Topological Twisting of 4d $\mathcal{N}=2$ Supersymmetric Field Theories

TL;DR

This paper explores the topological data necessary for defining partition functions of 4d $ abla$=2 supersymmetric theories after topological twisting, highlighting dependencies beyond the spacetime's diffeomorphism type, including gerbe connections and a new generalized spin-c structure.

Contribution

It introduces a comprehensive framework for topological twisting of 4d $ abla$=2 theories, including the novel concept of a generalized spin-c structure and its implications for class $ ext{S}$ theories.

Findings

Partition functions depend on spacetime topology, gerbe characteristic classes, and a generalized spin-c structure.

Different S-duality orbits can have distinct topological data.

The framework applies to both Lagrangian and class $ ext{S}$ theories.

Abstract

We discuss what topological data must be provided to define topologically twisted partition functions of four-dimensional $\mathcal{N}=2$ supersymmetric field theories. The original example of Donaldson-Witten theory depends only on the diffeomorphism type of the spacetime and 't Hooft fluxes (characteristic classes of background gerbe connections, a.k.a. "one-form symmetry connections.") The example of $\mathcal{N}=2^*$ theories shows that, in general, the twisted partition functions depend on further topological data. We describe topological twisting for general four-dimensional $\mathcal{N}=2$ theories and argue that the topological partition functions depend on (a): the diffeomorphism type of the spacetime, (b): the characteristic classes of background gerbe connections and (c): a "generalized spin-c structure," a concept we introduce and define. The main ideas are illustrated with both Lagrangian theories and class $\mathcal{S}$ theories. In the case of class $\mathcal{S}$ theories of $A_1$ type, we note that the different $S$-duality orbits of a theory associated with a fixed UV curve $C_{g,n}$ can have different topological data.