arXiv subjects

Yan Soibelman

Publications and source records attributed to Yan Soibelman.

At least 19 recordsLinked to original sources

arXiv2026

Motivic invariants of moduli stacks of Higgs bundles and bundles with connections: results and speculations

We review some results and techniques from our papers devoted to the computation of motivic classes of stacks of parabolic Higgs budles and bundles with connections on a curve. In the last section we present some directions for future work, as well as some speculations. The latter include a generalization of the P=W conjecture inspired by the work of Maxim Kontsevich and the third author on the Riemann--Hilbert correspondence for complex symplectic manifolds as well as our running project on the motivic classes of the moduli stacks of nilpotent pairs on the formal disk and geometric Satake correspondence for double affine Grassmannians.

math.AG↗

arXiv2025

Resurgence and perverse sheaves

We propose a point of view on resurgence theory based on the study of perverse sheaves on the complex line carrying an algebraic structure with respect to additive convolution. In particular, we lift the concept of alien derivatives introduced originally by J. \'Ecalle, to the framework of perverse sheaves and study its behavior under sheaf-theoretic convolution. The full fledged resurgence theory needs a (yet undeveloped) generalization of the concept of perverse sheaves allowing infinite, possibly dense, sets of singularities. We discuss possible approaches to defining such objects and some potential examples of them coming from Cohomological Hall Algebras, wall-crossing structures and Chern-Simons theory.

math.AG↗

arXiv2025

Algebra of the Infrared, secondary polytopes and perverse schobers

This survey paper, based on a talk at the International Congress of Basic Science in Beijing in July 2025, summarizes joint work of the authors with M. Kontsevich [1408.2673] establishing the relation between the ``Algebra of the Infrared" of D. Gaiotto, G. Moore and E. Witten [1506.04087] and the theory of secondary polytopes introduced in the 1990s in the study of higher-dimensional discriminants. It also summarizes subsequent work with L. Soukhanov [2011.00845] where the tunneling data were observed to be similar to linear algebra data describing perverse sheaves on the complex plane except that in the physical context vector spaces are replaced by triangulated categories. The relevant concept here is that of perverse schobers, which are conjectural categorical analogs of perverse sheaves proposed by M. Kapranov and V. Schechtman [1411.2772]. Finally, we sketch a research program of extending these ideas to $4$-dimensional theories and the resurgence formalism.

math.AG↗

arXiv2025

Moduli space of Conformal Field Theories and non-commutative Riemannian geometry

We discuss the analogy between collapsing Conformal Field Theories and measured Gromov-Hausdorff limit of Riemannian manifolds with non-negative Ricci curvature. Motivated by this analogy we propose the notion of non-commutative (``quantum") Riemannian d-geometry. We explain how this structure is related to Connes's spectral triples in the case d=1. In the Appendix based on the unpublished joint work with Maxim Kontsevich we discuss deformation theory of Quantum Field Theories as well as an approach to QFTs in the case when the space-time is an arbitrary compact metric space.

hep-th↗

arXiv2024

Motivic classes of irregular Higgs bundles and irregular connections on a curve

Let $X$ be a smooth projective curve over a field of characteristic zero and let $\mathcal D$ be an effective divisor on $X$. We calculate motivic classes of various moduli stacks of parabolic vector bundles with irregular connections on $X$ and of irregular parabolic Higgs bundles on $X$ with poles bounded by $\mathcal D$ and with fully or partially fixed formal normal forms. Along the way, we obtain several results about irregular connections and irregular parabolic Higgs bundles. In particular, we give a criterion for the existence of a connection on a higher level parabolic bundle and also develop homological algebra for irregular connections and irregular parabolic Higgs bundles. We also simplify our previous results in the regular case by re-writing the formulas for motivic classes in terms of the HLV generating function.

math.AG↗

arXiv2024

Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions

In the first of the series of papers devoted to our project ``Holomorphic Floer Theory" we discuss exponential integrals and related wall-crossing structures. We emphasize two points of view on the subject: the one based on the ideas of deformation quantization and the one based on the ideas of Floer theory. Their equivalence is a corollary of our generalized Riemann-Hilbert correspondence. In the case of exponential integrals this amounts to several comparison isomorphisms between local and global versions of de Rham and Betti cohomology. We develop the corresponding theories in particular generalizing Morse-Novikov theory to the holomorphic case. We prove that arising wall-crossing structures are analytic. As a corollary, perturbative expansions of exponential integrals are resurgent. Based on a careful study of finite-dimensional exponential integrals we propose a conjectural approach to infinite-dimensional exponential integrals. We illustrate this approach in the case of Feynman path integral with holomorphic Lagrangian boundary conditions as well as in the case of the complexified Chern-Simons theory. We discuss the arising perverse sheaf of infinite rank as well as analyticity of the corresponding ``Chern-Simons wall-crossing structure". We develop a general theory of quantum wave functions and show that in the case of Chern-Simons theory it gives an alternative description of the Chern-Simons wall-crossing structure based on the notion of generalized Nahm sum. We propose several conjectures about analyticity and resurgence of the corresponding perturbative series.

