Non-Semisimple Extended Topological Quantum Field Theories
Non-Semisimple Extended Topological Quantum Field Theories
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非半简单扩展拓扑量子场论
DOI：
10.1090/memo/1364
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
M. Renzi
中科院分区：
文献类型：
--
作者：
M. Renzi
我们发展了与闭三维流形的Costantino-Geer-Patureau量子不变量相关的扩展拓扑量子场论（ETQFT）的一般理论。为了做到这一点，我们引入了相对模范畴，一类带状范畴，它是以展开量子群的表示为模型的，可以被认为是模范畴的非半单模拟。我们的方法利用了由Blanchet，Habegger，Masbaum和Vogel引入的通用构造的2-分类版本。由此得到的1+1+1-EQFT是由定义在非刚性2-范畴上的对称monoidal 2-函子实现的，该2-函子是用有色带状图和上同调类装饰的，并且取值于2-范畴的完全分次线性范畴.特别是，我们的建设扩展了家庭的分级2+1-TQFT定义的展开版本的量子$\mathfrak{sl}_2$由Blanchet，Costantino，Geer，和Patureau到一个新的家庭的分级ETQFT。该理论的非半单性是由与临界1-流形相关的非半单分次线性范畴的存在所证明的。
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by symmetric monoidal 2-functors which are defined over non-rigid 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes, and which take values in 2-categories of complete graded linear categories. In particular, our construction extends the family of graded 2+1-TQFTs defined for the unrolled version of quantum $\mathfrak{sl}_2$ by Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The non-semisimplicity of the theory is witnessed by the presence of non-semisimple graded linear categories associated with critical 1-manifolds.