Journées de topologie quantique
The “journées de topologie quantique” (quantum topology days) are a regular event aiming at bringing together, in a somewhat informal setting, people thinking about quantum topology in a broad sense. As the name suggests this is a one day event, with three talks and lots of time for discussions, taking place either in Paris or in Dijon.
Next session: 18 March 2023
Registration
Registration is free but mandatory. Please fill the following form: https://framaforms.org/inscriptionjdtq1707227948
Location: Paris
Amphi Turing, bâtiment Sophie Germain
Program

09h30  Welcome/Coffee

10h15  Matthieu Faitg: Quantum moduli algebras and skein algebras
Quantum moduli algebras are a quantization of character varieties of surfaces which have been introduced by AlekseevGrosseSchomerus and BuffenoirRoche in the 90's. I will explain the definition of these algebras (based on quantum groups) and give structure results, namely they are finitely generated, Noetherian and do not have zero divisors. Then I will relate quantum moduli algebras with skein algebras, which is one the motivation for these results. I will finish by a few words on the specialization of quantum moduli algebras at roots of unity. Joint work with S. Baseilhac and P. Roche.
 11h30  Benjamin Haïoun: Nonsemisimple skein (3+1)TQFTs
I will recall some tools from nonsemisimple skein theory. The key players are modfied traces, which were introduced to obtain invariants of links with a projective strand for which the usual Reshetikhin–Turaev theory gives trivial invariants. I will explain how they give the first step into the definition of a skeintheoretic (3+1) Topological Quantum Field Theory, and give some elementary properties of this TQFT. This is joint work with Francesco Costantino, Nathan Geer and Bertrand PatureauMirand.
 14h30  Anna Beliakova: On algebraisation of lowdimensional Topology
The categories of $n$cobordisms are among the most studied objects in low dimensional topology. For $n=2$ we know that $2\operatorname{Cob}$ is a monoidal category freely generated by its commutative Frobenius algebra object: the circle. This result also classifies all TQFT functors on $2\operatorname{Cob}$. In this talk I will present similar classification results for special categories of 3 and 4cobordisms. Here Frobenius algebra is replaced by a socalled BobtchevaPiergallini Hopf algebra. The results are obtained in collaboration with Marco De Renzi, Ivelina Bobtcheva and Riccardo Piergallini.
Past events
21 November 2022 (Paris)
 Pedro Vaz: De la catégorification des modules de Verma aux homologies d'entrelacs
Abstract
Dans cet exposé j'expliquerai la catégorification des modules de Verma et leur utilisation dans la construction d'invariants d’entrelacs dans l'espace de dimension 3 et dans le tore solide. Les résultats présentés sont issus de collaborations avec G. Naisse et A. Lacabanne.
 Cristina PalmerAnghel: A globalisation of the Jones and Alexander polynomials from configurations on arcs and ovals in the punctured disc
Abstract
The Jones and Alexander polynomials are two important knot invariants and our aim is to see them from a topological model given by a graded intersection in a configuration space. Bigelow and Lawrence showed a topological model for the Jones polynomial, using arcs and figure eights in the punctured disc. On the other hand, the Alexander polynomial can be obtained from intersections between ovals and arcs. We present a common topological viewpoint which sees both invariants, based on ovals and arcs in the punctured disc. The model is constructed from a graded intersection between two explicit Lagrangians in a configuration space. It is a polynomial in two variables, recovering the Jones and Alexander polynomials through specialisations of coefficients. Then, we prove that the intersection before specialisation is (up to a quotient) an invariant which globalises these two invariants, given by an explicit interpolation between the Jones polynomial and Alexander polynomial. We also show how to obtain the quantum generalisation, coloured Jones and coloured Alexander polynomials, from a graded intersection between two Lagrangians in a symmetric power of a surface.
 Julien Korinman: Représentation d'algèbres d'écheveaux
Abstract
Le but de cet exposé est de présenter des progrès récents vers la classification des représentations des algèbres d'écheveaux aux racines de l'unité. Les représentations indécomposables de ces algèbres forment (conjecturalement) les briques de bases d'objets algébriques appelés TQFT qui contiennent des invariants de nœuds et de 3variétés et des représentations des groupes modulaires de surfaces. Je définirai d'abord les algèbres d'écheveaux, les relierait aux variétés de caractères et exposerai ensuite des méthodes générales de théorie des représentations (domaine Azumaya, théorie des ordres de Poisson), qui permettent de relier la théorie de représentations des algèbres d'écheveaux à la géométrie de Poisson des variétés de caractères. Au final, on obtiendra une description 'presque' complète des représentations à poids des algèbres d'écheveaux. C'est un travail en collaboration avec H.Karuo.
13 March 2023 (Dijon)
 Rinat Kashaev: Generalized TQFT's from local fields
Abstract
Based on the theory of quantum dilogarithms over locally compact Abelian groups, I will talk about a particular example of a quantum dilogarithm associated with a local field $F$ which leads to a generalized 3d TQFT based on the combinatorial input of ordered $\Delta$complexes. The associated invariants of 3manifolds are expected to be specific counting invariants of representations of $\pi_1$ into the group $PSL_2(F)$. This is an ongoing project in collaboration with Stavros Garoufalidis.
 Eilind Karlsson: Deformation quantisation and skein categories
Abstract
I will start by reviewing deformation quantisation of algebras, and explain how we in a similar spirit can define deformation quantisation of categories. The motivation is to understand how deformation quantisation interacts with categorical factorization homology, or more explicitly: how deformation quantisation interacts with “gluing” local observables to obtain global observables. One important and wellknown example of factorization homology is given by skein categories, which I will briefly introduce. We generalise the theory of skein categories to fit into the deformation quantisationsetting, and use it as a running example. This is based on joint work (in progress) with Corina Keller, Lukas Müller and Jan Pulmann.
