Abstract
We derive the analogues of the Harer-Zagier formulas for single- and double-trace correlators in the q-deformed Hermitian Gaussian matrix model. This fully describes single-trace correlators and opens a road to q-deformations of important matrix models properties, such as genus expansion and Wick theorem.
Highlights
Matrix models [1,2,3,4,5] are ubiquitous in mathematical and theoretical physics 1
Reduction of a problem in matrix model terms often leads to significant progress, be it in the domain of SUSY gauge theories [9,10,11,12,13,14,15,16,17,18,19,20,21,22], enumerative geometry [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], the theory of symmetric functions [39,40,41,42,43,44,45] or even the quantum computation [46, 47]
From the QFT point of view this expansion is nothing but WKB expansion, and in the simplest case of Hermitian Gaussian matrix model (HGMM, see Section 2 for a definition) it literally takes the form of the summation over fat graphs, each living on a particular Riemann surface
Summary
Matrix models [1,2,3,4,5] are ubiquitous in mathematical and theoretical physics 1. There are many ways to derive this formula in case of HGMM, for instance, using the Wick theorem, which in this case takes the form of gluing ribbons (MM propagator) to the discs with marked boundary points (for the details see, for instance, [26, 48]) This simplicity, is lost in the case of q-deformed Hermitian Gaussian matrix model (qHGMM, see Section 2), where the corresponding average is equal to. We interpret this as follows: the q-generalized Harer-Zagier provides a simple way to compute (1-point) correlators in qHGMM, by means of a partial fractions expansion This is the main practical result of the present paper, see Section 3, Equations (3-7),(3-8).
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