Spin-cobordisms, surgeries and fermionic modular bootstrap

TL;DR

This paper explores the relationship between fermionic quantum field theories with finite group symmetries, their anomalies, and modular properties in 2D, providing new methods to compute and classify these anomalies via spin-cobordisms and surgery techniques.

Contribution

It introduces a novel approach to determine modular transformations from bulk 3d invertible TQFTs and provides explicit combinatorial formulas for spin-bordism invariants for abelian groups.

Findings

Explicit formulas for spin-bordism invariants for abelian groups

Method to derive modular transformations from anomaly inflow

Constraints on IR spectra from 't Hooft anomalies

Abstract

We consider general fermionic quantum field theories with a global finite group symmetry $G$, focusing on the case of 2-dimensions and torus spacetime. The modular transformation properties of the family of partition functions with different backgrounds is determined by the 't Hooft anomaly of $G$ and fermion parity. For a general possibly non-abelian $G$ we provide a method to determine the modular transformations directly from the bulk 3d invertible topological quantum field theory (iTQFT) corresponding to the anomaly by inflow. We also describe a method of evaluating the character map from the real representation ring of $G$ to the group which classifies anomalies. Physically the value of the map is given by the anomaly of free fermions in a given representation. We assume classification of the anomalies/iTQFTs by spin-cobordisms. As a byproduct, for all abelian symmetry groups $G$, we provide explicit combinatorial expressions for corresponding spin-bordism invariants in terms of surgery representation of arbitrary closed spin 3-manifolds. We work out the case of $G=\mathbb{Z}_2$ in detail, and, as an application, we consider the constraints that 't Hooft anomaly puts on the spectrum of the infrared conformal field theory.