 Rhea Palak Bakshi: Skein modules, torsion, and framing changes of links
Abstract
Skein modules are invariants of 3manifolds which were introduced by Józef H. Przytycki (and independently by Vladimir Tuarev) in 1987 as generalisations of the Jones, HOMFLYPT, and Kauffman bracket polynomial link invariants in the 3sphere to arbitrary 3manifolds. Over time, skein modules have evolved into one of the most important objects in knot theory and quantum topology, having strong ties with many fields of mathematics such as algebraic geometry, hyperbolic geometry, and the WittenReshetikhinTuraev 3manifold invariants, to name a few. One avenue in the study of skein modules is determining whether they reflect the geometry or topology of the manifold, for example, whether the module detects the presence of incompressible or nonseparating surfaces in the manifold. Interestingly enough this presence manifests itself in the form of torsion in the skein module. In this talk we will discuss various skein modules which detect the presence of nonseparating surfaces. We will focus on the framing skein module and show that it detects the presence of nonseparating 2spheres in a 3manifold by way of torsion.
5 June 2023 (Paris)
 Emmanuel Wagner: Link homologies and spectral sequences
Abstract
After giving an overview of various link homologies and spectral sequences relating them, I will concentrate on a spectral sequence allowing for comparison of the triply graded link homology of Khovanov and Rozansky and the Knot Floer homology. The technics used involves in particular quantum Hochschild homology, foams and webs. This is a joint work with A. Beliakova, K. Putyra and LH. Robert.
 Patrick Kinnear: An invertible nonsemisimple TQFT varying over the character stack
Abstract
Lusztig's quantum group at a root of unity comes equipped with a quantum Frobenius map to the classical universal enveloping algebra, allowing us to pull back classical representations to representations of the quantum group. In fact, these pulledback representations braid trivially with all others in $\operatorname{Rep}_q(G)$, and so they lie in the Müger centre and the functor $\operatorname{Rep}(G) \rightarrow \operatorname{Rep}_q(G)$ makes $\operatorname{Rep}_q(G)$ into a module category for $\operatorname{Rep}(G)$. This makes $\operatorname{Rep}_q(G)$ into an endomorphism of $\operatorname{Rep}(G)$ in the Morita theory SymTens of symmetric tensor categories. In this talk, we will discuss invertibility of this endomorphism and its implications in TQFT. From the perspective of the Cobordism Hypothesis, this data defines an invertible nonsemisimple TQFT varying over the character stack (the moduli space of $G$local systems) which extends the CraneYetter theory. To 3manifolds this theory assigns a line bundle on the character stack with the fiber over the trivial local system being the nonsemisimple CraneYetter theory. To surfaces the theory assigns an invertible sheaf of categories on the character stack given by the skein category of the surface, yielding a categorified version of the fact that the skein algebra defines a sheaf on the character variety which is Azumaya over an open dense locus (the locus being precisely determined recently by KaruoKorinman). Building on recent work of Haïoun on dualizability, we expect to be able to define a nonsemisimple $G$relative version of WittenReshetikhinTuraev theory similar to the contemporary understanding of WRT as relative to CY.
 Nikita Markarian: Perturbative ChernSimons invariants and factorization homology
Abstract
I will discuss my approach to perturbative ChernSimons invariants of 3manifolds and knots, and their generalizations, such as RozanskyWitten invariants, via the factorization complex of Weyl $e_n$algebras. I will give a review of some results and will describe possible further applications of them.
13 November 2023 (Dijon)
 Ramanujan Santharoubane: On the kernel of $SO(3)$WittenReshetikhinTuraev quantum representations
Abstract
Given a surface $S$, $SO(3)$WittenReshetikhinTuraev TQFT's provide a sequence, indexed by odd integers, of complex finite dimensional representations of the mapping class group of $S$ called quantum representations. One major and difficult problem is to find the kernels of these quantum representations. For prime index, Gilmer and Masbaum proved that the representation preserves a lattice over the ring of integers of a cyclotomic field. This allows to approximate each quantum representation by homomorphisms from the mapping class group in to finite groups. In this talk, we will see that certain of these approximations can be completely understood and are related to the Johnson filtration of the mapping class group. As a consequence we can find some non trivial 'upper bound' for the kernels of quantum representations. This is a joint work with Renaud Detcherry.
 AnnaKatharina Hirmer: Generalised Kitaev models from Hopf monoids: topological invariance and examples
Abstract
Quantum double models were introduced by Kitaev to obtain a realistic model for a topological quantum computer. They are based on a directed ribbon graph and a finitedimensional semisimple Hopf algebra. The ground state of these models is a topological invariant of a surface, i.e. only depends on the homeomorphism class of the oriented surface but not the ribbon graph. Meusburger and Voß generalised part of the construction from Hopf algebras to pivotal Hopf monoids in symmetric monoidal categories. We explain the construction of the ground state for involutive Hopf monoids and show that it is topological invariant. We explicitly describe this construction for Hopf monoids in Set, Top, Cat and SSet.
 Emmanuel Graff: The Homotopy Braid Group is TorsionFree
Abstract
V. Lin, in the 'Kourkova notebook', questions the existence of a nontrivial epimorphism of the braid group onto a nonabelian torsionfree group. The homotopy braid group studied by Goldsmith in 1974 an known to be nilpotent, naturally appears as a potential candidate. In 2001, Humphries showed that this homotopy braid group is torsionfree for less than 6 strands. In this presentation, we will see a new approach based on the broader concept of welded braids, as well as algebraic techniques. This will allow us to demonstrate that the homotopy braid group is torsionfree for any number of strands.
Organizers
 Adrien Brochier
 Gwénaël Massuyeau
 Emmanuel Wagner
 Lukas Woike