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Secondary products in supersymmetric field theory.

Beem, C. and Ben-Zvi, D. and Bullimore, M. and Dimofte, T. and Neitzke, A. (2020) 'Secondary products in supersymmetric field theory.', Annales Henri Poincaré., 21 (4). pp. 1235-1310.

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.

Item Type:Article
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s00023-020-00888-3
Publisher statement:This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Date accepted:16 January 2020
Date deposited:24 January 2020
Date of first online publication:04 February 2020
Date first made open access:06 February 2020

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