On $W$-representations of $\beta$- and $q,t$-deformed matrix models

Abstract

$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- and $q,t$-deformations of the simplest rectangular complex matrix model. In the latter case, instead of the complicated definition in terms of multiple Jackson integrals, we define partition functions as the weight-two series, made from Macdonald polynomials, which are evaluated at different loci in the space of time variables. Resulting expression for the $\hat{W}$ operator appears related to the problem of simple Hurwitz numbers (contributing are also the Young diagrams with all but one lines of length two and one). This problem is known to exhibit nice integrability properties. Still the answer for $\hat{W}$ can seem unexpectedly sophisticated and calls for improvements. Since matrix models lie at the very basis of all gauge- and string-theory constructions, our exercise provides a good illustration of the jump in complexity between $\beta$- and $q,t$-deformations -- which is not always seen at the accidently simple level of Calogero-Ruijsenaars Hamiltonians (where both deformations are equally straightforward). This complexity is, however, quite familiar in the theories of network models, topological vertices and knots.

11 pages

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Cited by in corpus (7)

• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• Correlators in the supereigenvalue model in the Ramond sector
• Generalized cut operation associated with higher order variation in tensor models
• Higher order constraints for the ($\beta$-deformed) Hermitian matrix models
• Symmetric deformed 2D/3D Hurwitz-Kontsevich model and affine Yangian of gl(1)
• $\beta$-ensembles and higher genera Catalan numbers