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.

中文翻译：

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超对称场论中的次级产品

拓扑量子场论中局部算子的乘积大于1是可交换的，更普遍的是，余维的扩展算子的乘积大于1。在同调类型的理论中，这些可交换产品伴随着次级操作，这些操作捕获了操作员的链接或编织，并且相对于初级产品表现为（分级的）泊松括号。我们描述了所涉及的数学结构，并在由时空第二，第三和第四维的超对称场理论引起的一系列物理示例中说明了这种普遍现象。在Rozansky–Witten三维\（{\ mathcal {N}} = 4 \）中的扭曲理论上，这给出了真空模空间的全纯辛结构的内在实现。我们进一步给出了一个简单的数学推导，即引入\（\ Omega \）-背景精确变形可以量化此结构。然后，我们研究扩展运营商的二级产品结构，该产品结构包含本地运营商的二级产品结构，但通常要丰富得多。我们在Rozansky-Witten理论和四维\（{\ mathcal {N}} = 4 \）超Yang-Mills理论中计算线运算符的副括号的有趣情况，从而测量几何中球体类别的非对易性Langlands程序。