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Given a tensor category C, one constructs its Drinfeld center Z(C) which is abraided 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 newcategory Zel(C) which we call the Elliptic Drinfeld Center, which has objects(X, lambda1, lambda2), where the lambda i's are half-braidings that satisfysome compatibility conditions. We discuss an SL2(Z)-action on Zel(C) that isrelated to the anomaly appearing in Reshetikhin-Turaev theory. Thisconstruction is motivated from the study of the extended Crane-Yetter TQFT, inparticular the category associated to the once punctured torus.

The Elliptic Drinfeld Center of a Premodular Category