 ≠ 0: proposal for the instanton partition functions: If w and c turn on Q ≠ 0 as in Liouville/Toda, then we obtain them from the N=2 SU(kN) after replacing:

Zinst = Bw(w1, w2, w3, w4|q)

mSU(Nk)

j+Ns

7 ! mj e

2πi k s

aSU(Nk)

j+Ns

7 ! aj e

2πi k s

[1703.00736 Mitev,EP]

Instantons from D(p-4) branes

[1712.01288 Bourton, EP]

Instantons from D branes

[Witten 1995, Douglas 1995, Dorey 1999, ...]

N coincident Dp−

k D(p−

Tr

• Dp

dp+1x Cp−3 ∧ F ∧ F

Instanton on Dp brane = D(p-4) brane

N Dp

K D(p 4) The Dp-D(p-4) strings give the NxK I, J† and the D(p-4)-D(p-4) the KxK B1, B2 auxiliary matrices of the ADHM construction

µC := [B1, B2] + IJ = 0 , µR := [B1, B†

1] + [B2, B† 2] + II† J†J = 0

MDp

K-inst ' MD(p−4) Higgs

4) =

• B1, B2, I, J | µC = 0, µR = 0

/U(K)

4) =

n X⊥

Dp = 0, VD(p−4) = 0

• /U(K)

Instanton Partition Function from 2D SCI

[Gadde, Gukov, Putrov]

N coincident Dp−

N Dp

[Benini,Eager,Hori,Tachikawa]

Instanton’ s index = 2D SCI = flavoured elliptic genius

MDp

K-inst ' MD(p−4) Higgs

For p=5 and when the D1s wrapping a T2:

• ZD5

K-inst(a, m, ✏1, ✏2) = ZK D1 Higgs (a, m, ✏1, ✏2) = I2D = TrMK D1

Higgs(−1)F e✏1JLe✏2JRe⇠ReaJGemJF

(4.8)

• 2D Flavor

2D R-symmetry J69-J78 J01 , J23 J69+J78

ZK-inst(a, m, ✏1, ✏2) = TrMK-inst(−1)F e✏1JLe✏2JRe⇠ReaJGemJF

_______________________

Obtain the for mass deformed N=4 SYM (N=2*) from a 2D SCI computation: The 2D SCI of the gauge theory on the D1 branes = instantons of the 6D (2,0) on = M-strings. KK reducing: instantons of mass deformed 5D N=2 MSYM on and further KK reduce to mass deformed N=4 SYM in 4D.

Mass deformed N=4 SYM

Zinst

D3/D(1)

T-duality

• ! D5/D1

R4 ⇥ T 2 S1

K N

2D theory with (4,4) susy

“Orbifold” the 2D SCI of mass deformed N=4 SYM with one, say the ZM orbifold: we get M-strings on a transverse orbifold = instantons of an SU(N)M quiver when reduce down to 5D/4D

Instantons with an orbifold

Further “Orbifold” the 2D SCI with the Zk orbifold we get something new that should correspond to instantons of class Sk, the rational, trigonometric and elliptic uplift.

2D theory with (0,4) susy 2D theory with (0,2) susy

R4 × T 2 R4 × S1 R4 ID1 = Z(2,0)

inst KK

− − − − → ZN =2

inst KK

− − − − → ZN =4

inst

IZM

D1

= Z(1,0)

inst KK

− − − − → ZN =1

inst KK

− − − − → ZN =2

inst

IZM×Zk

D1

= Z(1,0)w/defect

inst KK

− − − − → ZN =1w/defect

inst KK

− − − − → ZN =1

inst

The same answer as in from conformal blocks!

[1703.00736 Mitev,EP] x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 N D5 – – – – – – · · · · ZM · · · · · · ⇥ ⇥ ⇥ ⇥ Zk · · · · ⇥ ⇥ · ⇥ ⇥ · K D1 · · · · – – · · · · ZD5

K-inst(a, m, ✏1, ✏2) = ZK D1 Higgs (a, m, ✏1, ✏2) = I2D = TrMK D1

Higgs(−1)F e✏1JLe✏2JRe⇠ReaJGemJF

(4.8)

Sk Instantons

Instantons from D0 branes

NS5 NS5 N D41

R

N D41

L

N D41 K1 D0 N D42

R

N D42

L

N D42 K2 D0

lim

β1,β2!0 I orb SCI ∝ k

Y

i=1

Z

Ki

Y

I=1

dui,I

Ki

Y

I=1

u0

ii,IJ

Q

j6=i

QKj

J=1 (uij,IJ − 2✏+)

Qk

j=1

QKj

J=1 (uij,IJ + ✏1) (uij,IJ + ✏2)

×

k

Y

j=1 Ki

Y

I=1 N

Y

A=1

(ui,I − e mL,j,A) (ui,I − e mR,j,A) (ui,I − e aj,A − ✏+) (ui,I − e aj,A + ✏+)

NS5 NS5 N D4R N D4L N D4 K D0

The Perturbative piece

Free trinion P.F. = 2D CFT 3pt functions

The free trinion theory on S4 : we can do the determinant! We know the conformal blocks: can write crossing equations Is the free trinion P.F. a solution of the crossing equations ?? 3pt functions (dynamics) + Blocks = AGTk

ZS4

free trinion =

+ V§(Œ)V•(1)V§(0) ,

[in progress Carstensen,EP,Mitev]

The Perturbative part

The free trinion theory on S4 : we can do the determinant! Guess via analytic continuation from 3D Deconstruction of (1,1) Little Strings

[Gorantis,Minahan,Naseer] [Hayling,Panerai,Papageorgakis]

We can write a reasonable guess for the perturbative part!

The orbifolded hyper part agrees with our determinant calculation! vector and chiral contributions precisely give the perturbative Little string!

No Localization: No Pestun for N=1 susy!

[in progress Carstensen,EP,Mitev]

Summary

We constructed spectral curves for N=1 theories in class Sk The curves: 2D symmetry algebra (WkN) and representations Conformal Blocks Instanton partition function Instanton partition function from Dp/D(p-4) on orbifold Free trinion partition functions on S4 = 3pt functions

[in progress Carstensen,EP,Mitev]

Future

Compute one, two instantons with standard QFT techniques Go away from the orbifold point Other N=1 theories The perturbative part? N=1 partition function on S4!

Get the AGTk from (1,0) 6D à la Cordova and Jafferis

[to appear Bourton, EP]

N N N N q q = 0 N 3N

Thank you!