Five-dimensional AGT conjecture and the deformed Virasoro algebra

Abstract

We study an analog of the AGT (Alday-Gaiotto-Tachikawa) relation in five dimensions. We conjecture that the instanton partition function of 5D N = 1 pure SU (2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four-dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.

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

“…In [39] it was shown that for the SU(2) theory with a (simple) surface operator, the dual conformal block satisfies a differential equation (the same differential equation was also found earlier…”

“…It was shown in [25] that the instanton partition function for the N = 2 * theory with k N = 0 is (up to a prefactor) an eigenfunction of the quantum trigonometric Calogero-Sutherland model. Connections between eigenfunctions of quantum integrable systems and instanton partition functions in the presence of surface operators have also been studied in [1,21,39]. In particular, in [1] (see also [40]) it was argued that the instanton partition function for the N = 2 * theory in the critical limit ǫ 2 → 0 is an eigenfunction of the quantum elliptic Calogero-Moser model.…”

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

“…The spectral dual interpretation of the resulting amplitude is the partition function of a 6d linear quiver gauge theory compactified on a two-dimensional torus. The AGT relations in this case [121][122][123][124][125][126][127][128][129][130][131][132][133][134] give the conformal block of the q-deformed W -algebras on torus, or the spherical conformal block of the affine W -algebra [98,99].…”

Volume conjecture: refined and categorified

“…In particular, as explained in [83] (see also [84,88,89]), the twisted superpotential is given by the integral of the Seiberg-Witten differential vdu = log y dx x over a path: Now, let us explain how our discussion in this paper compares to the Nekrasov-Shatashvili limit and its relation to quantum integrable systems. There are some similarities and some differences, and both are important.…”