Frobenius代数上的三层CFTs
Three-tier CFTs from Frobenius algebras
摘要：
这些是2012年5月29日至6月2日在印第安纳州圣母大学数学中心举行的拓扑学和场论暑期学校的课程讲义。将量子场论推广到低维流形的想法最早是由丹·弗里德在90年代提出的。在共形场论（CFT）的情况下，我们讨论的是Atiyah-Segal公理的一个扩展，其中用一个合适的边形双范畴代替黎曼曲面的边形范畴，其对象是点，其态射是1-流形，其2-态射是黎曼曲面的片。由于Fuchs，Runkel和Schweigert，有一个完整的（理性的）CFTs的美丽分类，大致如下。固定一个手征代数A（=顶点代数）。然后用Rep（A）内部的Frobenius代数对左、右手征代数与A一致的满集进行分类。一个著名的例子，其中一个可以成功地应用这是情况下，手征代数A是仿射su（2）在水平k，为某些k在N。在这种情况下，Rep（A）中的Frobenius代数被分类为A_n，D_n，E_6，E_7，E_8，相应的CFTs也是如此。最近，Kapustin和Saulina给出了一个概念上的FRS分类的三维Chern-Simons理论的缺陷。这些缺陷也由Rep（A）中的Frobenius代数对象给出。受Kapustin和Saulina建议的启发，我们将（部分）构建与Frobenius代数对象相关联的三层CFT。
英文摘要：
These are lecture notes of a course given at the Summer School on Topology and Field Theories held at the Centre for Mathematics of the University of Notre Dame, Indiana, from May 29 to June 2, 2012. The idea of extending quantum field theories to manifolds of lower dimension was first proposed by Dan Freed in the nineties. In the case of conformal field theory (CFT), we are talking of an extension of the Atiyah-Segal axioms, where one replaces the bordism category of Riemann surfaces by a suitable bordism bicategory, whose ob jects are points, whose morphisms are 1-manifolds, and whose 2-morphisms are pieces of Riemann surface. There is a beautiful classification of full (rational) CFTs due to Fuchs, Runkel and Schweigert, which roughly says the following. Fix a chiral algebra A (= vertex algebra). Then the set of full cfts whose left and right chiral algebras agree with A is classified by Frobenius algebras internal to Rep(A). A famous example to which one can successfully apply this is the case where the chiral algebra A is affine su(2) at level k, for some k in N. In that case, the Frobenius algebras in Rep (A) are classified by A_n, D_n, E_6, E_7, E_8, and so are the corresponding CFTs. Recently, Kapustin and Saulina gave a conceptual interpretation of the FRS classification in terms of 3-dimensional Chern-Simons theory with defects. Those defects are also given by Frobenius algebra ob ject in Rep(A). Inspired by the proposal of Kapustin and Saulina, we will (partially) construct the three-tier CFT associated to a Frobenius algebra object.
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