The finiteness conjecture for skein modules

Abstract

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.

Relating stated skein algebras and internal skein algebras

“…The structure of internal Hom object comes with its own composition and endows A Σ with an algebra structure. In [GJS19], [BBJ18] one considers a right Sk V (R 2 )-action ¡ on Sk V (Σ) which induces a braided opposite algebra structure compared to the left action.…”

“…However, such an object does not always exist in Sk V (R 2 ), and actually lives in its cocompletion. The internal skein algebra A Σ of the surface is the internal endomorphism algebra of the empty set Hom(∅, ∅), see [GJS19] or [BBJ18] together with [Coo19]. This means one can understand ribbon graphs in Sk V (Σ) with boundary points on the bottom and near the boundary edge as morphisms in (the Free cocompletion of) Sk V (R 2 ) V with target A Σ .…”

“…The goal of this paper is to link stated skein algebras of [CL19] and internal skein algebras of [GJS19] and [BBJ18] in a way that would be understandable by researchers from both fields. For this purpose we adopt the point of view of [GJS19] using the results of [Coo19] to compute factorisation homology used in [BBJ18] by skein categories, so we do not use factorisation homology.…”

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We give an explicit correspondance 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 [BBJ18] or in [GJS19]. For the sake of accessibility, we do not use factorisation homology but compute it using skein categories following [Coo19].

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Higher Deformation Quantization for Kapustin-Witten Theories

“…Subsequently, there has been an enormous amount of work at this crossroads between quantum field theory, representation theory, topology, and higher category theory. To give just a few examples, we refer to the research program of Ben-Zvi, Gunningham, Nadler and collaborators [BN09; BN18; BGN19], the work of Ben-Zvi, Brochier and Jordan [BBJ18a;BBJ18b] and the results on skein algebras that have followed them [Coo20;GJS19], and the work of Frenkel and Gaiotto [Gai18;FG20]. Much of the mathematical work has used methods involving higher categories and derived geometry, in much the same spirit as homological mirror symmetry reworks the physicists' view on duality for N = (2, 2) supersymmetric sigma models.…”