Abstract
Many eigenvalue matrix models possess a peculiar basis of observables that have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property langle characterrangle ,sim character. This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated Q Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.
Highlights
In the original definition, matrix models are defined as averages over matrix ensembles, often described by integrals over matrices or eigenvalues
We extended the superintegrability relation character ∼ character to the second kind of matrix models, depending on background fields
This provides a very nice generalization for the complex matrix model, where dimensions of matrices are lifted to more general traces of background matrices
Summary
Matrix models are defined as averages over matrix ensembles, often described by integrals over matrices or eigenvalues. Simple in terms of the box coordinates is the expression on the topological locus: where X is an N × N matrix, and the generating function of all correlators is exp − 1 Tr X 2 + pk Tr X k d X k k χ { pk }χR{N } χR{δk,2}. Further deformation of the measure to the knot eigenvalue model gives basically the same expression for the correlators [22,23]: Page 3 of 11 270. Important feature of formulas (22)–(24) is that they depend on the matrix size only implicitly, through the traces of powers of background fields This is the crucial feature, which allows one to forget about the matrix-integral origin/realization of these models, in particular to treat them in terms of τ -functions of integrable hierarchies [32,33]
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