Abstract
One of the main features of eigenvalue matrix models is that the averages of characters are again characters, what can be considered as a far-going generalization of the Fourier transform property of Gaussian exponential. This is true for the standard Hermitian and unitary (trigonometric) matrix models and for their various deformations, classical and quantum ones. Arising explicit formulas for the partition functions are very efficient for practical computer calculations. However, to handle them theoretically, one needs to tame remaining finite sums over representations of a given size, which turns into an interesting conceptual problem. Already the semicircle distribution in the large-N limit implies interesting combinatorial sum rules for characters. We describe also implications to W-representations, including a character decomposition of cut-and-join operators, which unexpectedly involves only single-hook diagrams and also requires non-trivial summation identities.
Highlights
Which are dimensions of the symmetric and antisymmetric representations [2] and [1,1]
One of the main features of eigenvalue matrix models is that the averages of characters are again characters, what can be considered as a far-going generalization of the Fourier transform property of Gaussian exponential
Looks like a statement that integration over M is reduced to the substitution of the “mean field” M = Id
Summary
Besides (1.2), there are other explicit generating functions like the Harer-Zagier formula [25,26,27]. Substituting (2.2) into (2.1), we obtain a non-trivial sum rule for characters: 2|R|z|R|+2 (|R| − 1)!! (m + 1)(35m2 − 77m + 12) , The simplest here is the p21m term: it comes from the contribution χR{δk,1}p|1R| to χR{p}, and the coefficient cancels the denominator (1.2) so that the remaining sum is calculated with the help of the Cauchy formula, χR(N )χR{δk,2}p|1R| = eNp21/2 =. Already at the Hermitian matrix model level, it has important applications, the currently fashionable ones being related to localization formulas [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61] in conformally invariant supersymmetric field theories, which reduce perturbative contributions to certain correlators to those in the Gaussian matrix model averages [62,63,64]
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