Secondary Products in Supersymmetric Field Theory
Secondary Products in Supersymmetric Field Theory
复制标题
超对称场论中的二次积
DOI：
10.1007/s00023-020-00888-3
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Neitzke, Andrew
中科院分区：
文献类型：
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作者：
Beem, Christopher;Ben-Zvi, David;Bullimore, Mathew;Dimofte, Tudor;Neitzke, Andrew
在拓扑量子场论中，局部算符在大于1维的情况下的乘积是可交换的，更一般地，余维大于1的扩展算符的乘积也是可交换的。在上同调类型的理论中，这些交换乘积伴随着次级运算，次级运算捕获算子的链接或编织，并表现为相对于初级乘积的（分次）泊松括号。我们描述了所涉及的数学结构，并说明了这种普遍现象所产生的超对称场理论在时空二维，三维和四维的物理例子范围。在三维理论的Rozansky-Witten扭曲中，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的断言，引入一个背景精确变形量化这种结构。然后，我们研究了扩展运营商的二次产品结构，其中包含的本地运营商，但往往是更丰富。我们计算了Rozansky-Witten理论和四维超杨-米尔斯理论中线算子的次括号的有趣情况，测量了几何Langlands程序中球形范畴的非交换性。
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-dimensionaltheories, 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-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-dimensionalsuper-Yang–Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.