Non-perturbative double scaling limits

TL;DR

This paper introduces a non-perturbative approach to random surface and polymer models by generalizing matrix and vector integrals to gauge theories and sigma models, enabling definitions beyond Borel summability.

Contribution

It proposes a novel non-perturbative framework using gauge and sigma models to extend matrix and vector models for random surfaces and polymers.

Findings

Constructed (multi)critical metrics for O(N) sigma models

Provided non-perturbative definitions of partition functions

Demonstrated solutions to limitations of classic approaches

Abstract

Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path) integrals by gauge theory or non-linear sigma model (path) integrals. We explain how this solves one of the most fundamental limitation of the classic approach: we automatically obtain non-perturbative definitions in non-Borel summable cases. This is exemplified on the simplest possible examples involving O(N) symmetric non-linear sigma models with N-dimensional target spaces, for which we construct (multi)critical metrics. The non-perturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path) integrals.