Ding–Iohara–Miki symmetry of network matrix models

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.

Section: Jhep10(2016)047mentioning

confidence: 87%

Section: Refined Topological Strings and Rt T Relationsmentioning

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

“…Taking into account that the functions c n (τ ) are quasimodular forms of weight 2n, we realize them as polynomials of three generators E 2 , E 4 and E 6 : 18) where β 1 = 0 and γ 1 = γ 2 = 0. The first coefficient α 1 is defined by the first equation of (6.16):…”

Section: Jhep11(2017)023mentioning

confidence: 99%

“…One of the research directions here is the interpretation of the corresponding Nekrasov functions in terms of the representation theory of DIM algebras [20,21] and network models [18,22], which generalize the Dotsenko-Fateev (conformal matrix model [23][24][25][26][27][28]) realization of conformal blocks, manifest an explicit spectral duality [16,17,[29][30][31][32][33][34] and satisfy the Virasoro/W-constraints in the form of the qq-character equations [18,21,[35][36][37]. Another direction is study of the underlying integrable systems, where the main unknown ingredient is the double-elliptic (DELL) generalization [38][39][40][41][42][43] of the Calogero-Ruijsenaars model [44][45][46][47][48][49][50][51].…”

Section: Introductionmentioning

confidence: 99%

Modular properties of 6d (DELL) systems

Aminov¹,

Миронов

Морозов

2017

Self Cite

If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge τ = θ 2π + 4πı g 2 −→ − 1 τ . The low-energy Seiberg-Witten prepotential F (a), however, is not explicitly invariant, because the flat moduli also change a −→ a D = ∂F /∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E 2 . This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N ) theory with two independent modular parameters τ andτ , the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift ofτ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

“…W N -algebra generators are built from the currents of the DIM algebra [39][40][41] and vertex operators are combinations of topological vertices intertwining the action of the DIM algebra and therefore of the W N -algebra [42]. On the geometric side the 5d gauge theory is obtained by compactifying M-theory on the toric CY three-fold.…”

Section: Jhep09(2017)070mentioning

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.