Secondary Products in Supersymmetric Field Theory

The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type, these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect to the primary product. We describe the mathematical structures involved and illustrate this general phenomenon in a range of physical examples arising from supersymmetric field theories in spacetime dimension two, three, and four. In the Rozansky-Witten twist of three-dimensional N = 4 theories, this gives an intrinsic realization of the holomorphic symplectic structure of the moduli space of vacua. We further give a simple mathematical derivation of the assertion that introducing an O-background precisely deformation quantizes this structure. We then study the secondary product structure of extended operators, which subsumes that of local operators but is often much richer. We calculate interesting cases of secondary brackets of line operators in Rozansky-Witten theories and in four-dimensional N = 4 super-Yang-Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.