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 {mathcal {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 {mathcal {N}}=4 super-Yang–Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.
Highlights
Mathematical perspectives on topological quantum field theory (TQFT) have evolved significantly since their initial axiomatization by Atiyah [1], inspired by Segal’s approach to conformal field theory (CFT) [2]
It takes into account homological structures that express the local constancy of the theory over spaces of bordisms of manifolds, thereby capturing aspects of the topology of these spaces. Such structures are ubiquitous in “Witten-type” or “cohomological” TQFTs obtained via topological twist from supersymmetric QFTs [3,4], and in the setting of two-dimensional topological conformal field theories (TCFTs) [5,6,7]
In the context of the BRST cohomology of topological conformal field theories, this structure was constructed by Lian and Zuckerman in the early 1990s [43]. (See [44].) The relevance of operads in topological field theory was discovered by Kontsevich, and Getzler proved that the operad controlling the structure of operators in oriented two-dimensional TQFT is identified with that of BV algebras [5]
Summary
Mathematical perspectives on topological quantum field theory (TQFT) have evolved significantly since their initial axiomatization by Atiyah [1], inspired by Segal’s approach to conformal field theory (CFT) [2]. It takes into account homological structures that express the local constancy of the theory over spaces of bordisms of manifolds, thereby capturing aspects of the topology of these spaces. One may consider “extended” TQFTs, which express the locality of field theories in the language of higher categories of manifolds with corners and capture additional physical entities such as extended operators and defects. For cohomology classes of local operators, the most important secondary operation turns out to be a Lie bracket of degree 1−d, which acts as a derivation with respect to the primary product. This secondary product has a surprisingly simple and concrete physical definition in terms of topological descent. This is again a Q-closed local operator and represents a well-defined cohomology class in the topological operator algebra
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