Matrix Models as Integrable Systems

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101

There is now a renewed interest [1]-[4] to a Hurwitz τ -function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz τ -functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model τ -functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W -representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz τ -functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.

Section: Jhep11(2014)080mentioning

confidence: 99%

On KP-integrable Hurwitz functions

Alexandrov

Миронов

Морозов

et al. 2014

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101

“…(2) by a change of integration variables δx i = x n+1 i in the multiple integral (2.28), [17,18,[73][74][75][76][77][78][79]: in this case we get the identities in a slightly different form:…”

Section: Jhep07(2016)103mentioning

confidence: 99%

“…In other words, the full symmetry of the Seiberg-Witten theory seems to be the Pagoda triple-affine elliptic DIM algebra (not yet fully studied and even defined), and particular models (brane patterns or Calabi-Yau toric varieties labeled by integrable systems a la [3,4]) are associated with its particular representations. The ordinary DF matrix models arise when one specifies "vertical" and "horizontal" directions, then convolutions of topological vertices can be split into vertex operators and screening charges, and the DIM algebra constraints can be attributed in the usual way [70][71][72][73][74][75][76][77][78][79] to commutativity of screening charges with the action of the algebra in the given representation. Dualities are associated with the change of the vertical/horizontal splitting, or, more general, with the choice of the section, where the algebra acts [80][81][82].…”

Section: Introductionmentioning

confidence: 99%

Explicit examples of DIM constraints for network matrix models

Awata

Kanno

Matsumoto

et al. 2016

118

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.

“…However, for the purpose of present paper this definition is not enough. The case when the integrand f (U ) depends only on the eigenvalues, does not cover all the eigenvalue models [76][77][78][79][113][114][115][116]. In particular, the main object of the present paper, which we use to describe the PGL of the Selberg integrals, has an integrand which is not a function of the eigenvalues of U only, it involves an "external field" matrix Ψ in the following way…”

Section: Jhep03(2011)102mentioning

confidence: 99%

“…A promising approach to origins of the AGT relations is through their reformulation as relations between matrix models [76][77][78][79] and Seiberg-Witten theory [80][81][82][83], see [84,85] for a concise review of this idea (which is a new application of the topological recursion [86][87][88][89]- [98]). …”

Section: Introductionmentioning

confidence: 99%

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

Миронов

Морозов

Shakirov

2011

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Abstract:The AGT relation identifies the Nekrasov functions for various N = 2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (β-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.