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Topological and conformal field theory as Frobenius algebras

Ingo Runkel, Jens Fjelstad, Jurgen Fuchs, Christoph Schweigert

TL;DR

This work links two-dimensional conformal field theory to topological field theory by showing that the algebraic input of a full open/closed CFT can be captured by symmetric special Frobenius algebras in a braided (modular tensor) category. Using a three-dimensional TFT, correlators are constructed as invariants of connecting manifolds, guaranteeing independence from triangulations and compatibility with world-sheet cuts, while the closed and open sectors are encoded by $H_{ m cl}=Z(A)$ and $H_{ m op}=A$. The main result proves that any symmetric special Frobenius algebra in a modular tensor category yields a consistent full CFT and provides a precise, categorical framework for classifying RCFTs up to Morita equivalence via module categories. The approach encompasses both chiral and full theories, with potential extensions to unoriented settings and connections to broader algebraic structures such as centres and weak Hopf algebras.

Abstract

Two-dimensional conformal field theory (CFT) can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. The construction of a (rational) CFT can be divided into two steps, of which one is complex-analytic and one purely algebraic. We realise the algebraic part of the construction with the help of three-dimensional topological field theory and show that any symmetric special Frobenius algebra in the appropriate braided monoidal category gives rise to a solution. A special class of examples is provided by two-dimensional topological field theories, for which the relevant monoidal category is the category of vector spaces.

Topological and conformal field theory as Frobenius algebras

TL;DR

This work links two-dimensional conformal field theory to topological field theory by showing that the algebraic input of a full open/closed CFT can be captured by symmetric special Frobenius algebras in a braided (modular tensor) category. Using a three-dimensional TFT, correlators are constructed as invariants of connecting manifolds, guaranteeing independence from triangulations and compatibility with world-sheet cuts, while the closed and open sectors are encoded by and . The main result proves that any symmetric special Frobenius algebra in a modular tensor category yields a consistent full CFT and provides a precise, categorical framework for classifying RCFTs up to Morita equivalence via module categories. The approach encompasses both chiral and full theories, with potential extensions to unoriented settings and connections to broader algebraic structures such as centres and weak Hopf algebras.

Abstract

Two-dimensional conformal field theory (CFT) can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. The construction of a (rational) CFT can be divided into two steps, of which one is complex-analytic and one purely algebraic. We realise the algebraic part of the construction with the help of three-dimensional topological field theory and show that any symmetric special Frobenius algebra in the appropriate braided monoidal category gives rise to a solution. A special class of examples is provided by two-dimensional topological field theories, for which the relevant monoidal category is the category of vector spaces.

Paper Structure

This paper contains 17 sections, 2 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Two-dimensional conformal field theory
  3. What is a two-dimensional CFT?
  4. Special case: Two-dimensional topological field theory
  5. Open/closed CFT
  6. Chiral CFT and full CFT
  7. The construction of full CFT as an algebraic problem
  8. Statement of the problem
  9. Three-dimensional topological field theory
  10. The assignment $X \,{\mapsto}\, V(X)$
  11. Cobordism for ${\rm tr}_{\rm last}$
  12. Frobenius algebras in ${\mathcal{C}}$
  13. Construction of the correlator $\hbox{\sl Cor}_{\!A}({\mathrm X}) \,{\in}\, V({\mathrm X})$
  14. Main theorem
  15. Comments
  16. ...and 2 more sections

Key Result

Proposition 3

Let $A$ be a symmetric special Frobenius algebra in ${\mathcal{C}}$. Then the object $Z(A)$ of ${\mathcal{C}} \,{\boxtimes}\, \overline{{\mathcal{C}}}$ inherits from $A$ the structure of a commutative symmetric Frobenius algebra.

Figures (1)

  • Figure 1: a) An example of a world sheet and b) the result of cutting that world sheet along a circle.

Theorems & Definitions (3)

  • Remark 2
  • Proposition 3
  • Theorem 4