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Non-semisimple CFT/TFT correspondence I: General setup

Aaron Hofer, Ingo Runkel

TL;DR

The paper extends the TFT-based construction of CFT correlators from semisimple to finite non-semisimple (logarithmic) CFTs by leveraging a defect-enhanced 3d TFT built from a finite modular tensor category C. It sets up a rigorous algebraic/topological framework: finite ribbon categories with their coend L, Drinfeld center, and boundary/defect data, together with a 2-categorical open-closed world-sheet formalism and defect bordism categories. The central achievement is defining full CFTs as braided monoidal oplax natural transformations Cor from a world-sheet 2-category to Prof, with correlators arising from connecting manifolds evaluated in the defect TFT Z_C; the diagonal/Cardy case is worked out explicitly, including non-invertible line defects and the Grothendieck-ring defect algebra. The work lays the groundwork for non-semisimple bulk-defect interplay and provides tools to study topological line defects and their action on bulk fields in finite CFTs, linking to modular functor constructions and the symTFT program. Overall, the paper provides a general, topological construction of full CFTs in the non-semisimple setting and demonstrates that Cardy-type theories admit nontrivial, non-invertible defect structures with nonsemisimple fusion rules.

Abstract

We extend the TFT construction of CFT correlators of [arXiv:hep-th/0204148] to so-called finite logarithmic CFTs for which the algebraic input data is no longer semisimple but still finite. More specifically, starting from the data of a chiral CFT given in the form of a not necessarily semisimple modular tensor category C we use a three dimensional topological field theory with surface defects based on the surgery TFT of [arXiv:1912.02063] to construct a full CFT as a braided monoidal oplax natural transformation. We make our construction explicit in the example of the transparent surface defect, resulting in the so-called Cardy case. In particular, we consider topological line defects and their action on bulk fields in these logarithmic CFTs, providing a source of examples for non-invertible and non-semisimple topological symmetries.

Non-semisimple CFT/TFT correspondence I: General setup

TL;DR

The paper extends the TFT-based construction of CFT correlators from semisimple to finite non-semisimple (logarithmic) CFTs by leveraging a defect-enhanced 3d TFT built from a finite modular tensor category C. It sets up a rigorous algebraic/topological framework: finite ribbon categories with their coend L, Drinfeld center, and boundary/defect data, together with a 2-categorical open-closed world-sheet formalism and defect bordism categories. The central achievement is defining full CFTs as braided monoidal oplax natural transformations Cor from a world-sheet 2-category to Prof, with correlators arising from connecting manifolds evaluated in the defect TFT Z_C; the diagonal/Cardy case is worked out explicitly, including non-invertible line defects and the Grothendieck-ring defect algebra. The work lays the groundwork for non-semisimple bulk-defect interplay and provides tools to study topological line defects and their action on bulk fields in finite CFTs, linking to modular functor constructions and the symTFT program. Overall, the paper provides a general, topological construction of full CFTs in the non-semisimple setting and demonstrates that Cardy-type theories admit nontrivial, non-invertible defect structures with nonsemisimple fusion rules.

Abstract

We extend the TFT construction of CFT correlators of [arXiv:hep-th/0204148] to so-called finite logarithmic CFTs for which the algebraic input data is no longer semisimple but still finite. More specifically, starting from the data of a chiral CFT given in the form of a not necessarily semisimple modular tensor category C we use a three dimensional topological field theory with surface defects based on the surgery TFT of [arXiv:1912.02063] to construct a full CFT as a braided monoidal oplax natural transformation. We make our construction explicit in the example of the transparent surface defect, resulting in the so-called Cardy case. In particular, we consider topological line defects and their action on bulk fields in these logarithmic CFTs, providing a source of examples for non-invertible and non-semisimple topological symmetries.

Paper Structure

This paper contains 11 sections, 9 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and previous results
  3. Statement of the main results
  4. Algebraic preliminaries
  5. Finite ribbon categories
  6. (Co)modules of the canonical Hopf algebra
  7. 3d defect TFTs and topological world sheets
  8. Defect TFTs
  9. Topological world sheets
  10. Algebraic structure underlying full 2d CFT
  11. Modular functors

Key Result

Theorem 1

Evaluating the $3$d defect TFT $\mathrm{Z}_\mathcal{C}$ on the connecting manifolds induces a braided monoidal oplax natural transformation where $\Delta_\mathbbm{k} \colon \mathfrak{WS} \to \mathcal{P}\mathrm{rof}_{\mathbbm{k}}^{\mathrm{L} \mathrm{ex}}$ is the constant symmetric monoidal $2$-functor sending every object to $\mathrm{vect}_{\mathbbm{k}}$.

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3: The dinatural transformations defining the Hopf algebra structure morphisms and the Hopf pairing on $\mathrm{L}$. We use red for the ribbons labelled with $X;Y \in \mathcal{C}$ here to highlight that the structural morphism of $\mathrm{L}$ do not depend on the specific labels.

Theorems & Definitions (15)

  • Theorem : \ref{['thm:fullcft']}
  • Proposition : \ref{['prop:algofdefs']}
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3: Lyu95invproj
  • Proposition 2.4: Shimizu17ribbondrinf
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 2.7: SHIMIZU2019nondeg
  • Lemma 2.8: GR17projmod
  • ...and 5 more