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Maxim Kontsevich

Publications and source records attributed to Maxim Kontsevich.

At least 19 recordsLinked to original sources
arXiv2026

Towards Categorical K\"ahler Geometry

We outline the contours of an emerging theory of K\"ahler metrics in derived noncommutative geometry. This is a refinement of the theory of Bridgeland stability conditions encoding underlying differential-geometric structures. We propose elements of such a structure in both Archimedean and non-Archimedean settings, including metrized objects, mass measures satisfying a BPS inequality, harmonic metrics, minimizing flows, and complexified K\"ahler potentials. We develop the framework through examples and constructions involving Fukaya categories, quiver representations and associated C$^*$-algebras, spectral networks, and comonadic adjunctions of stable $\infty$-categories.

math.AG↗
arXiv2026

Non-commutative cluster Lagrangians

The space Loc(m,S) of rank m flat bundles on a closed surface S is K_2-symplectic. A threefold M bounding S gives rise a K_2-Lagrangian in Loc(m,S) given by the flat bundles on S extending to M. We generalize this, replacing the zero section in the cotangent bundle to M by certain singular Lagrangians. First, we introduce Q-diagrams in threefolds. They are collections Q of smooth cooriented surfaces, intersecting transversally everywhere but in a finite set of quadruple crossing points. We require that shifting any surface of the collection from such a point in the direction of its coorientation creates a simplex with the cooriented out faces. The Q-diagrams are 3d analogs of bipartite ribbon graphs. Let L be the Lagrangian in the cotangent bundle to M given by the union of the zero section and the conormal bundles to the cooriented surfaces of Q. Let X(L) be the stack of admissible dg-sheaves on M with the microlocal support in L, whose microlocalization at the conormal bundle to each cooriented surface of Q is a rank one local system. We introduce the boundary dL of L. It is a singular Lagrangian in a symplectic space, providing a symplectic stack X(dL), and a restriction functor from X(L) to X(dL). The image of the latter is Lagrangian. We show that, under mild conditions on Q, this Lagrangian has a cluster description, and so it is a K_2-Lagrangian. It also has a simple description in the non-commutative setting.

math.AG↗
arXiv2025

Atoms meet symbols

This paper introduces a novel framework for constructing invariants in $G$-equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun. We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications. In addition, we develop a separate class of purely geometric invariants for $\mathbb{Z}/2$- and $\mathbb{Z}/3$-actions on surfaces and threefolds. We provide many examples of non-$G$-linearizable $G$-actions on projective varieties treated with these new techniques.

math.AG↗
arXiv2025

Birational Invariants from Hodge Structures and Quantum Multiplication

We introduce new invariants of smooth complex projective varieties, called Hodge atoms. Their construction combines rational Gromov-Witten invariants with classical Hodge theory and relies on the notion of an F-bundle, which is a non-archimedean version of a non-commutative Hodge structure. The Hodge atoms arise from the spectral decomposition of the F-bundle under the Euler vector field action, and behave additively under blowups, in accordance with Iritani's blowup theorem. We compute several examples and demonstrate applications to birational geometry. In particular, we prove that a very general cubic fourfold is not rational. We also obtain a new proof of the equality of Hodge numbers of birational Calabi-Yau manifolds in any dimension. Furthermore, we show that the framework naturally extends to representations of other motivic Galois groups. This enables the theory of atoms to produce new obstructions to rationality over non-algebraically closed fields of characteristic zero as well.

math.AG↗
arXiv2025

Explicit formulas for arithmetic support of differential and difference operators

We compute arithmetic support of the formal deformations $D=P+tQ_1+t^2Q_2+...$ of the differential operator $P=(x\partial_x-r_1)...(x\partial_x-r_k)$, where $r_1,...,r_k\in\mathbb{Q}$ for sufficiently large primes $p$ in terms of the monodromy of $D$ in characteristic zero. An analog of these results is also provided in the case of $q$-difference operators.

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↗
arXiv2024

The miracle of integer eigenvalues

For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the linear order $Q$ with respect to linear order $P$. We show that all the eigenvalues of any such matrix $M^{X}$ are $\mathbb{Z}$-linear combinations of those variables.

math.CO↗
arXiv2023

When the Fourier transform is one loop exact?

