  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
Omega-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.
