Spin-cobordism and fermionic d=2 anomalies

The aim of this work is to improve our description of global anomalies and the tools we have at our disposal for their computation. In particular, we focus on general fermionic quantum field theories with a global finite group symmetry G^f in 2-dimensions, with a special regard for the torus spacetime. The modular transformation properties of the family of partition functions with different backgrounds is determined by the ’t Hooft anomaly of G^f. For a general G^f, possibly non-abelian or twisted, 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^f 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, we provide explicit combinatorial expressions for corresponding spin-bordism invariants of abelian symmetry groups G^f in terms of surgery representation of arbitrary closed spin 3-manifolds. As an application, we compute the constraints that ’t Hooft anomaly puts on the spectrum of infrared conformal field theories for various symmetry groups. In particular, we provide a first of such analysis for discrete non-abelians G^f or with a non-trivial twist of the Z_2^f subgroup.