Non-semisimple Crane-Yetter theory varying over the character stack

Abstract

We construct a relative version of the Crane-Yetter topological quantum field theory in four dimensions, from non-semisimple data. Our theory is defined relative to the classical $G$-gauge theory in five dimensions -- this latter theory assigns to each manifold $M$ the appropriate linearization of the moduli stack of $G$-local systems, called the character stack. Our main result is to establish a relative invertibility property for our construction. This invertibility generalizes the key invertibility property of the original Crane-Yetter theory which allowed it to capture the framing anomaly of the celebrated Witten-Reshetikhin-Turaev theory. In particular our invertibilty statement at the level of surfaces implies a categorical, stacky version of the unicity theorem for skein algebras; at the level of 3-manifolds it equips the character stack with a canonical line bundle. Regarded as a topological symmetry defect of classical gauge theory, our work establishes invertibility of this defect by a gauging procedure.

v5: final version incorporating reviewer feedback. v4: conjectures introduced in section 4.3 to precisely handle SO-fixed point structures. v3: Corrections around 3.18, 3.19 and where these are used. v2: Important corrections made throughout sections 3.1 and 3.3. v1: Comments welcome!