Spin-Cobordisms, Surgeries and Fermionic Modular Bootstrap

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 are 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 = 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.

Automated summary

We study 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 partition functions are determined by the 't Hooft anomaly of G and fermion parity. For non-abelian G, the modular transformations can be directly determined from the corresponding 3D invertible topological quantum field theory (iTQFT) through inflow. We also explore the evaluation of the character map from the real representation ring of G to the group classifying anomalies.