4-Dimensional BF Theory as a Topological Quantum Field Theory
John C. Baez
TL;DR
The paper addresses the limited understanding of 4D TQFTs by formulating 4D BF theory with a nonzero cosmological term and showing it defines a TQFT-like functor $Z_{BF}$ on cobordisms equipped with trivializable G-bundles. Specializing to the frame bundle case with $G = GL(4,\mathbb{R})$ yields a genuine 4D TQFT with partition function $Z(M)= e^{-36 \pi^2 i \sigma(M)/\Lambda}$, linking the theory to the Crane-Yetter-Broda state-sum. It proves that, when the CY parameter satisfies $y = \exp(-36 \pi^2 i / \Lambda)$, the BF-derived TQFT is monoidally naturally isomorphic to the CY-Broda theory, bridging differential-geometric and combinatorial TQFT frameworks. The work also discusses extended TQFT considerations and conjectures a broader BF-CY correspondence for general simply-connected compact semisimple groups, suggesting a unified perspective on framing and 4-manifold invariants.
Abstract
Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G = GL(4,R) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4-manifolds.