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Register a new user$b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
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Added by
Maciej Do{\\l}\\k{e}ga
Publication date
2021
fields
Physics
and research's language is
English
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We study a $b$-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. It is a special case of the $b$-deformed weighted Hurwitz numbers recently introduced by the last two authors and has an interpretation in terms of generalized branched coverings of the sphere by non-oriented surfaces. We give an evolution (cut-and-join) equation for this model and we derive, by a method of independent interest, explicit Virasoro constraints from it, for arbitrary values of the deformation parameter $b$. We apply them to prove a conjecture of Feray on Jack characters. We also provide a combinatorial model of non-oriented monotone Hurwitz maps, which generalizes monotone transposition factorizations. In the case $b=1$ we show that the model obeys the BKP hierarchy of Kac and Van de Leur. As a consequence of our analysis we prove a recent conjecture of Oliveira and Novaes relating zonal polynomials with the dimensions of irreducible representations of $O(N)$. We also relate the model to an $O(N)$ version of the Brezin-Gross-Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.
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A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.
We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction on the rank of the matrix model.
We show how q-Virasoro constraints can be derived for a large class of (q,t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for beta-ensembles. From free-field point of view the models considered have zero momentum of the highest weight, which leads to an extra constraint T_{-1} Z = 0. We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on D^2 cross S^1 and S^3.
The Virasoro constraints play the important role in the study of matrix models and in understanding of the relation between matrix models and CFTs. Recently the localization calculations in supersymmetric gauge theories produced new families of matrix models and we have very limited knowledge about these matrix models. We concentrate on elliptic generalization of hermitian matrix model which corresponds to calculation of partition function on $S^3 times S^1$ for vector multiplet. We derive the $q$-Virasoro constraints for this matrix model. We also observe some interesting algebraic properties of the $q$-Virasoro algebra.
In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalised Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection theory of some analogues of the moduli space of curves. The structure of these conjectural intersection numbers is discussed in detail.
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