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Branched polymers with loops coupled to the critical Ising model

Jan Ambjørn, Yukimura Izawa, Yuki Sato

TL;DR

This work addresses the non-perturbative continuum limit of branched polymers with loops coupled to the critical Ising model at zero temperature, showing that it maps to a renormalized Hermitian two-matrix model and can be described by a string field theory whose Dyson–Schwinger equation matches the loop equation of the continuum theory. A finite-N, non-perturbative formulation with N=1 leads to a convergent two-dimensional integral satisfying a third-order linear differential equation, in contrast to the Airy equation governing pure BP with loops; the disk and all-genus loop amplitudes are encoded in a Wheeler–DeWitt framework and connected to stochastic quantization. The results reveal how critical Ising fluctuations alter the WD structure and produce a γ-dependent generalization of CDT, while preserving the same string susceptibility γ_str = 1/2 as in pure BPs. The combination of matrix-model techniques, a field-theoretic string description, and stochastic quantization provides a robust toolkit for exploring quantum gravity aspects of BP-like geometries with spin couplings and their non-perturbative regimes.

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.

Branched polymers with loops coupled to the critical Ising model

TL;DR

This work addresses the non-perturbative continuum limit of branched polymers with loops coupled to the critical Ising model at zero temperature, showing that it maps to a renormalized Hermitian two-matrix model and can be described by a string field theory whose Dyson–Schwinger equation matches the loop equation of the continuum theory. A finite-N, non-perturbative formulation with N=1 leads to a convergent two-dimensional integral satisfying a third-order linear differential equation, in contrast to the Airy equation governing pure BP with loops; the disk and all-genus loop amplitudes are encoded in a Wheeler–DeWitt framework and connected to stochastic quantization. The results reveal how critical Ising fluctuations alter the WD structure and produce a γ-dependent generalization of CDT, while preserving the same string susceptibility γ_str = 1/2 as in pure BPs. The combination of matrix-model techniques, a field-theoretic string description, and stochastic quantization provides a robust toolkit for exploring quantum gravity aspects of BP-like geometries with spin couplings and their non-perturbative regimes.

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.
Paper Structure (17 sections, 141 equations, 3 figures)

This paper contains 17 sections, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Two-matrix model
  3. Critical Ising model
  4. Absence of Ising coupling
  5. Saddle-point equation in the large-$N$ limit
  6. Loop equation
  7. String field theory
  8. Sphere case
  9. General amplitudes
  10. Non-perturbative formulation
  11. Integration domains
  12. The third-order differential equation
  13. Free energy
  14. Non-perturbative effects
  15. Loop functions
  16. ...and 2 more sections

Figures (3)

  • Figure 1: A ribbon graph consisting of vertices of degree one and three where the red and blue arrows corresponding to the up and down spins are placed at vertices.
  • Figure 2: Skeleton vertices, shown in white, and black vertices belonging to the local tree structure.
  • Figure 3: Domains of convergence in the complex $y$ plane: The branch cut is placed on the positive real axis.