Compactification on the Ω-background and the AGT correspondence Junya Yagi Universität Hamburg

June 28, 2012 @ LMU München

AGT correspondence Relates two kinds of theories: 4d N = 2 gauge theory ⇐⇒ 2d Liouville/Toda CFT Mappings between various objects: É

partition function = correlator

É

Wilson-’t Hooft loops ↔ Verlinde loops

É

surface ops ↔ degenerate fields

CFT symmetry seen in 4d: ˆ

W

H• (M)

M: instanton moduli space

6d derivation Start with the (2, 0) theory on R4 × C. Compactify it in two ways: (2, 0) theory on R4 × C C→0 N = 2 theory on R4 The C → 0 part is understood.

“R4 → 0” Toda theory on C [Gaiotto, Gaiotto–Moore–Neitzke]

We want to understand the “R4 → 0” part.

6d derivation Twisting (2, 0) SUSY on M4 × C2 in two steps: 1. Twist along C. This breaks half of SUSY (16 → 8), giving N = 2 SUSY on M. 2. Twist along M. This is the Donaldson–Witten twist. This leaves a single scalar supercharge Q with Q2 = 0 up to a gauge transformation. The Q-invariant sector is a topological field theory on M.

6d derivation Or one can reverse the order: 1. Twist along M. This preserves 1/ 8 SUSY (16 → 2), and we get N = (0, 2) SUSY on C. 2. Twist along C. This produces a holomorphic field theory on C. The local observables form a chiral (or vertex) algebra, and may contain Virasoro, affine, etc. This is a rich but largely unexplored subject, related to geometric Langlands etc., and as we will see, also AGT.

6d derivation We conclude: twisted (2, 0) theory on M × C has a supercharge Q with Q2 = 0 up to a “gauge transformation,” and is topological along M and holomorphic along C. We will exploit this property.

6d derivation Q-invariant quantities are protected. Place half-BPS codim-2 defect operators at points on C: x

x

x

x x

They become local operators in 2d, but invisible in 4d: 〈· · ·〉 C→0 Z = 〈1〉4d

M→0 〈· · ·〉2d

6d derivation 6d Q-cohomology ⊃ a chiral algebra on C ˆ

Suppose W ⊂ HQ (6d). Then, W

HQ (6d) naturally.

The action will descend to 4d: W ⊂ HQ (6d) C→0 ˆ

W

HQ (4d)

M→0 W ⊂ HQ (2d)

and HQ (4d) = HT• (M). Some cases proven.

[Maulik–Okounkov, Schiffmann–Vasserot]

6d derivation Questions: 1. Compactify on M = R4 ? 2. 2d theory is not supersymmetric! Answers: 1. We need Ω-deformation to confine quantum effects near 0 ∈ R4 . (“compactification on Ω-background”) 2. 2d d.o.f. are Q-invariant but purely bosonic.

(2, 0) theory on the Ω-background Consider N = 2 theory on a cylinder × M2 :

×

Ω-deform it using rotations about the axis of the cylinder:

The Ω-deformation can be canceled by a change of variables & rescaling of e and R! [Nekrasov–Witten]

(2, 0) theory on the Ω-background Apply this to twisted N = 2 theory on R2 × M2 . Bend R2 into a cigar D:

−→

The Ω-deformation can be canceled on the flat region. The Ω-deformation is equivalent to the insertion of a Q-invariant operator supported near the tip {0} × M2 .

(2, 0) theory on the Ω-background Similarly, for a theory on R4 , we can bend R4 = R2 × R2 into the product of cigars D1 × D2 :

×

The Ω-deformation is represented by Q-invariant operators supported near the tips {0} × D2

and D1 × {0}.

(2, 0) theory on the Ω-background Now the case of the (2, 0) theory on R4 × C ∼ D1 × D2 × C:

×

×

In this case, we define the Ω-deformation by lifting these Q-invariant operators from 4d to 6d. They are supported near the tips {0} × D2 × C

and D1 × {0} × C.

(2, 0) theory on the Ω-background To better understand the situation, let’s compactify on the circle fibers of the cigars. Then D1 × D2 × C becomes y

× x

and the theory becomes N = 4 SYM, with Q-invariant boundary couplings. To identify them, we need to determine the boundary conditions.

Boundary conditions The boundaries are endowed with half-BPS BCs. To determine the BCs, brane construction is helpful. Take N M5-branes wrapped on D1 × D2 × C:

×

×

At low energies, we get the (2, 0) theory of type AN−1 . Let’s take R1 , R2 → 0.

Boundary conditions Look at 0 ∈ D1 , regarding D2 as a cylinder. Take R2 → 0. We get N D4 branes on D1 × R × C.

D4 x

Boundary conditions Look at 0 ∈ D1 , regarding D2 as a cylinder. Take R2 → 0. We get N D4 branes on D1 × R × C.

NS5 D3 x

Take R1 → 0. We get a D3-NS5 system.

Boundary conditions N = 4 SYM has 6 adjoint scalars. The NS5 brane divides them into two triplets ~ = (X1 , X2 , X3 ), X

~ = (Y1 , Y2 , Y3 ), Y

coming from fluctuations tangent/normal to the NS5. ~ and A: The D3-NS5 BCs are Neumann for X ~ = Fxμ = 0, Dx X ~: while Dirichlet for Y

~ = 0. Y

Boundary conditions Next, look at 0 ∈ D2 , regarding D1 as a cylinder. Take R2 → 0. We get D4s ending on a D6.

y D4

D6

Boundary conditions Next, look at 0 ∈ D2 , regarding D1 as a cylinder. Take R2 → 0. We get D4s ending on a D6.

y D3

D5 Take R1 → 0. We get a D3-D5 system.

