Comment on integrability in Dijkgraaf-Vafa beta-ensembles

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.

6 pages

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• New Insights into Superintegrability from Unitary Matrix Models
• On determinant representation and integrability of Nekrasov functions
• Superintegrability as the hidden origin of Nekrasov calculus
• Tensorial generalization of characters
• q-Painleve equation from Virasoro constraints
• Check-Operators and Quantum Spectral Curves
• Faces of matrix models
• Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
• AGT correspondence, (q-)Painlevè equations and matrix models