Publications

16/06/2024

Publications:

Skein Categories in Non-semisimple Settings, June 2024, arXiv preprint, with Jennifer Brown.

We introduce a version of skein categories which depends on a tensor ideal in a ribbon category, thereby extending the existing theory to the setting of non-semisimple TQFTs. We obtain modified notions of skein algebras of surfaces and skein modules of 3-cobordisms for non-semisimple ribbon categories. We prove that these skein categories built from ideals coincide with factorization homology, shedding new light on the similarities and differences between the semisimple and non-semisimple settings. As a consequence, we get a skein-theoretic description of factorization homology for a large class of balanced braided categories in Pr, precisely all those which are expected to induce an oriented categorified 3-TQFT.

Skein (3+1)-TQFTs from non-semisimple ribbon categories, June 2023, arXiv preprint, with Francesco Costantino, Nathan Geer and Bertrand Patureau-Mirand.

Using skein theory very much in the spirit of the Reshetikhin–Turaev constructions, we define a (3+1)-TQFT associated with possibly non-semisimple finite unimodular ribbon tensor categories. State spaces are given by admissible skein modules, and we prescribe the TQFT on handle attachments. We give some explicit algebraic conditions on the category to define this TQFT, namely to be « chromatic non-degenerate ». As a by-product, we obtain an invariant of 4-manifolds equipped with a ribbon graph in their boundary, and in the « twist non-degenerate » case, an invariant of 3-manifolds. Our construction generalizes the Crane–Yetter–Kauffman TQFTs in the semi-simple case, and the Lyubashenko (hence also Hennings and WRT) invariants of 3-manifolds. The whole construction is very elementary, and we can easily characterize invertibility of the TQFTs, study their behavior under connected sum and provide some examples.

Unit inclusion in a (non-semisimple) braided tensor category and (non-compact) relative TQFTs, April 2023, arXiv preprint.

The inclusion of the unit in a braided tensor category V induces a 1-morphism in the Morita 4-category of braided tensor categories BrTens. We give criteria for the dualizability of this morphism. When V is a semisimple (resp. non-semisimple) modular category, we show that the unit inclusion induces under the Cobordism Hypothesis a (resp. non-compact) relative 3-dimensional topological quantum field theory. Following Jordan–Safronov, we conjecture that these relative field theories together with their bulk theories recover Witten–Reshetikhin–Turaev (resp. De Renzi–Gainutdinov–Geer–Patureau-Mirand–Runkel) theories, in a fully extended setting. In particular, we argue that these theories can be obtained by the Cobordism Hypothesis.

Relating stated skein algebras and internal skein algebras, June 2022, published in SIGMA, arXiv.

We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in  CL19, and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in BBJ18 or in GJS19. Stated skein algebras are defined on surfaces with multiple boundary edges and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edges internal skein algebras using excision properties of skein categories, and agreeing with excision properties of stated skein algebras when V=Uq(sl2)-mod^fin. Our proofs are mostly based on skein theory and we do not require the reader to be familiar with the formalism of higher categories.

My PhD Thesis is available here.

Notes:

2021 — HOMFLY-PT skein invariants and the Deligne interpolation Repq(GLt), internship report.

2020 — Skein algebras and Factorisation homology, M2 master thesis.

Diffusion:

2020 — Une illustration du théorème de Brouwer, published in Quadrature N°117.

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