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Spin from defects in two-dimensional quantum field theory

Sebastian Novak, Ingo Runkel

TL;DR

The paper develops a general framework to construct 2D spin quantum field theories from 2D defect QFTs by encoding spin structures with triangulations and defect networks labeled by a $\Delta$-separable Frobenius algebra $A$ whose Nakayama automorphism satisfies $N^2=\text{id}$. It provides a precise correspondence between spin structures and defect data, proves independence from triangulation through algebraic identities, and identifies NS/R sectors via projectors in the defect bimodule category; within rational CFT, the construction becomes explicit via Cardy theories and defect bimodule categories, with detailed Ising and $\mathrm{so}(n)_1$ examples illustrating the graded (fermionic) extension. This unifies orbifold and spin constructions as generalized state-sum TFTs embedded in defect QFTs and yields concrete spin CFTs from rational defect data, offering a path to systematic classification and analysis of spin 2D QFTs and their supersymmetric variants. The approach highlights deep connections between spin geometry, defect categories, and RCFT modular data, enabling explicit computation of spin-state spaces and Dehn-twist actions in fermionic theories.

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.

Spin from defects in two-dimensional quantum field theory

TL;DR

The paper develops a general framework to construct 2D spin quantum field theories from 2D defect QFTs by encoding spin structures with triangulations and defect networks labeled by a -separable Frobenius algebra whose Nakayama automorphism satisfies . It provides a precise correspondence between spin structures and defect data, proves independence from triangulation through algebraic identities, and identifies NS/R sectors via projectors in the defect bimodule category; within rational CFT, the construction becomes explicit via Cardy theories and defect bimodule categories, with detailed Ising and examples illustrating the graded (fermionic) extension. This unifies orbifold and spin constructions as generalized state-sum TFTs embedded in defect QFTs and yields concrete spin CFTs from rational defect data, offering a path to systematic classification and analysis of spin 2D QFTs and their supersymmetric variants. The approach highlights deep connections between spin geometry, defect categories, and RCFT modular data, enabling explicit computation of spin-state spaces and Dehn-twist actions in fermionic theories.

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.

Paper Structure

This paper contains 14 sections, 11 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A combinatorial model for spin surfaces
  3. Two-dimensional QFT with defects
  4. Surfaces with defects, state spaces, amplitudes
  5. The trivial defect TFT and products of defect theories
  6. Amplitudes on spin surfaces from amplitudes on defect surfaces
  7. From spin structures to defect networks
  8. Algebraic description
  9. State spaces
  10. Application to two-dimensional rational conformal field theory
  11. Topological defects in rational conformal field theory
  12. Spin theories from defects in the graded case
  13. Example: Ising model
  14. Example: affine so(n) at level 1

Key Result

Proposition 4.1

Suppose $Q$ and $A,t,c_{\pm 1}$ satisfy conditions (1)--(5) above. Let $\Sigma$ be a spin surface and $\Sigma^d_1$, $\Sigma^d_2$ surfaces with defects obtained by steps (a)--(c) above. Then $Q(\Sigma^d_1)=Q(\Sigma^d_2)$.

Figures (3)

  • Figure 2.1: a) The six configurations a path can transverse a triangle relative to the marked edge of the triangle, together with the sign used to determine the lifting property of the path. b) The two configurations the oriented path (horizontal, solid blue) can cross an oriented edge (vertical, dashed black), again with corresponding signs.
  • Figure 4.1: a) Insertion of the topological junction field $c_{s(e)}$ on an oriented edge $e$ with edge sign $s(e)$. b) On a triangle $t$ is inserted such that the left-most $A$-line entering $t$ comes from the marked edge of the triangle. c) The defect network inserted at each boundary component. Here the boundary triangulation with edges $e_0$, $e_1$, $e_2$ has been moved into the surface to make the defect network easier to draw.
  • Figure 4.2: Graphical notation for duality maps and shorthand graphical notation for frequently used morphisms.

Theorems & Definitions (18)

  • Proposition 4.1
  • Theorem 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • Lemma 4.6
  • proof
  • Lemma 4.7
  • ...and 8 more