Topological field theories on manifolds with Wu structures
Topological field theories on manifolds with Wu structures
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Wu 结构流形的拓扑场论
DOI：
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发表时间：
2016
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通讯作者：
Samuel Monnier
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文献类型：
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作者：
Samuel Monnier
我们构造了可逆场论，将阿贝尔前量子自旋chen - simons理论推广到具有2k+2次Wu结构的4k+3维流形。在分析了某个离散对称的异常之后，我们测量了它，产生了路径积分减少到有限和的拓扑场理论，类似于Dijkgraaf-Witten理论。我们采用一般的观点，将chen - simons规范群及其耦合编码在一个局部的积分格系统中。这些理论的拉格朗日量必须被解释为广义上同调理论中的一类，才能得到规范不变作用。我们为这种广义上同调建立了一个计算友好的协链模型，并将其用于详细研究Wu chen - simons作用的性质。在三维自旋情况下，后者提供了最近在费米子对称保护拓扑相的文献中引入的“费米子修正”的定义。为了构造量规理论的状态空间，我们建立了具有斜对称对的有限阿贝尔群的几何量子化模拟。这项工作的物理动机来自于这样一个事实，即在k = 1的情况下，这里构建的测量7维拓扑场论本质上是具有（2,0）超对称性的6维共形场论的异常场论，这将在其他地方讨论。
We construct invertible field theories generalizing abelian prequantum spin Chern-Simons theory to manifolds of dimension 4k+3 endowed with a Wu structure of degree 2k+2. After analysing the anomalies of a certain discrete symmetry, we gauge it, producing topological field theories whose path integral reduces to a finite sum, akin to Dijkgraaf-Witten theories. We take a general point of view where the Chern-Simons gauge group and its couplings are encoded in a local system of integral lattices. The Lagrangian of these theories has to be interpreted as a class in a generalized cohomology theory in order to obtain a gauge invariant action. We develop a computationally friendly cochain model for this generalized cohomology and use it in a detailed study of the properties of the Wu Chern-Simons action. In the three-dimensional spin case, the latter provides a definition of the "fermionic correction" introduced recently in the literature on fermionic symmetry protected topological phases. In order to construct the state space of the gauged theories, we develop an analogue of geometric quantization for finite abelian groups endowed with a skew-symmetric pairing. The physical motivation for this work comes from the fact that in the k = 1 case, the gauged 7-dimensional topological field theories constructed here are essentially the anomaly field theories of the 6-dimensional conformal field theories with (2,0) supersymmetry, as will be discussed elsewhere.