Spin from defects in two-dimensional quantum field theory

TL;DR

This paper constructs two-dimensional quantum field theories on spin surfaces using defect networks and triangulations, providing a combinatorial approach that ensures independence from choices and applies to rational conformal field theories.

Contribution

It introduces a novel combinatorial method to define spin surface amplitudes via defect networks derived from Frobenius algebras, extending the framework to rational CFTs.

Findings

Explicit description of defect categories for rational CFTs

Worked out examples for Ising model and so(n) WZW model at level 1

Demonstrated independence from triangulation choices

Abstract

We build two-dimensional quantum field theories on spin surfaces starting from theories on oriented surfaces with networks of topological defect lines and junctions. The construction uses a combinatorial description of the spin structure in terms of a triangulation equipped with extra data. The amplitude for the spin surfaces is defined to be the amplitude for the underlying oriented surface together with a defect network dual to the triangulation. Independence of the triangulation and of the other choices follows if the line defect and junctions are obtained from a Delta-separable Frobenius algebra with involutive Nakayama automorphism in the monoidal category of topological defects. For rational conformal field theory we can give a more explicit description of the defect category, and we work out two examples related to free fermions in detail: the Ising model and the so(n) WZW model at level 1.