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Durham University

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Publication details for Mathew Bullimore

Beem, C., Ben-Zvi, D., Bullimore, M., Dimofte, T. & Neitzke, A. (2020). Secondary products in supersymmetric field theory. Annales Henri Poincaré 21(4): 1235-1310.

Author(s) from Durham

Abstract

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 Ω-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.