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Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras

Aaron D. Lauda, Hendryk Pfeiffer

TL;DR

The paper establishes a complete algebraic description of 2-dimensional open-closed TQFTs by proving an equivalence between the open-closed cobordism category $2\mathrm{Cob}^{\mathrm{ext}}$ and the theory $\mathrm{Th}(\mathrm{K\text{-}Frob})$ freely generated by a knowledgeable Frobenius algebra. This yields a precise correspondence: open-closed TQFTs valued in a symmetric monoidal category $\mathcal{C}$ are in bijection with knowledgeable Frobenius algebras in $\mathcal{C}$, via a constructive generators-and-relations framework and a normal form for open-closed cobordisms. The authors develop Morse-like theory for manifolds with corners to prove sufficiency of a finite set of generators and relations, and extend the framework to boundary-labeled (D-brane) settings, yielding $S$-coloured generalizations. The results connect topological structure directly to algebraic data, offering a robust platform for extended TQFTs and suggesting routes to higher-dimensional extensions using manifolds with faces. Overall, the work provides both a computable algebraic model and a constructive topological normal form for open-closed 2D TQFTs with potential applications to boundary conformal field theory and related extended field theories.

Abstract

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an algebra homomorphism i:C->A with dual i^*:A->C, subject to some conditions. This result is achieved by providing a generators and relations description of the category of open-closed cobordisms. In order to prove the sufficiency of our relations, we provide a normal form for such cobordisms which is characterized by topological invariants. Starting from an arbitrary such cobordism, we construct a sequence of moves (generalized handle slides and handle cancellations) which transforms the given cobordism into the normal form. Using the generators and relations description of the category of open-closed cobordisms, we show that it is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra. Our formalism is then generalized to the context of open-closed cobordisms with labeled free boundary components, i.e. to open-closed string worldsheets with D-brane labels at their free boundaries.

Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras

TL;DR

The paper establishes a complete algebraic description of 2-dimensional open-closed TQFTs by proving an equivalence between the open-closed cobordism category and the theory freely generated by a knowledgeable Frobenius algebra. This yields a precise correspondence: open-closed TQFTs valued in a symmetric monoidal category are in bijection with knowledgeable Frobenius algebras in , via a constructive generators-and-relations framework and a normal form for open-closed cobordisms. The authors develop Morse-like theory for manifolds with corners to prove sufficiency of a finite set of generators and relations, and extend the framework to boundary-labeled (D-brane) settings, yielding -coloured generalizations. The results connect topological structure directly to algebraic data, offering a robust platform for extended TQFTs and suggesting routes to higher-dimensional extensions using manifolds with faces. Overall, the work provides both a computable algebraic model and a constructive topological normal form for open-closed 2D TQFTs with potential applications to boundary conformal field theory and related extended field theories.

Abstract

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an algebra homomorphism i:C->A with dual i^*:A->C, subject to some conditions. This result is achieved by providing a generators and relations description of the category of open-closed cobordisms. In order to prove the sufficiency of our relations, we provide a normal form for such cobordisms which is characterized by topological invariants. Starting from an arbitrary such cobordism, we construct a sequence of moves (generalized handle slides and handle cancellations) which transforms the given cobordism into the normal form. Using the generators and relations description of the category of open-closed cobordisms, we show that it is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra. Our formalism is then generalized to the context of open-closed cobordisms with labeled free boundary components, i.e. to open-closed string worldsheets with D-brane labels at their free boundaries.

Paper Structure

This paper contains 26 sections, 24 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Knowledgeable Frobenius algebras
  3. Definitions
  4. The category $\hbox{\bfseries {\upshape {Th}}}(\hbox{\bfseries {\upshape {K-Frob}}})$
  5. The category of open-closed cobordisms
  6. $\left<2\right>$-manifolds
  7. Manifolds with corners
  8. Manifolds with faces
  9. $\left<n\right>$-manifolds
  10. Collars
  11. Open-closed cobordisms
  12. Cobordisms
  13. Gluing
  14. The category $\hbox{\bfseries {\upshape {2Cob}}}^{{\rm ext}}(S)$
  15. Invariants for open-closed cobordisms
  16. ...and 11 more sections

Key Result

Proposition 2.6

Let $\mathcal{C}$ be a symmetric monoidal category. The category $\hbox{\bfseries {\upshape {K-Frob}}}(\mathcal{C})$ forms a symmetric monoidal category as follows. The tensor product of two knowledgeable Frobenius algebra objects ${\mathbb A}=(A,C,\imath,\imath^{\ast})$ and ${\mathbb A}'=(A',C',\im

Figures (1)

  • Figure 1: This figure depicts the normal form of an open-closed cobordism in $\hbox{\bfseries {\upshape {2Cob}}}^{{\rm ext}}_{\mathrm{O\rightarrow C}}[\vec{n},\vec{m}]$ without precomposition with a permutation, i.e. it shows the open-closed cobordism \ref{['eq_prenormal']}.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Lemma 3.1: Lemma 2.1.6 of Lr
  • ...and 49 more