Non-semisimple Crane–Yetter theory varying over the character stack
Patrick Kinnear · Advances in Mathematics · 2026
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 invertibility 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.