On the Relation Between Non-Semisimple TQFTs

Abstract

The goal of this paper is to clarify the relation between the Kerler-Lyubashenko TQFT JE associated with the adjoint end ℰ of a (not necessarily semisimple) modular category 𝒞 and its renormalized version VE based on the modified trace supported by the ideal Proj(𝒞) of projective objects of 𝒞. More precisely, we construct a $3$-dimensional ETQFT ÂE that contains both JE and VE. The ETQFT ÂE is given by a $2$-functor whose source is the $2$-category of admissible $3$-dimensional cobordisms, and whose target is the $2$-category of finitely complete linear categories. We improve on earlier versions of the construction by dropping the admissibility condition for surfaces. By doing so, we obtain an ETQFT whose circle category ÂE(S¹) is equivalent to the category 𝒞, as opposed to the ideal Proj(𝒞).

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BibTeX RIS

BibTeX

@misc{renzi2026relation,
  title = {{On the Relation Between Non-Semisimple TQFTs}},
  author = {Marco De Renzi},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31449},
  eprint = {2609.31449},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05799431v1},
}

RIS

TY  - UNPB
TI  - On the Relation Between Non-Semisimple TQFTs
AU  - Marco De Renzi
PY  - 2026
DA  - 2026/09/25
PB  - HAL
DO  - 10.48550/arxiv.2609.31449
UR  - https://hal.science/hal-05799431v1
AB  - The goal of this paper is to clarify the relation between the Kerler-Lyubashenko TQFT J_E associated with the adjoint end ℰ of a (not necessarily semisimple) modular category 𝒞 and its renormalized version V_E based on the modified trace supported by the ideal Proj(𝒞) of projective objects of 𝒞. More precisely, we construct a $3$-dimensional ETQFT Â_E that contains both J_E and V_E. The ETQFT Â_E is given by a $2$-functor whose source is the $2$-category of admissible $3$-dimensional cobordisms, and whose target is the $2$-category of finitely complete linear categories. We improve on earlier versions of the construction by dropping the admissibility condition for surfaces. By doing so, we obtain an ETQFT whose circle category Â_E(S^1) is equivalent to the category 𝒞, as opposed to the ideal Proj(𝒞).
ER  -

Abstract

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