math.SG↗

arXiv2022

Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and W-algebras

A W-algebra action is constructed on the equivariant Borel-Moore homology of the Hilbert scheme of points on a nonreduced plane in three dimensional affine space, identifying it to the vacuum W-module. This is based on a generalization of the ADHM construction as well as the W-action on the equivariant Borel-Moore homology of the moduli space of instantons constructed by Schiffmann and Vasserot.

math.AG↗

arXiv2022

Analyticity and resurgence in wall-crossing formulas

We introduce the notion of analytic stability data on the Lie algebra of vector fields on a torus. We prove that the subspace of analytic stability data is open and closed in the topological space of all stability data. We formulate a general conjecture which explains how analytic stability data give rise to resurgent series. This conjecture is checked in several examples.

math.AG↗

arXiv2021

Spherical adjunctions of stable $\infty$-categories and the relative S-construction

We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and equip it with a natural paracyclic structure (``rotational symmetry''). This fulfills a part of the general program to provide a rigorous account of perverse schobers which are (thus far conjectural) categorifications of perverse sheaves. Namely, in terms of our previous identification of perverse sheaves on Riemann surfaces with Milnor sheaves, the relative $S$-construction with its paracyclic symmetry amounts to a categorification of the stalks of a Milnor sheaf at a singularity of the corresponding perverse sheaf. The action of the paracyclic rotation is a categorical analog of the monodromy on the vanishing cycles of a perverse sheaf. Having this local categorification in mind, we may view the S-construction of a spherical functor as defining a schober locally at a singularity. Each component $S_n(F)$ can be interpreted as a partially wrapped Fukaya category of the disk with coefficients in the schober and with $n+1$ stops at the boundary.

math.AG↗

arXiv2021

Perverse sheaves on Riemann surfaces as Milnor sheaves

Constructible sheaves of abelian groups on a stratified space can be equivalently described in terms of representations of the exit-path category. In this work, we provide a similar presentation of the abelian category of perverse sheaves on a stratified surface in terms of representations of the so-called paracyclic category of the surface. The category models a hybrid exit-entrance behaviour with respect to chosen sectors of direction, placing it "in between" exit and entrance path categories. In particular, this perspective yields an intrinsic definition of perverse sheaves as an abelian category without reference to derived categories and t-structures.

math.AG↗

arXiv2020

Perverse schobers and the Algebra of the Infrared

We relate the Algebra of the Infrared of Gaiotto-Moore-Witten with the theory of perverse schobers which are (conjectural, in general) categorical analogs of perverse sheaves. A perverse schober on a complex plane C can be seen as an algebraic structure that can encode various categories of D-branes of a 2-dimensional supersymmetric field theory, as well as the interaction (tunnelling) between such categories. We show that many constructions of the Algebra of the Infrared can be developed once we have a schober on C. These constructions can be seen as giving various features of the analog, for schobers, of the geometric Fourier transform well known for D-modules and perverse sheaves.

math.AG↗

arXiv2020

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds

We study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi-Yau 3-fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via "raising operators". Conjecturally the COHA action extends to an action of the Drinfeld double by adding the "lowering operators". In this paper, we show that the Drinfeld double is a generalization of the notion of the Cartan doubled Yangian defined earlier by Finkelberg and others. We extend this "$3d$ Calabi-Yau perspective" on the Lie theory furthermore by associating a root system to certain families of $X$. We formulate a conjecture that the above-mentioned action of the Drinfeld double factors through a shifted Yangian of the root system. The shift is explicitly determined by the moduli problem and the choice of stability conditions, and is expressed explicitly in terms of an intersection number in $X$. We check the conjectures in several examples, including a special case of an earlier conjecture of Costello.

math.QA↗

arXiv2020

Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve

Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$ and motivic classes of moduli stacks of semistable parabolic Higgs bundles on $(X,D)$. As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.

math.AG↗

arXiv2019

Cohomological Hall algebras, vertex algebras and instantons

We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.

math.QA↗

arXiv2017

On 2d-4d motivic wall-crossing formulas

In this paper we propose definitions and examples of categorical enhancements of the data involved in the $2d$-$4d$ wall-crossing formulas which generalize both Cecotti-Vafa and Kontsevich-Soibelman motivic wall-crossing formulas.

math.AG↗

arXiv2017

Motivic classes of moduli of Higgs bundles and moduli of bundles with connections

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to $-\infty$.

math.AG↗

arXiv2017

Airy structures and symplectic geometry of topological recursion

We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. Few other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from the general point of view of Airy structures.

math.AG↗

arXiv2016

Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for $2$-dimensional Calabi-Yau categories (with an appendix by Ben Davison)

We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the "dimensional reduction" from 3-dimensional Calabi-Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson-Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi-Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi-Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).

math.QA↗