Reduced Tensor Product on the Drinfeld Center

Abstract

The annulus comes with a "stacking" operation which glues two annuli into one. This provides a tensor product structure on the category of boundary values $Z_{\text{CY}}(\text{Ann})$ associated to the annulus in an extended Crane-Yetter TQFT. It is known that $Z_{\text{CY}}(\text{Ann}) \simeq \mathcal{Z}(\mathcal{A})$, the Drinfeld center of the premodular category $\mathcal{A}$ from which $Z_{\text{CY}}$ is constructed. We give an explicit formula for the tensor product on $\mathcal{Z}(\mathcal{A})$ that corresponds to the stacking operation on $Z_{\text{CY}}(\text{Ann})$.