Machine Learning Topological Order from Defect Partition Functions

Abstract

We introduce a machine learning framework for extracting Ising topological order from defect partition functions of the two-dimensional Ising model on a torus. Restricted Boltzmann Machines (RBMs) are trained on Ising model data sampled at criticality across topological sectors. We take a component-wise square-root map of the learned distributions which naturally produces candidate wavefunctions for the (2+1)-dimensional Ising TQFT. As a nontrivial consistency check, we extract the modular S-matrix from overlaps of the resulting states and recover the expected Ising modular data. Our results demonstrate that neural network representations can capture both critical fluctuations and emergent topological structure, providing a data-driven route from lattice statistical mechanics to topological quantum field theory.