Five-Dimensional AGT Relation and the Deformed -Ensemble

118

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.

Section: Jhep07(2016)103mentioning

confidence: 85%

Section: Jhep07(2016)103mentioning

confidence: 99%

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Explicit examples of DIM constraints for network matrix models

Awata

Kanno

Matsumoto

et al. 2016

118

“…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].…”

Section: Jhep10(2016)047mentioning

confidence: 99%

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

Awata

Kanno

Миронов

et al. 2016

R-matrix is explicitly constructed for simplest representations of the DingIohara-Miki algebra. Calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e. of refined topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2, Z) group of DIM algebra automorphisms. We also construct the T -operators satisfying the RT T relations with the R-matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the reflection matrices of the Liouville/Toda theories.

“…The second famous example comes from the gauge theory: the equivariant cohomology of the instanton moduli spaces (captured by Nakajima quiver varieties [12][13][14] and the corresponding Nekrasov partition functions [15][16][17][18][19][20][21][22]) is acted on by a certain vertex operator algebra, which turns out to be the W N -algebra of two dimensional conformal field theory. This correspondence between the geometric (moduli space) and algebraic (W N -algebra) objects is known as the AGT relation [23][24][25] and has many known implications and generalizations [26][27][28][29][30][31][32][33][34][35][36]. These two examples are in fact directly related to each other and their relation can be understood on both sides of the algebro-geometric correspondence.…”

Section: Introductionmentioning

confidence: 99%

Refined toric branes, surface operators and factorization of generalized Macdonald polynomials

Zenkevich

2017

We find new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix model averages, which are themselves consequences of Ding-Iohara-Miki constraints. Factorized expressions for generalized Macdonald polynomials are identified with refined topological string amplitudes containing a toric brane on an intermediate preferred leg, surface operators in gauge theory and certain degenerate CFT vertex operators.