Explicit examples of DIM constraints for network matrix models

Abstract

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.

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

“…(1.1), the job of the R-matrix is to permute the components in the tensor product of representations of the algebra G. This is the property we will use in refined topological strings. The representations in question are going to be Fock modules [37][38][39] and their permutation exchanges the legs of the toric diagram corresponding to a DIM intertwiner [40,41]. The permutation of the legs performed by the R-matrix has a simple interpretation in terms of the corresponding conformal blocks of the q-Virasoro or qW N -algebras.…”

“…The permutation of the legs performed by the R-matrix has a simple interpretation in terms of the corresponding conformal blocks of the q-Virasoro or qW N -algebras. Ratios of the spectral parameters on the horizontal legs determine the Liouville-like momenta of the primary states [41]. By exchanging the spectral parameters, the R-matrix inverts the momenta, and therefore acts exactly as the Liouville reflection matrix introduced in [42].…”

“…The generators, x + n , ψ ± n , x − n and their commutators form a lattice, which is sketched in figure 1. The exact definition of the DIM algebra can be found in [41,[63][64][65][66] (see also [67][68][69][70][71] for elliptic DIM algebra).…”

“…−n )|u 2 (in order to see this, one should use the DIM coproduct [41] and the fact that, for the vertical representations, C 1 = 1). Thus, the generalized Macdonald polynomials…”

Modular properties of 6d (DELL) systems

“…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].…”