标题： 具有吴结构的流形上的拓扑场论 显示英文标题

标题： Topological field theories on manifolds with Wu structures

摘要： 我们构建了可逆场论，推广了阿贝尔预量子自旋Chern-Simons理论到具有度数2k+2的Wu结构的4k+3维流形上。在分析了一定离散对称性的反常性之后，我们规范化了它，产生了路径积分约化为有限和的拓扑场论，类似于Dijkgraaf-Witten理论。我们采取了一种一般观点，其中Chern-Simons规范群及其耦合被编码在一个局部积分格系中。这些理论的Lagrangian必须被解释为广义上同调理论中的一个类，以便得到规范不变的作用。我们发展了一个计算友好的上链模型来描述这个广义上同调，并用它详细研究了Wu Chern-Simons作用的性质。在三维自旋情形下，后者为最近文献中关于费米子对称保护拓扑相的研究引入的“费米子修正”提供了一个定义。为了构造规范化的理论状态空间，我们发展了有限阿贝尔群上带有斜对称配对的几何量化的一个类似物。这项工作的物理动机来源于k=1的情形，在此情况下，这里构造的7维拓扑场论本质上是6维具有(2,0)超对称的共形场论的反常场论，这一点将在别处讨论。 显示英文摘要

摘要： 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.