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={\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.

Note:

86 pages, 27 figures. v2: minor corrections

bootstrap: modular
anomaly
field theory: topological
transformation: modular
fermion: field theory
spin
kink
crossing
space-time: torus
partition function

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