Supersymmetry, differential operators of infinite order and theta-functions

• Mikhail Kapranov (Kavli Institute)

Room: Centre de conférences Marilyn et James Simons Differential operators of infinite order (DOI) are infinite series in derivatives with holomorphic coefficients decaying so fast that the action on holomorphic functions converges and preserves the domain of definition. Thus exp(d/dx) (shift operator) is not a DOI but cos(√(d/dx)) is. In 1973 M. Sato gave a characterization of theta-zerovalues by a manifestly modular invariant system of DOI in the modular variables alone, thus deducing modularity from local conditions. This has been developed by several authors (Kashiwara, Kawai, Takei, Yoshida and others) since. I will present a "supersymmetric" approach to Sato's theory based on two observations: The exponential of any odd supersymmetry generator is a DOI. In some cases such odd generators, acting "on-shell" (in the space of solutions of equations of motion), satisfy even-style commutation relations

Riemann-Hilbert correspondence, representations of spherical DAHA, and P=W phenomenon

• Yan Soibelman (Kansas State University)

Room: Centre de conférences Marilyn et James Simons My talk is devoted to some results and speculations which are in a sense inspired by physics. Main goal of the talk is to discuss how generalized Riemann-Hilbert correspondence proposed by Maxim Kontsevich and myself in 2015 brings new perspectives to some questions in representation theory as well as to the P=W phenomenon in nonabelian Hodge theory.

Chi-independence for moduli spaces of one-dimensional sheaves on symplectic surfaces

• Olivier Schiffmann (CNRS-Laboratoire de Mathématiques d'Orsay)

Room: Centre de conférences Marilyn et James Simons Moduli spaces $M(\beta; \chi)$ of one-dimensional sheaves on a complex K3 or abelian surface S have a rich and well-studied enumerative geometry. In this work, we prove that the so-called BPS cohomology (or Donaldson-Thomas invariants) of $M(\beta;\chi)$ is independent of $\chi$ --the Euler characteristic--for any curve class $\beta$. We establish a relative version of this statement, conjectured by Toda in 2019, over the Chow variety of $1$-cycles on $S$. We do this by constructing an action of the cohomological Hall algebra of zero-dimensional sheaves on the BPS Lie algebra of the stack of coherent sheaves on $S$. This is joint work with B. Davison, L. Hennecart, T. Kinjo and E. Vasserot.

Explicit formulas for the noncommutative Cartan calculus

• Bruno Vallette (Université Sorbonne Paris-Nord)

Room: Centre de conférences Marilyn et James Simons The commutative/classical Cartan calculus amonts to a Lie type action of vector fields of a smooth manifold on its de Rham complex of differential forms. Its noncommutative analogue is expected to take the form of a homotopy Lie type action of the Hochschild cochain complex of a (homotopy) associative algebra on its the Hochschild chain complex compatible with Connes' boundary map. Such a structure implies a noncommutative chain-level version of the Gauss-Manin connexion. In this talk, I will explain how the operadic calculus allows one to solve this problem and I will provide fully explicit formulas in terms of an operad introduced by Kontsevich—Soibelman.

Quantum spectra and quantum integrable systems

• Alexander Soibelman (IHES)

Room: Centre de conférences Marilyn et James Simons A classical problem in quantum mechanics involves computing the spectrum of a Schrödinger-type differential operator. One can, for example, find the asymptotic expansions of the eigenvalues using the WKB method. Another approach, due to Sjöstrand, obtains such expansions directly from the operator, using its quantum normal form. We provide a geometric interpretation for this normal form, encoding it as a section of a vector bundle associated with the quantization of a complex integrable system. We also propose a number of conditions that allow us to determine this section uniquely. This is joint work with Maxim Kontsevich.

On actions of braid groups on triangulated categories arising in cluster theory

• Bernhard Keller (Institut de mathématiques Jussieu-Paris Rive Gauche)

Room: Centre de conférences Marilyn et James Simons We will present an approach to the construction of actions of braid groups on triangulated categories arising in the (additive) categorification of cluster algebras and varieties. The examples will be inspired by combinatorial braid group actions constructed by Fraser, Fock-Goncharov, Goncharov-Shen and others. The results we will present were obtained in several joint projects notably involving, in chronological order, Chris Fraser, Yilin Wu, Alessandro Contu, Miantao Liu, Haoyu Wang and Xiaofa Chen.

On resurgence and summability of Andersen-Kashaev states integrals

• Veronica Fantini (Laboratoire de Mathématiques d'Orsay)

Room: Centre de conférences Marilyn et James Simons Given a hyperbolic knot, the Andersen-Kashaev state integrals are convergent integrals built from certain triangulations of the knot complement. Their asymptotic expansion is a perturbative topological invariant of the knot, conjectured to be resurgent and Borel summable by Garoute falidis, Gu, and Mariño. In this talk, I will present the main ideas of the proofs of these conjectures, based on a joint project with Wheeler (arXiv:2410.20973) and our ongoing work together with J. E. Andersen and M. Kontsevich.

Sheaves for spacetimes

• Pierre Schapira (Institut de mathématiques Jussieu-Paris Rive Gauche)

Room: Amphithéâtre Léon Motchane A causal manifold $(M,\lambda)$ is a real manifold $M$ endowed with a closed convex proper cone $\lambda$ in its cotangent bundle $T^*M$. On such a manifold, one defines the $\lambda$-topology and the past or the future of any subset. A time function is a smooth surjective causal map $q: M\to\mathbb R$ proper on the past or future of any compact subset of $M$. Using a time function, we show that if the micro-support of a sheaf $F$ does not intersect $\lambda\cup-\lambda$ outside of the zero-section, then for any Cauchy hypersurface $N_t=q^{-1}(t)$, the restriction morphism $\mathrm{R}\Gamma(M;F)\to\mathrm{R}\Gamma(N_t;F\vert_{N_t})$ is an isomorphism. As an application, we get that the Cauchy problem is globally well-posed for hyperfunction solutions of hyperbolic systems.

Stability conditions in Fukaya categories: old and new ideas

• Maxim Kontsevich (IHES)

Room: Amphithéâtre Léon Motchane For the Fukaya category associated to a symplectic manifold with vanishing c₁, it is expected that closed complex-valued differential forms of middle degree subject to an open constraint each give rise to a Bridgeland stability structure (often called a stability condition in the literature). I will discuss related questions in differential geometry and attempts to formulate precise conjectures. The talk is based on earlier ideas developed together with Yan Soibelman, and on a current joint project with Fabian Haiden, Ludmil Katzarkov and Pranav Pandit.