The Elliptic Drinfeld Center of a Premodular Category

Abstract: Given a tensor category C, one constructs its Drinfeld center Z(C) which is a braided tensor category, having as objects pairs (X, lambda), where X in Obj(C) and lambda is a half-braiding. For a premodular category C, we construct a new category Zel(C) which we call the Elliptic Drinfeld Center, which has objects (X, lambda1, lambda2), where the lambda i's are half-braidings that satisfy some compatibility conditions. We discuss an SL2(Z)-action on Zel(C) that is related to the anomaly appearing in Reshetikhin-Turaev theory. This construction is motivated from the study of the extended Crane-Yetter TQFT, in particular the category associated to the once punctured torus.