Derived Algebraic Geometry, CoHAs and Operads

Angers, 16-19 Juin 2026

How to reach the lab.

We want to emphasize the use of derived algebraic geometry and operads as tools to make significant progress in two directions:

• geometric representation theory, and specifically with the aim of constructing and studying cohomological Hall algebras ;
• topological field theories, via the use of mapping stacks and Simpson shapes.

These two axes are related, at least via (shifted) symplectic geometry, and the crucial use of lagrangian structures. More specifically, in some situations may naturally appear moduli of curves, that can often be dealt with using operadic methods. The first axis is extremely active at present, and tackled through various closely related angles by several teams, see eg [DHM, KPS, DPSSV]. The second axis morally relies on the fully extended AKSZ construction performed in [CHS] associated to the Betti shape, as well as on the recent proof of the Moore-Tachikawa conjecture in [CM].

• [CHS] D. Calaque, R. Haugseng & C. Scheimbauer. The AKSZ construction in derived algebraic geometry as an extended topological field theory, Mem. Amer. Math. Soc. 308 (2025), no. 1555, v+173 pp, arXiv:2108.02473.
• [CM] P. Crooks & M. Mayrand. The Moore-Tachikawa conjecture via shifted symplectic geometry, arXiv:2409.03532.
• [DHM] B. Davison, L. Hennecart & S. S. Mejia. BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories, arXiv:2303.12592.
• [DPSSV] D-E. Diaconescu, M. Porta, F. Sala, O. Schiffmann & E. Vasserot. Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians, arXiv:2502.19445.
• [KPS] T. Kinjo, H. Park & P. Safronov. Cohomological Hall algebras for 3-Calabi-Yau categories, arXiv:2406.12838.

Organisers

• Tristan Bozec - Université Angers
• Damien Calaque - Université Montpellier
• Julien Grivaux - Sorbonne Université

Speakers

• Ben Davison - University Edinburgh
• Lucien Hennecart - Université Picardie Jules Verne
• Šarūnas Kaubrys - Kavli IPMU
  Loop CoHAs and factorization coalgebras

  In this talk I will report on work in progress studying certain CoHAs of 3-Calabi-Yau categories, as defined by Kinjo-Park-Safronov, Descombes. Our main motivation is the study of the CoHA of the 3-Calabi-Yau category of local systems of a 3-manifold of the form Sigma_g x S^1. The moduli stack of such local systems can be identified with the loop or inertia stack of local systems of the surface Sigma_g. One of the central tools is the use of the factorization coproduct of Davison-Hennecart-Kinjo-Schiffmann-Vasserot, adapted to the loop setting. This is joint with Descombes and Latyntsev.

• David Kern
Higher-genus categorified Gromov–Witten invariants and modular quantisation

Gromov–Witten invariants are K-theory classes used to quantise the Frobenius algebra structure on the K-theory of a target stack to a Cohomological Field Theory, an algebra over the modular operad of stable curves. Mann–Robalo showed how to categorify the genus-0 part of this structure to one at the level of derived categories through the use of derived moduli stacks of stable maps to geometrise the virtual structures, and of the general phenomenon of brane actions, providing actions of ∞-operads on their spaces of extensions of identity, discovered by Toën.

In joint work with Hugo Pourcelot, we extend this construction from operads to modular operads so as to categorify higher-genus Gromov–Witten invariants. I will explain how the formalism of ∞-properads developed by Barkan–Steinebrunner allows us to construct modular brane actions, and how we can further use them to view Gromov–Witten invariants as expressing a quantisation by genus.

• Alyosha Latyntsev - Beijing Institute of Mathematical Sciences and Applications
• Joost Nuiten - Université Toulouse
  Tannakian duality and symmetric powers

  Classical Tannakian duality over a field k provides a relation between certain symmetric monoidal k-linear categories and stacks over k. In this talk, I will discuss an extension of this correspondence to more general base rings, in the setting of infinity categories. It turns out that symmetric monoidal infinity categories are not very well-adapted to this kind of algebro-geometric problem. I will present a proposal for an enhancement of the notion of symmetric monoidal infinity category, that includes not only tensor products, but also derived symmetric towers. All of this is based on joint work with Bertrand Toën.

