Explicit examples of DIM constraints for network matrix models

TL;DR

This paper demonstrates how Dotsenko-Fateev and Chern-Simons matrix models, representing Nekrasov functions, can be embedded into network matrix models with DIM symmetry, unifying their Ward identities into a single identity.

Contribution

It introduces explicit DIM constraints for network matrix models, unifies various Ward identities under a single framework, and simplifies the understanding of these models' symmetries.

Findings

Embedding of matrix models into network models with DIM symmetry

Unification of Ward identities into a single identity

Simplification for balanced networks

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.