Anomaly in RTT relation for DIM algebra and network matrix models

TL;DR

This paper investigates the RTT relation in network matrix models, revealing an abelian anomaly cocycle that affects braiding and modular properties, with implications for gauge theories and Nekrasov functions.

Contribution

It explicitly evaluates the anomaly cocycle in quantum toroidal algebra representations and shows how to cancel it, elucidating the modular structure of related gauge theories.

Findings

Identifies an abelian anomaly cocycle in RTT relations for network matrix models.

Shows how the cocycle influences braiding and modular properties of conformal blocks.

Describes the limit to affine Yangian, revealing breaking of spectral duality.

Abstract

We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which we explicitly evaluate for the free field representations of the quantum toroidal algebra. This cocycle is responsible for the braiding, which permutes the external legs in the q-deformed conformal block and its 5d/6d gauge theory counterpart, i.e. the non-perturbative Nekrasov functions. Thus, it defines their modular properties and symmetry. We show how to cancel the anomaly using a construction somewhat similar to the anomaly matching condition in gauge theory. We also describe the singular limit to the affine Yangian (4d Nekrasov functions), which breaks the spectral duality.