Mathematics – Quantum Algebra
Scientific paper
2010-03-29
Mathematics
Quantum Algebra
54 pages, 3 figures; revision 3 includes major new theorems, new last section, covers twisted Dijkgraaf-Witten model Submitted
Scientific paper
In this paper, we describe a relation between a categorical quantization construction, called "2-linearization", and extended topological quantum field theory (ETQFT). This is a weak 2-functor Z_G : nCob2 \rightarrow 2Vect valued in (Kapranov-Voevodsky) 2-vector spaces. The 2-linearization process assigns 2-vector 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 stacks for topological gauge theory with finite group G, the ETQFT obtained is the (untwisted) Dijkgraaf-Witten model associated to G, extended to manifolds with boundary as described by Freed. This construction is related to 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 give some explicit examples and calculations of invariants. We then describe how to extend the 2-linearization process to accommodate the full DW model, including twisting by a 3-cocycle {\omega} on the classifying space BG, by using a generalization of the symmetric monoidal 2-category of groupoids and spans.
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