Summing up perturbation series around superintegrable point

Abstract

We work out explicit formulas for correlators in the Gaussian matrix model perturbed by a logarithmic potential, i.e. by inserting Miwa variables. In this paper, we concentrate on the example of a single Miwa variable. The ordinary Gaussian model is superintegrable, i.e. the average of the Schur functions SQ is an explicit function of the Young diagram Q. The question is what happens to this property after perturbation. We show that the entire perturbation series can be nicely summed up into a kind of Borel transform of a universal exponential function, while the dependence on R enters through a polynomial factor in front of this exponential. Moreover, these polynomials can be described explicitly through a single additional structure, which we call “truncation” of the Young diagram Q. It is unclear if one can call this an extended superintegrability, but at least it is a tremendously simple deformation of it. Moreover, the vanishing Gaussian correlators remain vanishing and, hence, are not deformed at all.