A proof of AGT conjecture forgeneral βS. Kanno, Y. Matsuo and H. Zhang, ArXiv:1304.????

AGT conjecture
Inspired by the idea that the 6d theory can be compactified in two different way

M5 brane
6d theory on 𝐑4 × 𝐓 2

CFT
2d theory on 𝐓 2[Gaiotto ’09]?

N =2 gauge
4d theory on 𝐑4[Alday-Gaiotto-Tachikawa ’09]

General idea
2D CFT
Conformal
blocks4D Gauge
AGT conjecture

Satisfy Wardidentity

Partitionfunction

Constrained by arecursion Relation

MAIN RESULT
Selberg Integral and SU(N) AGT Conjecture
Virasoro constraint for Nekrasov instanton partition function

N =2 theories[Gaiotto ’09]
4d N =2 theories are constructed by wrapping M5 branes over a Riemann surface.

Circles indicate gauge groups,and squares fundamental flavors.

The corresponding space of gauge
couplings is parameterized by a spherewith six punctures.

the same theory depicted as a generalized
quiver, with the SU(2) factors of the flavor
group made evident and labeled.

Nekrasov partition function
The low energy effective Limit of 4d N=2 SUSY gauge theory is described bySeiberg-Witten prepotential. Its generating function is given by Nekrasov.
we focus on the linear quiver gauge theories with gauge group
SU(N1)×···×SU(Nn).

For this case, Nekrasov partition function is written in the form of matrix
multiplication

2D CFT
We illustrate the conformal block as below. It can be reduced to themultiplication of three point functions by inserting a complete basis of theHilbert space at the intermediate channel. Insertion points of such operators aredepicted by arrows.

Method: Variation of Nekrasov formula
Through former works, we get familiar with the action of adding one box

Construction of the constraints
We would like to establish that the partition function is written as an inner product,

We use the following trivial identity of operator O,

Then the Nekrasov function should also satisfy the relation, namely,

Previous work
there exist simple recursion formulae for Nekrasov’s instanton partition function
for SU(N) gauge theories

Extend the constraint to W1+∞ algebra.
Consider general β.
modied vertex operator
Evaluate the action on bra and ket basis

β = 1 case: Action on bra and ket basis

The generator
flips one blackbox at ℓ to white and one white box at −ℓ − n to black.

Commutator with the vertex

algebra
to evaluate the action on bra and ket basis, different from the former β = 1case, we need the introduction of SHc algebra

where cl are central charges and Ek are some nonlinear combinations of D.

Definition of generators on the basis
The action on the ket state basis is defined as

Virasoro and U(1) algebra as subalgebras of SHcU(1) current and Virasoro generators are embedded in SHc as,

These satisfy the following standard relations

Conformal property of modied vertex operator
We need a modified vertex operator

Commutate with U(1) current

U(1)By using the proposal

These are compatible with the fact that

So we can see k is the momentum and iQ =β - 1
We have also established a similar expression of J2 case.

Virasoro=And there is a similar relation of L-1 .

Conclusion

The constraint equations give a direct support to (SU(N) generalization of)
AGT conjecture, with arbitrary β.

Extend the constraint to W1+∞ algebra.