On the Category of Boundary Values in the Extended Crane-Yetter TQFT

Abstract

The Crane-Yetter state sum is an invariant of closed 4-manifolds, defined in terms of a triangulation, based on 15-j symbols associated to the category A of representations over quantum sl2 (at a root of unity). In this thesis, we define the state sum in terms of a 'PLCW decomposition', which generalizes triangulations, and generalize A to an arbitrary premodular category. We extend the state sum to 4-manifolds with corners, making it an extended TQFT. We also develop a parallel theory based on skeins, which are essentially A-colored graphs, and we show that the two theories are equivalent. Focusing on the 2-dimensional part, we prove several properties of skein categories, the most important of which is that they satisfy excision. We provide explicit algebraic descriptions of the category associated to the once-punctured torus and the annulus, giving rise to a new tensor product on the Drinfeld center of a premodular category. As it is well-known that, when A is modular, the Crane-Yetter state sum computes the signature of a closed 4-manifold, we connect the Crane-Yetter theory to the signature of a 4-manifold with boundary and even corners. Finally, we show that the Reshetikhin-Turaev TQFT is a boundary theory of the Crane-Yetter theory (up to a normalization).