[Paper Review] Extended TQFT, Gauge Theory, AND 2-Linearization

Jeffrey C. Morton|arXiv (Cornell University)|Mar 30, 2010

Homotopy and Cohomology in Algebraic Topology50 references8 citations

TL;DR

This paper extends the 2-linearization functor to incorporate cohomological twisting, enabling a categorical construction of the full Dijkgraaf-Witten topological gauge theory with a 3-cocycle. By generalizing spans of groupoids to include cocycle data, it realizes the extended TQFT as a functor from a symmetric monoidal bicategory of twisted groupoids to 2-vector spaces, thereby categorically deriving the DW model with gauge group G and classifying cocycle ω.

ABSTRACT

In this paper, we describe a relation between a categorical construction, called 2-linearization, and extended topological quantum field (ETQFT). We then describe an extension of the 2-linearization process which incorporates cohomological twisting. 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 spaces for topological gauge with finite group G, the ETQFT obtained is the untwisted Dijkgraaf-Witten (DW) model associated to G. This illustrates the factorization of TQFT into 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 2-linearization functor which acts on groupoids and spans which have associated cohomological data. We show how the 3-cocycle {\omega} on the classifying space BG which appears in the action for the DW model induces a classical field valued in this bicategory.

Motivation & Objective

• To extend the 2-linearization process to include cohomological twisting, enabling a categorical description of the full Dijkgraaf-Witten model.
• To generalize the symmetric monoidal bicategory of groupoids and spans to incorporate groupoid 2-cocycles.
• To show how the 3-cocycle ω on BG naturally arises in the classical field theory component of the extended TQFT.
• To establish a factorization of ETQFT into field theory valued in twisted groupoids and a quantization functor via 2-linearization.
• To provide a categorical framework that unifies the field-theoretic and quantization components of topological gauge theories.

Proposed method

• Generalize the symmetric monoidal bicategory of groupoids and spans to include cohomological data via 2-cocycles on groupoids.
• Define a twisted 2-linearization functor that assigns 2-vector spaces to twisted groupoids, functors to twisted spans, and natural transformations to morphisms of twisted spans.
• Construct a classical field theory valued in the extended bicategory of twisted groupoids and spans, parameterized by a 3-cocycle ω on BG.
• Use the classifying space BG of a finite group G to interpret the 3-cocycle ω as a characteristic class in H^3(BG, U(1)).
• Apply the twisted 2-linearization functor to the moduli groupoid of flat G-bundles (discrete), yielding the full DW TQFT as a 3-functor into 2-vector spaces.
• Demonstrate that the resulting TQFT recovers the Dijkgraaf-Witten invariant with action functional weighted by ω.

Experimental results

Research questions

• RQ1 How can the 2-linearization process be generalized to incorporate cohomological twisting in the context of topological quantum field theories?
• RQ2 What is the categorical structure of twisted groupoids and spans that supports a symmetric monoidal bicategory for gauge theories?
• RQ3 How does the 3-cocycle ω on BG manifest in the classical field theory component of the extended TQFT?
• RQ4 Can the full Dijkgraaf-Wilden model be reconstructed via a factorization of ETQFT into field theory and quantization functors using twisted structures?
• RQ5 What is the role of the classifying space BG in encoding the gauge theory action via cohomological data?

Key findings

• The 2-linearization functor is successfully extended to include cohomological twisting via 2-cocycles on groupoids, forming a generalized symmetric monoidal bicategory.
• The twisted 2-linearization assigns 2-vector spaces to twisted groupoids, functors to twisted spans, and natural transformations to morphisms of twisted spans.
• The classical field theory component of the ETQFT is realized as a value in the bicategory of twisted groupoids and spans, with the 3-cocycle ω on BG naturally encoded in the structure.
• The resulting TQFT from the twisted 2-linearization reproduces the full Dijkgraaf-Witten model with gauge group G and action weighted by ω.
• The construction provides a categorical realization of the factorization of ETQFT into field theory and quantization, as proposed by Freed, Hopkins, Lurie, and Teleman.
• The 3-cocycle ω on BG is shown to induce a classical field in the extended bicategory, directly linking the action functional to higher categorical data.