We investigate the question: for which functions $f(x_1,...,x_n),~g(x_1,...,x_n)$ the asymptotic expansion of the integral $\int g(x_1,...,x_n) e^{\frac{f(x_1,...,x_n)+x_1y_1+...+x_ny_n}{\hbar}}dx_1...dx_n$ consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form $\{(1:x_1:...:x_n:f)\}$. We also construct various examples, in particular we prove that Kummer surface in $\mathbb{P}^3$ gives a solution to our problem.

math.AG↗
arXiv2023

Burnside rings and volume forms with logarithmic poles

We develop a theory of Burnside rings in the context of birational equivalences of algebraic varieties equipped with logarithmic volume forms. We introduce a residue homomorphism and construct an additive invariant of birational morphisms. We also define a specialization homomorphism. -- Nous proposons une th\'eorie d'anneaux de Burnside dans le contexte de la g\'eom\'etrie birationnelle des vari\'et\'es alg\'ebriques munies d'une forme volume \`a p\^oles logarithmiques. Nous introduisons un homomorphisme {\guillemotleft} r\'esidu {\guillemotright}, construisons un invariant additif des morphismes birationnels. Nous d\'efinissons aussi un homomorphisme de sp\'ecialisation.

math.AG↗
arXiv2023

Smooth Calabi-Yau structures and the noncommutative Legendre transform

We elucidate the relation between smooth Calabi-Yau structures and pre-Calabi-Yau structures. We show that, from a smooth Calabi-Yau structure on an $A_\infty$-category $A$, one can produce a pre-Calabi-Yau structure on $A$; as defined in our previous work, this is a shifted noncommutative version of an integrable polyvector field. We explain how this relation is an analogue of the Legendre transform, and how it defines a one-to-one mapping, in a certain homological sense. For concreteness, we apply this formalism to chains on based loop spaces of (possibly non-simply connected) Poincar\'e duality spaces, and fully calculate the case of the circle.

math.AT↗
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

Pre-Calabi-Yau algebras and topological quantum field theories

We introduce a notion generalizing Calabi-Yau structures on A-infinity algebras and categories, which we call pre-Calabi-Yau structures. This notion does not need either one of the finiteness conditions (smoothness or compactness) which are required for Calabi-Yau structures to exist. In terms of noncommutative geometry, a pre-CY structure is as a polyvector field satisfying an integrability condition with respect to a noncommutative analogue of the Schouten-Nijenhuis bracket. We show that a pre-CY structure defines an action of a certain PROP of chains on decorated Riemann surfaces. In the language of the cobordism perspective on TQFTs, this gives a partially defined extended 2-dimensional TQFT, whose 2-dimensional cobordisms are generated only by handles of index one. We present some examples of pre-CY structures appearing naturally in geometric and topological contexts.

math.AG↗
arXiv2021

Spectral description of non-commutative local systems on surfaces and non-commutative cluster varieties

Let R be a non-commutative field. We prove that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with (m-1)(m-2)/2 holes. Generalising this, we prove that non-commutative stacks X(m,S) of framed rank m flat R-vector bundles of on decorated surfaces S are birationally identified with the moduli spaces of flat line bundles on a spectral surface assigned to certain bipartite graphs on S. We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry canonical non-commutative Poisson structure. The space X(m,S) has a structure of a non-commutative cluster Poisson variety, equivariant under the action of the mapping class group. For bipartite graphs on a torus, we get the non-commutative dimer cluster integrable system. We define non-commutative cluster A-varieties related to bipartite ribbon graphs. They carry canonical non-commutative 2-form. The non-commutative stack A(m,S) of twisted decorated local systems on S carries a cluster A-variety structure, equivariant under the action of the mapping class group. The non-commutative cluster A-coordinates on the space A(m,S) are ratios of Gelfand-Retakh quasideterminants. In the case m=2 this recovers the Berenstein-Retakh non-commutative cluster algebras related to surfaces. We introduce stacks of admissible dg-sheaves, and use them to give an alternative proof of main results. We show that any stack of Stokes data is a stack of admissible dg-sheaves of certain type. Using this we prove that all stacks of framed Stokes data carry a cluster Poisson structure, equivariant under the wild mapping class group. Therefore they can be equivariantly quantized. Similar stacks of decorated Stokes data carry an equivariant cluster A-variety structure.

math.AG↗
arXiv2021

Wick rotation and the positivity of energy in quantum field theory

We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.

hep-th↗
arXiv2021

Multiplication kernels

We introduce the notion of multiplication kernels of birational and $D$-module type and give various examples. We also introduce the notion of a semi-classical multiplication kernel associated with an integrable system and discuss its quantization. Finally, we discuss geometric and algebraic aspects of method of separation of variables, and describe hypothetically a cyclic $D$-module for the generalized multiplication kernels for Hitchin systems for groups $GL_r$.

math.AG↗
arXiv2020

Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $ξδ$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $ξδ$-monomials. This subcomplex and a basis of $ξδ$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $ξδ$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.

math.RA↗
arXiv2020

p-Determinants and monodromy of differential operators

We prove that $p$-determinants of a certain class of differential operators can be lifted to power series over $\mathbb{Q}$. We compute these power series in terms of monodromy of the corresponding differential operators.

math.AG↗