Invertible bimodule categories and generalized Schur orthogonality
Jacob C. Bridgeman, Laurens Lootens, Frank Verstraete
TL;DR
This work addresses how to decide when a bimodule category between fusion categories is invertible by establishing a generalized Schur orthogonality framework within the annular algebra of the dual category. The main approach reduces invertibility to comparing Frobenius-Perron dimensions and requiring a nondegenerate inner product on annular- algebra characters derived from the Haar integral of a weak Hopf algebra, i.e., $(\chi_a,\chi_b) = \langle \chi_a^*\chi_b,\Lambda\rangle = \delta_{a}^{b}$. It further provides an algorithm to compute the full skeletal data of the invertible bimodule in unitary gauge, enabling explicit Morita equivalences, and proves the invertibility criterion is equivalent to MPO-injectivity in tensor-network models of topological order, with implications for generalized symmetries and a generalized Wigner-Eckart theorem.
Abstract
The Schur orthogonality relations are a cornerstone in the representation theory of groups. We utilize a generalization to weak Hopf algebras to provide a new, readily verifiable condition on the skeletal data for deciding whether a given bimodule category is invertible and therefore defines a Morita equivalence. As a first application, we provide an algorithm for the construction of the full skeletal data of the invertible bimodule category associated to a given module category, which is obtained in a unitary gauge when the underlying categories are unitary. As a second application, we show that our condition for invertibility is equivalent to the notion of MPO-injectivity, thereby closing an open question concerning tensor network representations of string-net models exhibiting topological order. We discuss applications to generalized symmetries, including a generalized Wigner-Eckart theorem.