Morita classes of algebras in modular tensor categories

We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.

💡 Research Summary

The paper investigates algebras A living inside a modular tensor category C, focusing on those whose trace pairing is non‑degenerate. For such an algebra the authors construct the “full centre” Z(A), a commutative algebra in the doubled category C ⊠ C^rev (equivalently the Drinfeld centre of C). The main theorem states that for simple algebras with non‑degenerate trace pairing, Morita equivalence is exactly characterised by isomorphism of their full centres: A and B are Morita‑equivalent ⇔ Z(A) ≅ Z(B) as algebras.

The construction of Z(A) proceeds by embedding A into the product category C ⊠ C^rev, then taking the centraliser of the image with respect to the braiding. Non‑degeneracy guarantees that the resulting object inherits a well‑defined multiplication and unit, and that it is commutative. Moreover, Z(A) is “full” in the sense that every A‑module can be recovered from a Z(A)‑module via induction, establishing a tight link between the module category Mod_A(C) and the Drinfeld centre.

The proof of the main theorem has two directions. Assuming A and B are Morita‑equivalent, one exhibits an A‑B‑bimodule X and its dual Y. These bimodules acquire natural Z(C)‑module structures, and the induced maps X⊗_B Y → A and Y⊗_A X → B are isomorphisms. From these data one builds an explicit algebra isomorphism Z(A) → Z(B). Conversely, given an isomorphism Z(A) ≅ Z(B), the authors pull back the Z(C)‑module structure to construct a bimodule that implements a Morita equivalence. The non‑degenerate trace pairing is crucial in both directions, ensuring the existence of the required duals and the invertibility of the constructed maps.

From a physical perspective, the result has a clear interpretation in rational two‑dimensional conformal field theory (RCFT). In RCFT an algebra A encodes a consistent set of boundary conditions (open‑string sector), while Z(A) encodes the bulk (closed‑string) data. The theorem therefore implies that a given bulk theory cannot admit two incompatible families of boundary conditions: any two boundary algebras that give rise to the same bulk must be Morita‑equivalent. This resolves a long‑standing question about the uniqueness of compatible boundary data for a fixed bulk RCFT.

Overall, the paper provides a powerful categorical invariant— the full centre— that completely classifies Morita classes of simple algebras with non‑degenerate trace pairing in modular tensor categories, and it bridges this mathematical classification with the physical consistency conditions of rational conformal field theories.