Proving AGT conjecture as HS duality: Extension to five dimensions

Abstract

We extend the proof from [25],which interprets the AGT relation as the Hubbard-Stratonovich duality relation to the case of 5d gauge theories. This involves an additional q-deformation. Not surprisingly, the extension turns out to be trivial: it is enough to substitute all relevant numbers by q-numbers in all the formulas, Dotsenko-Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β = 1, i.e. for q = t in MacDonald's notation. For β = 1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams. * Lebedev Physics Institute and ITEP

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

“…In [40,41] (see also [42,43]) the Dotsenko-Fateev (DF) integrals for conformal blocks of the q-deformed CFT [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] have been rewritten as sums over residues, each corresponding to a fixed point in the instanton moduli space of a 5d gauge theory. However, this gauge theory turned out to be not the theory related to the conformal block by the AGT duality, but rather its spectral dual.…”

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

“…3d Chern − Simons new AGT ←→ NS limit of 5d LMNS prepotential ordinary AGT [49] ←→ q − Virasoro conformal blocks More concretely, in accordance with [46], one associates with the solution to the Baxter equation, i.e. with < K > R , the SW differential, its monodromies around the A-and B-cycles on the spectral curve (32) giving rise to the 5d Nekrasov functions in the NS limit via the SW equations.…”

Explicit examples of DIM constraints for network matrix models

“…This corresponds to a composite vertex operator having two parts: the Virasoro part depending onã n and the Heisenberg part depending on the orthogonal linear combination of the oscillators,ā n . This is exactly as prescribed by the AGT relation [175][176][177][178][179][180][181][183][184][185], where the Nekrasov functions for the gauge group U(N ) correspond to the conformal block of the algebra Vir q,t ⊗ Heis q,t .…”

“…In fact, one could repeat this matrix model consideration in the deformed case with non-unit q, following the lines of [175][176][177][178][179][180][181]. However, the actual symmetry in this case becomes much larger than the Virasoro algebra: it is the DIM algebra, and we start its general description in the next section.…”