Non-Semisimple Extended Topological Quantum Field Theories

Research paper by Marco De Renzi

Indexed on: 22 Mar '17Published on: 22 Mar '17Published in: arXiv - Mathematics - Geometric Topology


In this paper we provide the general theory for the construction of 3-dimensional ETQFTs extending the Costantino-Geer-Patureau quantum invariants defined in arXiv:1202.3553. Our results rely on relative modular categories, a class of non-semisimple ribbon categories modeled on representations of unrolled quantum groups, and they exploit a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum and Vogel. The 1+1+1 TQFTs thus obtained are realized by symmetric monoidal functors defined over 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes and taking values in 2-categories of complete graded linear categories. In particular our construction extends the family of graded TQFTs defined for unrolled quantum $\mathfrak{sl}_2$ by Blanchet, Costantino, Geer and Patureau in arXiv:1404.7289 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-dimensional manifolds.