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Morita classes of algebras in modular tensor categories

Liang Kong, Ingo Runkel

TL;DR

This work shows that in a modular tensor category, Morita classes of simple non-degenerate algebras are completely captured by their full centres in the doubled category $\\mathcal{C} \\boxtimes \\tilde{\\mathcal{C}}$. The key technical development is that Morita equivalence implies isomorphism of full centres, and conversely, an isomorphism of full centres implies Morita equivalence, with a constructive path via the functor $T$ and haploid representatives. The results have a direct interpretation in 2D rational conformal field theory: for a given bulk theory, there cannot be multiple incompatible sets of boundary conditions. The paper also connects centre data to simple left modules through a transported centre $C_A=T(Z(A))$ and the algebra $T_A$, clarifying the role of boundary and defect data in the categorical framework.

Abstract

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.

Morita classes of algebras in modular tensor categories

TL;DR

This work shows that in a modular tensor category, Morita classes of simple non-degenerate algebras are completely captured by their full centres in the doubled category . The key technical development is that Morita equivalence implies isomorphism of full centres, and conversely, an isomorphism of full centres implies Morita equivalence, with a constructive path via the functor and haploid representatives. The results have a direct interpretation in 2D rational conformal field theory: for a given bulk theory, there cannot be multiple incompatible sets of boundary conditions. The paper also connects centre data to simple left modules through a transported centre and the algebra , clarifying the role of boundary and defect data in the categorical framework.

Abstract

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.

Paper Structure

This paper contains 11 sections, 17 theorems, 61 equations.

Table of Contents

  1. Introduction and summary
  2. Preliminaries
  3. Properties of non-degenerate algebras
  4. Modular tensor categories
  5. Properties of the full centre
  6. Morita equivalence implies isomorphic full centre
  7. Isomorphic full centre implies Morita equivalence
  8. The functor $T$
  9. The full centre transported to $\mathcal{C}$ and simple modules
  10. A surjection from $T(Z(A))$ to $A$
  11. Haploid representatives of Morita classes

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a modular tensor category and let $A$, $B$ be simple non-degenerate algebras in $\mathcal{C}$. Then the following two statements are equivalent. (i) $A$ and $B$ are Morita equivalent. (ii) $Z(A)$ and $Z(B)$ are isomorphic as algebras.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Proposition 2.7
  • Lemma 3.1
  • ...and 13 more