We explain how to perform topological twisting of supersymmetric field theories in the language of factorization algebras. Namely, given a supersymmetric factorization algebra with a choice of a topological supercharge we construct an algebra over the operad of little disks. We also explain the role of the twisting homomorphism allowing us to construct an algebra over the operad of framed little disks. Finally, we give a complete classification of topological supercharges and twisting homomorphisms in dimensions 1 through 10.
Section 1 Introduction• We can construct E n -algebras from topologically twisted factorization algebras subject to a single condition: that the map Obs(B r (0)) → Obs(B R (0)) on concentric balls of radii R > r is an equivalence.• This condition is automatically satisfied in many natural examples, including those coming from superconformal field theories.• In many of these natural examples these E n -algebras are additionally compatible with the action of SO(n). This means the topological field theories can be defined on any oriented n-manifold.Remark 1.1. We do not know whether the twisted factorization algebra Obs Q is locally-constant, so our arguments will not use Lurie's result comparing locally-constant factorization algebras and E n -algebras. Instead we will equip the local observables of the topologically twisted factorization algebra directly with an action of a specific model for the E n -operad.In addition to understanding twisted factorization algebras abstractly we will give a complete classification of the possible twists coming from super Poincaré algebra actions in dimensions 1 to 10.
Factorization Algebras and E n -AlgebrasThroughout the paper we work over the ground field C of complex numbers. Thus, all complexes, Lie algebras and so on are over C. The chain complexes of observables will be differentiable chain complexes, see [CG17, Appendices B, C].