Cohomological Twisting of 2Linearization and Extended TQFT
Abstract
In this paper, we describe a relation between a categorical quantization construction, called "2linearization", and extended topological quantum field theory (ETQFT). We then describe an extension of the 2linearization process which incorporates cohomological twisting. The 2linearization process assigns 2vector spaces to (finite) groupoids, functors between them to spans of groupoids, and natural transformations to spans between these. By applying this to groupoids which represent the (discrete) moduli spaces for topological gauge theory with finite group G, the ETQFT obtained is the untwisted DijkgraafWitten (DW) model associated to G. This illustrates the factorization of TQFT into "classical field theory" valued in groupoids, and "quantization functors", which has been described by Freed, Hopkins, Lurie and Teleman. We then describe how to extend this to the full DW model, by using a generalization of the symmetric monoidal bicategory of groupoids and spans which incorporates cocycles. We give a generalization of the 2linearization functor which acts on groupoids and spans which have associated cohomological data. We show how the 3cocycle {\omega} on the classifying space BG which appears in the action for the DW model induces a classical field theory valued in this bicategory.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 DOI:
 10.48550/arXiv.1003.5603
 arXiv:
 arXiv:1003.5603
 Bibcode:
 2010arXiv1003.5603M
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Category Theory;
 18E10;
 20L05;
 57M27;
 57R56
 EPrint:
 54 pages, 3 figures