Branched polymers with loops coupled to the critical Ising model

TL;DR

This paper explores the continuum limit of branched polymers with loops coupled to the critical Ising model, developing a string field theory and analyzing non-perturbative partition functions satisfying differential equations.

Contribution

It introduces a string field theory for branched polymers with loops coupled to the Ising model and derives a non-perturbative partition function satisfying a third-order differential equation.

Findings

The continuum partition function satisfies a third-order linear differential equation.

The non-perturbative loop amplitude solves the Wheeler-DeWitt equation with all genera contributions.

The Wheeler-DeWitt equation can be derived via stochastic quantization.

Abstract

We study the continuum limit of branched polymers (BPs) with loops coupled to Ising spins at the zero-temperature critical point. It is known that the continuum partition function can be represented by a Hermitian two-matrix model, and we propose a string field theory whose Dyson-Schwinger equation coincides with the loop equation of this continuum matrix model. By setting the matrix size to one, we analyze a convergent non-perturbative partition function expressed as a two-dimensional integral, and show that it satisfies a third-order linear differential equation. In contrast, in the absence of coupling to the critical Ising model, the continuum partition function of pure BPs with loops is known to satisfy the Airy equation. From the viewpoint of two-dimensional quantum gravity, we introduce a non-perturbative loop amplitude that serves as a solution to the Wheeler-DeWitt equation incorporating contributions from all genera. Furthermore, we demonstrate that the same Wheeler-DeWitt equation can also be derived through the stochastic quantization.