Comment on integrability in Dijkgraaf-Vafa beta-ensembles

TL;DR

The paper critically examines recent claims of integrability in Dijkgraaf-Vafa beta-ensembles, arguing that these claims are based on specific reformulations and may not be consistent with established theories like AGT relations and Nekrasov-Shatashvili quantization.

Contribution

It clarifies that current claims of integrability are based on particular representations and questions their consistency with broader theoretical frameworks.

Findings

Recent claims rely on specific reformulations of partition functions.

Unclear if these reformulations align with AGT relations.

Potential inconsistency with Nekrasov-Shatashvili quantization.

Abstract

We briefly discuss the recent claims that the ordinary KP/Toda integrability, which is a characteristic property of ordinary eigenvalue matrix models, persists also for the Dijkgraaf-Vafa (DV) partition functions and for the refined topological vertex. We emphasize that in both cases what is meant is a particular representation of partition functions: a peculiar sum over all DV phases in the first case and hiding the deformation parameters in a sophisticated potential in the second case, i.e. essentially a reformulation of some questions in the new theory in the language of the old one. It is at best obscure if this treatment can be made consistent with the AGT relations and even with the quantization of the underlying integrable systems in the Nekrasov-Shatashvili limit, which seem to require a full-scale beta-deformation of individual DV partition functions. Thus, it is unclear if the story of integrability is indeed closed by these recent considerations.