arXiv subjects

Mikhail Kapranov

Publications and source records attributed to Mikhail Kapranov.

At least 19 recordsLinked to original sources

arXiv2026

Supersymmetry, differential operators of infinite order and theta functions

In 1972, M. Sato proposed an approach to proving modularity of forms like Thetanullwerte by characterizing them via certain differential operators of infinite order (DOI) in the modular variable(s) alone. A DOI is an infinite series in derivatives decreasing so fast that it acts on holomorphic functions by a sheaf morphism. This approach was developed by several authors including Kashiwara, Kawai, Takei and Yoshida. We give an interpretation of this approach using supersymmetry which provides a natural source of DOIs: the naive exponential of any odd supersymmetry generator is a DOI. The case of the Riemann theta function of genus n is governed by the supergroup OSp(1|2n) (and its metaplectic cover) acting on a natural super-thickening of the Siegel plane. For n=2 this is the 3-dimensional N=1 superconformal group and the structure at hand is precisely the free massless scalar supermultiplet (combining the Laplace and Dirac equations). For n>2 we get a super-extension of the generalized conformal structure existing on the Lagrangian Grassmannian as on any Hermitian symmetric space. An additional interesting feature here is that the odd supersymmetry generators acting ``on-shell'' (i.e., in the space of solutions of the equations of motion) satisfy even-style Heisenberg commutation relations. These equations of motion upgrade to a complex of differential operators corresponding to a natural BGG-type resolution of the super-Weil representation of osp(1|2n).

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↗

arXiv2024

The Langlands formula and perverse sheaves

For a complex reductive Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$ we consider the category $\text{Perv}(W \backslash \mathfrak{h})$ of perverse sheaves on $W \backslash \mathfrak{h}$ smooth w.r.t. the natural stratification. We construct a category $\boldsymbol{\mathcal{C}}$ such that $\text{Perv}(W\backslash \mathfrak{h})$ is identified with the category of functors from $\boldsymbol{\mathcal{C}}$ to vector spaces. Objects of $\boldsymbol{\mathcal{C}}$ are labelled by standard parabolic subalgebras in $\mathfrak{g}$. It has morphisms analogous to the operations of parabolic induction (Eisenstein series) and restriction (constant term) of automorphic forms. In particular, the Langlands formula for the constant term of an Eisenstein series has a counterpart in the form of an identity in $\boldsymbol{\mathcal{C}}$. We define $\boldsymbol{\mathcal{C}}$ as the category of $W$-invariants (in an appropriate sense) in the category $Q$ describing perverse sheaves on $\mathfrak{h}$ smooth w.r.t. the root arrangement. This matches, in an interesting way, the definition of $W \backslash \mathfrak{h}$ itself as the spectrum of the algebra of $W$-invariants.

math.RT↗

arXiv2023

N-spherical functors and categorification of Euler's continuants

Euler's continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their categorical lifts which are natural complexes (more precisely, coherently commutative cubes) of functors involving compositions of a given functor and its adjoints of various orders, with the differentials built out of units and counits of the adjunctions. In the stable infinity-categorical context these complexes/cubes can be assigned totalizations which are new functors serving as higher analogs of the spherical twist and cotwist. We define N-spherical functors by vanishing of the twist and cotwist of order N-1 in which case those of order N-2 are equivalences. The usual concept of a spherical functor corresponds to N=4. We characterize N-periodic semi-orthogonal decompositions of triangulated (stable infinity-) categories in terms of N-sphericity of their gluing functors. The procedure of forming iterated orthogonals turns out to be analogous to the procedure of forming a continued fraction.

math.CT↗

arXiv2022

PROBs and perverse sheaves II. Ran spaces and 0-cycles with coefficients

We consider the space Z(C,L) of 0-cycles on the complex line C with coefficients in a commutative monoid L subject to certain conditions. Such spaces include the symmetric products (for L=Z_+) and the Ran space (for L=T={ True, False} being the Boolean algebra of truth values). We describe the appropriately defined category of perverse sheaves on Z(C,L) in terms of the braided category (PROB) generated by the components of the universal $L$-graded bialgebra. We give another description in terms of so-called Janus sheaves which are objects of mixed functoriality (data covariant in one direction and contravariant in the other) on a category formed by certain matrices with entries in L. The matrices in question are analogs of contingency tables familiar in statistics.

