Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $W$-representations of $\beta$- and $q,t$-deformed matrix models   $W$-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.

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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org