  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.