Boundary conditions The roles of D1 and D2 were interchanged, so we applied S-duality. The D3-D5 BCs are: ~= X

~t y

+ less singular

with [ti , tj ] = εijK tk (“Nahm pole”). ~t give a principal embedding sl2 → gC . ~ obey Dirichlet: A and Y ~ = 0. A=Y

Boundary conditions All in all, we have:

y NS5 D3

D5 x ~ = (X1 , X2 , X3 ) obey Three scalars X É

Neumann BCs at x = 0 (D3-NS5)

É

Nahm pole BCs at y = 0 (D3-D5)

Boundary couplings The two boundary couplings are related by S-duality. Focus on the D3-NS5 boundary at x = 0. The 6d theory is topological along D1 × D2 and holomorphic along C, so the boundary theory must be 1. holomorphic along C 2. independent of e2 ∝

R1 R2

.

The bulk N = 4 theory realizes these by the topological-holomorphic twist, introduced by Kapustin.

Boundary couplings The Kapustin twist gives a Q-invariant gC -connection: ¯. A = (Ax + iYx )dx + (Ay + iXy )dy + Xz dz + Az¯ dz A natural candidate is then the Chern-Simons term Z   k 2 Tr A ∧ dA + A ∧ A ∧ A 4πi x=0 3 with level k independent of e. Others are possible, but do not seem very interesting; they will not lead to 2d d.o.f.

Boundary couplings The boundary at x = 0 is R≥0 × C. The boundary has a boundary! We get a WZW model on the 2d boundary: Z Z DA e−kSCS [A] =

M3

Dg e−kSWZW [g] ,

∂M3

where g: ∂M3 → GC satisfy g−1 Az g + g−1 ∂z g = 0 on ∂M3 . Affine currents are given by J = −k∂z gg−1 = kAz = kXz .

2d theory On the 2d boundary, we have the Nahm pole BC J=

kt+ y

+ less singular,

with t+ = t1 + it2 is the raising operator of the sl2 ⊂ gC . Let’s make this regular. Take a basis {ta } of g s.t. [it3 , ta ] = sa ta , and split gC = g+ ⊕ g0 ⊕ g − . A singular gauge transf. by e−it3 ln y multiplies ta by ysa : X J = kt+ + ysa f a ta . ta ∈g0 ⊕g−

2d theory The constraints J = kt+ +

X

Ja ta

ta ∈g0 ⊕g−

are first class, generating the G− gauge symmetry. This is quantum Drinfeld–Sokolov reduction! So we get this gauged WZW model. It’s purely bosonic, but Q-invariant since A is. The symmetry of the model is the W-algebra Wk (g). The 2d d.o.f. are confined on the boundary of the boundaries, located at the origin of R4 , as expected.

2d theory Comparing the partition function and a correlator fixes k = −h∨ −

ε2

.

ε1

S-duality interchanges ε1 ↔ ε2 . Notice the (2, 0) theory has type A, D, E. It’s true that W−h∨ − ε2 (g) ∼ = W−h∨ − ε1 (g) ε1

for g = A, D, E.

ε2

2d theory Example: g = AN−1 . The W-algebra is WN . For N = 2, W2 = Vir and we get Liouville theory. More generally, Vir ⊂ WN and c ∼ #N3 for N  1. Gauge fix: 

0 0 .. .

1 0 .. .

0 1 .. .

    J/ k =   0 0 0   0 0 0 WN WN−1 WN−2

... ... .. .

0 0 .. .

... ... ...

1 0 W2

 0 0  ..  .  0  1 0

Semiclassically (k  1), the Wi generate WN .

Seiberg–Witten curve (2, 0) theory compactified on C is an N = 2 theory. Its SW curve is a branched cover Σ ⊂ T ∗ C of C

given in the limit ε1 , ε2 → 0 by   X € 〈det x − Xz ) · · ·〉 = xN + ui (z)xN−i 〈· · ·〉 = 0. (x, z): coordinates on T ∗ C.

Seiberg–Witten curve But recall J = kXz . So X € LHS = 〈det x − J/ k) · · ·〉 = xN 〈· · ·〉 − 〈Wi · · ·〉xN−i . Comparing this with the SW curve equation   X xN + ui (z)xN−i 〈· · ·〉 = 0, we find 〈Wi · · ·〉 ∼ ui 〈· · ·〉 in the limit ε1 , ε2 → 0. 〈Wi 〉 determine the SW curve.

[Alday–Gaiotto–Tachikawa]

Concluding remarks The (2, 0) theory “compactified on the Ω-background” is a gauged WZW model with W symmetry. Our argument É

works for general ε1 , ε2 ∈ C.

É

can incorporate a surface operator, by placing a codim-2 defect at the tip of D1 . This changes the sl2 embedding and the resulting W, explaining the conjecture of Braverman et al. and Wyllard.

É

can deal with the non-ADE case, by including outer-automorphism twists on C. [Tachikawa]

Concluding remarks Future work: É

É

Study the case M = C2 / Zk . One should find para-Toda theory. [Belavin–Feigin, Nishioka–Tachikawa] Connect to Nekrasov–Shatashvili. The connection is more or less clear; W∞ (g) quantizes the Hitchin hamiltonians, while the SUSY configurations of N = 4 SYM form the Hitchin moduli space.

É

Connect to geometric Langlands.

É

Carry out the compactification. We can lift the Ω-deformation to M-theory. [Hellerman–Orlando–Reffert]