A Taxonomy of Twists of Supersymmetric Yang--Mills Theory

Abstract: We give a complete classification of twists of supersymmetric Yang--Mills theories in dimensions $2\leq n \leq 10$. We formulate supersymmetric Yang--Mills theory classically using the BV formalism, and then we construct an action of the supersymmetry algebra using the language of $L_\infty$ algebras. For each orbit in the space of square-zero supercharges in the supersymmetry algebra, under the action of the spin group and the group of R-symmetries, we give a description of the corresponding twisted theory. These twists can be described in terms of mixed holomorphic-topological versions of Chern--Simons and BF theory.