Proving the AGT relation for N f = 0, 1, 2 antifundamentals

Abstract

Using recursive relations satisfied by Nekrasov partition functions and by irregular conformal blocks we prove the AGT correspondence in the case of N = 2 superconformal SU(2) quiver gauge theories with N f = 0, 1, 2 antifundamental hypermultiplets.

Affine sl(N)conformal blocks from $ \mathcal{N} = 2 $ SU(N) gauge theories

“…The result (5.4) can also be obtained from the AGT relation by sending the masses to infinity in a conformal SU(2) theory [10,47], and was proven in [48]. The extension to higher rank SU(N ) theories was discussed in [49].…”

Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial

“…Note that this degenerate version of the AGT conjecture is proved by the method of Zamolodchikovtype recursive formula in the papers [6] and [10].…”

Seiberg-Witten curve via generalized matrix model

“…However, if we defineρ(z) by the function in (4.24) under the usual convention that the function f (z) = √ z has cut on the negative real axis, it changes the sign accross the imaginary axis. Since the original eigenvalue function ρ(λ) does not have such discontinuity on C 1 , we rewrite (4.24) and defineρ(z) as 28) so that it is smooth on the path C 1 . The sign was determined so that −iρ(z) is positive at z ∼ im/a.…”

“…( See also [25,26] for the M-theory considerations.) A proof has been given in [27] and [28] for SU(2) gauge theory with adjoint and (N f = 0, 1, 2) fundamental hypermultiplets by using the recursion relation [29][30][31][32].…”