math.CT↗

arXiv2022

Fourier transform on hyperplane arrangements

We consider the category of perverse sheaves on a complex vector space smooth with respect to a stratification given by an arrangement of hyperplanes with real equations. As shown in an earlier wotk of two of the authors, this category can be described in terms of certain diagrams of vector spaces labelled by all the faces of the real arrangement (we call such diagrams hyperbolic sheaves). In this paper we calculate, in these terms, several fundamental operations of sheaf theory such as forming the space of vanishing cycles, specialization and the Fourier-Sato transform.

math.AT↗

arXiv2022

The cohomological Hall algebra of a surface and factorization cohomology

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall multiplication. This multiplication involves data (virtual pullbacks) governing the derived moduli stack, i.e., the perfect obstruction theory naturally existing on the non-derived stack. By restricting to sheaves with support of given dimension, we obtain several types of Hecke operators. In particular, we study R(S), the Hecke algebra of 0-dimensional sheaves. For the flat case S=A^2, we show that R(S) is an enveloping algebra and identify it, as a vector space, with the symmetric algebra of an explicit graded vector space. For a general S we find the graded dimension of R(S), using the techniques of factorization cohomology.

math.AG↗

arXiv2022

Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras

Let X be a smooth affine variety over a field k of characteristic 0 and T(X) be the Lie algebra of regular vector fields on X. We compute the Lie algebra cohomology of T(X) with coefficients in k. The answer is given in topological terms relative to any embedding of k into complex numbers and is analogous to the classical Gelfand-Fuks computation for smooth vector fields on a C-infinity manifold. Unlike the C-infinity case, our setup is purely algebraic: no topology on T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.

math.AG↗

arXiv2021

Parabolic induction and perverse sheaves on h/W

For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h and Weyl group W, we describe the category of perverse sheaves on h/W smooth w.r.t the natural stratification. The answer is given in terms of mixed Bruhat sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed Bruhat sheaves with the properties of various procedures of parabolic induction and restriction that connect different Levi subgroups in G.

math.AT↗

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

PROBs and perverse sheaves I. Symmetric products

Algebraic structures involving both multiplications and comultiplications (such as, e.g., bialgebras or Hopf algebras) can be encoded using PROPs (categories with PROducts and Permutations) of Adams and MacLane. To encode such structures on objects of a braided monoidal category, we need PROBs (braided analogs of PROPs). Colored PROBs correspond to multi-sorted structures. In particular, we have a colored PROB B governing non-negatively graded bialgebras in braided categories. As a category, B splits into blocks B_n according to the grading. We relate B_n with the category P_n of perverse sheaves on the n-th symmetric product of the complex line, smooth with respect to the natural stratification by multiplicities. More precisely, we show that P_n is equivalent to the category of functors from B_n to vector spaces. This gives a natural quiver description of P_n.

math.CT↗

arXiv2021

Conformal maps in higher dimensions and derived geometry

By Liouville's theorem, in dimensions 3 or more conformal transformations form a finite-dimensional group, an apparent drastic departure from the 2-dimensional case. We propose a derived enhancement of the conformal Lie algebra which is an infinite-dimensional dg-Lie algebra incorporating not only symmetries but also deformations of the conformal structure. Our approach is based on (derived) deformation theory of the ambitwistor space of complex null-geodesics.

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

Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof)

Motivated by applications to perverse sheaves, we study combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition, goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the ''contingency decomposition'' to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system $A_n$) introduced by T.K. Petersen. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the ''meta-matrix'' formed by the numbers of contingency matrices of various sizes, is totally positive.

math.GT↗

arXiv2020

Shuffle algebras and perverse sheaves

We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct an equivalence between: (i) Braided Hopf algebras of a certain type. (ii) Factorizable collections of perverse sheaves on all the symmetric products. Under this eqiuvalence, the Nichols algebra associated to an object V corresponds to the collection of the intersection cohomology extensions of the local systems on the open configuration spaces associated to the tensor powers of V. Our approach is based on using real skeleta of complex configuration spaces.

math.AT↗

arXiv2019

Perverse sheaves over real hyperplane arrangements II

Let H be an arrangement of hyperplanes in R^n and Perv(C^n,H) be the category of perverse sheaves on C^n smooth with respect to the stratification given by complexified flats of H. We give a description of Perv(C^n,H) in terms of "matrix diagrams", i.e., diagrams formed by vector spaces E_{AB} labelled by pairs A,B of real faces of H (of all dimensions) or, equivalently, by the cells iA+B of a natural cell decomposition of C^n. A matrix diagram is formally similar to a datum describing a constructible (non-perverse) sheaf but with the direction of one half of the arrows reversed.

math.AT↗