• Emanuele Pavia - Université Luxembourg
  Categorified Representations Of Topological Groups

  In his celebrated 2014 ICM address, Costantin Teleman suggested that one first step towards a theory of G-gauged TQFTs, where G is a compact Lie group, is provided by studying categorified representation theory of Lie groups. Building on the work of Jacob Lurie and showing evidence provided by concrete, computable examples, he hinted at some of the desirable properties that such categorical representations should enjoy.

  In this talk, we will see how the powerful formalism of higher category theory allows us to pursue these insights of Teleman: we provide rigorous proofs of some of his claims and deduce analogs of fundamental statements in classical representation theory. In particular, we are able to formulate a categorified Koszul duality statement in the topological setting.

  This is based on joint work with James Pascaleff and Nicolò Sibilla.

• Hugo Pourcelot - Université Angers
• Anja Švraka - Technical University Munich
  Additivity of constructible factorization algebras

  Factorization algebras, introduced by Costello and Gwilliam, encode the structure of observables in perturbative quantum field theory and capture concepts such as the operator product and correlation functions. Beyond this, their local structures encompass familiar algebraic objects including associative and A_{\infty}-algebras, vertex algebras, bimodules, and E_n-algebras. Examples of the latter include (possibly braided) tensor categories and n-fold loop spaces. In many situations of interest, particularly in the presence of defects or boundaries, it is natural to consider constructible factorization algebras. Examples are given by factorization homology. In this talk, I will present recent joint work with Victor Carmona proving an additivity theorem for constructible factorization algebras on manifolds with corners, resolving a conjecture of Ginot. These results address the behavior of factorization algebras under products of spaces. They are an important step toward a more general additivity theorem for constructible factorization algebras on conically smooth stratified spaces in the sense of Ayala–Francis–Tanaka. Furthermore, results of this kind provide an essential step towards comparing the models for higher Morita categories developed by Scheimbauer and Haugseng, two widely used frameworks in the study of dualizability and topological field theories. If time permits, I will briefly discuss the extension to conical manifolds.

• Nikola Tomić - Université Montpellier
  AKSZ theories from shifted Poisson structures and shifted coisotropic reduction

  The AKSZ procedure is a construction of field theories described by Alexandrov–Kontsevich–Schwarz–Zaboronsky. Given a 2-dimensional manifold M, they construct a geometrical object 𝓕_T(M) corresponding to fields on M with a symplectic target T. Such field theory is obtained as a mapping space Map(M,T) and is called a σ-model. This construction has a natural interpretation in shifted symplectic geometry, where the mapping space will have the structure of a derived stack and T will be a derived stack with an n-shifted symplectic form. In their seminal paper, Pantev–Toën–Vaquié–Vezzosi showed that for any compact oriented manifold M of dimension d, it is possible to endow the mapping stack Map(M,T) with a (n-d)-shifted symplectic form. This process has been extended into an extended TFT taking values in shifted Lagrangian correspondences by Calaque–Haugseng–Scheimbauer. Given a shifted symplectic structure, it is natural to study quantization of it, but the correct theory were quantization takes place is the theory of shifted Poisson structures. Shifted Poisson derived stacks are more general that shifted symplectic derived stacks and then harder to study. Such object is expected to satisfies the same properties than shifted symplectic ones. In particular, there should be a way to construct (n-d)-shifted Poisson structures on Map(M,T), given an n-shifted Poisson structure on T. In this talk, I will explain how my recent structure result on shifted Poisson structures gives such a construction. I will also explain how fuctoriality of such a construction is expected to behave and sketch a construction of this functor. In the process, I will explain how we extract a construction of shifted coisotropic reductions, a shifted analog of classical coisotropic reductions. If time permits, I will try to discuss expected results for quantizations of such theories.

Schedule

Tuesday 16	Wednesday 17	Thursday 18	Friday 19
9h30 - 10h45 Nuiten	9h30 - 10h45 Latyntsev	9h30 - 10h45 Hennecart	9h30 - 10h45 Pourcelot
10h45 - 11h15 coffee break	10h45 - 11h15 coffee break	10h45 - 11h15 coffee break	10h45 - 11h00 coffee break
11h15 - 12h30 Pavia	11h15 - 12h30 Kaubrys	11h15 - 12h30 Davison	11h00 - 12h00 Švraka
Lunch // discussions	Lunch // discussions	Lunch // discussions
15h30 - 16h45 Kern		16h00 - 17h15 Tomić
		19h - Dinner La Réserve