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Vladimir Rubtsov

Publications and source records attributed to Vladimir Rubtsov.

At least 19 recordsLinked to original sources
arXiv2026

Two-Valued Groups, Chazy Equation, Dubrovin-Frobenius Structures, and QYBE

We show that the associativity condition of the universal symmetric 2-algebraic 2-valued group defined by the Buchstaber polynomial admits several mutually equivalent interpretations from the viewpoints of the Chazy equation, Gauss-Manin connections, Dubrovin-Frobenius structures, and the quantum Yang-Baxter equation. These results place the universal 2-valued law in a unified framework linking geometry, algebraic topology, group theory, and mathematical physics.

math.AG↗
arXiv2026

Microlocal index theorems and analytic torsion invariants in the geometric theory of partial differential equations

We develop a microlocal and derived-geometric framework for index theory and analytic torsion of nonlinear PDEs. By integrating Spencer hypercohomology, microlocal sheaf theory, and factorization algebras, we establish new connections between classical index theorems, BCOV invariants of Calabi-Yau manifolds, and the geometry of configuration spaces. We prove sheaf-theoretic index formulas for families of formally integrable PDEs, a microlocal index theorem for D-algebras generalizing Atiyah-Singer, and a mixed-type index theorem via microlocal stratification. We construct Ray-Singer analytic torsion for involutive systems and show that the BCOV invariant equals the Spencer torsion of the de Rham system. A categorical trace interpretation leads to a virtual index theory for derived moduli spaces of solutions. Finally, we extend the theory to configuration spaces using factorization algebras, with applications to renormalization and QFT. These results unify geometric perspectives on PDEs, torsion invariants, and moduli theory, with implications for mirror symmetry, quantum fields, and Calabi-Yau degenerations.

math.AG↗
arXiv2024

Algebraic $2-$valued group structures on $\mathbb P^1$, Kontsevich-type polynomials, and multiplication formulas, I

The theory of a two-valued algebraic group structure on a complex plane and complex projective line is developed. In this theory, depending on the choice of the neutral element, the local multiplication law is given by the Buchstaber polynomial or the generalized Kontsevich polynomial. One of the most exciting results of our studies is a simple construction of a two-valued algebraic group on $\mathbb C$ different from known coset-construction.

math.AG↗
arXiv2024

Product formulas for the Higher Bessel functions

We consider the generating function $\Phi^{(N)}$ for the reciprocals $N$-th power of factorials. We show a connection of product formulas for such series with the periods for certain families of algebraic hypersurfaces. For these families we describe their singular loci. We show that these singular loci are given by zeros of the Buchstaber-Rees polynomials, which define $N$-valued group laws. We describe a generalized Frobenius method and use it to obtain special expansions for multiplication kernels in the sense of Kontsevich. Using these expansions we provide some experimental results that connect $N$-Bessel kernels and the hierarchies of the palindromic unimodal polynomials. We study the properties of such polynomials and conjecture positivity of their roots. We also discuss the connection with Kloosterman motives as a version of the mirror duality.

math.AG↗
arXiv2023

Non-Abelian discrete Toda chains and related lattices

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several well-known (1+1) and one-dimensional lattices is discussed. In particular, we consider a non-commutative analog of the scheme: discrete Toda equations $\rightarrow$ Somos-$N$ sequences $\rightarrow$ discrete Painlev\'e equations.

nlin.SI↗
arXiv2023

Monge-Ampere Geometry and Vortices

We introduce a new approach to Monge-Ampere geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampere geometry to the Poisson equation for the pressure that arises for incompressible Navier-Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampere structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge-Ampere geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier-Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd-Beltrami-Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

math-ph↗
arXiv2022

Frolicher structures, diffieties, and a formal KP hierarchy

We propose a definition of a diffiety based on the theory of Frolicher structures. As a consequence, we obtain a natural Vinogradov sequence and, under the assumption of the existence of a suitable derivation, we can form on it a Kadomtsev-Petviashvili hierarchy which is well-posed.

nlin.SI↗
arXiv2022

Isomonodromic deformations: Confluence, Reduction $\&$ Quantisation

In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the Takiff algebra (i.e. truncated current algebra). Our motivation is to produce confluent versions of the celebrated Knizhnik--Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic $τ$-function. In order to achieve this, we study the confluence cascade of $r+ 1$ simple poles to give rise to a singularity of arbitrary Poincaré rank $r$ as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians.

math.AG↗
arXiv2022

On $(t_2,t_3)-$Zakharov-Shabat equations of generalized Kadomtsev-Petviashvili hierarchies

We review the integration of the KP hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation; an algebra of functions equipped with a derivation that generalizes the gradient operator; an algebra of quaternion-valued functions; a differential Lie algebra; an algebra of polynomials equipped with the Pincherle differential; a Moyal algebra. In all these cases we can formulate and solve the Cauchy problem of the KP hierarchy. Also, in each of these cases we derive different Zakharov-Shabat $(t_2,t_3)$-equations -- that is, different Kadomtsev-Petviashvili equations -- and we prove existence of solutions arising from solutions to the corresponding KP hierarchy.

nlin.SI↗
arXiv2021

On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators

We study the existence and uniqueness of the Kadomtsev-Petviashvili (KP) hierarchy solutions in the algebra of $\F Cl(S^1,\K^n)$ of formal classical pseudo-differential operators. The classical algebra $ΨDO(S^1,\K^n)$ where the KP hierarchy is well-known appears as a subalgebra of $\F Cl(S^1,\K^n).$ We investigate algebraic properties of $\F Cl(S^1,\K^n)$ such as splittings, r-matrices, extension of the Gelfand-Dickii bracket, almost complex structures. Then, we prove the existence and uniqueness of the KP hierarchy solutions in $\F Cl(S^1,\K^n)$ with respect to extended classes of initial values. Finally, we extend this KP hierarchy to complex order formal pseudo-differential operators and we describe their Hamiltonian structures similarly to previously known formal case.

math-ph↗
arXiv2020

Dualities for rational multi-particle Painlevé systems: Spectral versus Ruijsenaars

The extension of the Painlevé-Calogero coorespondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlevé equations which may be obtained by the procedure of Hamiltonian reduction applied to the matrix or non-commutative Painlevé systems, which also gives isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide here dual systems for the rational multi-particle Painlevé systems (PI,PII and PIV) by reduction from another intersection a coadjoint orbit of GL(n) action with the level set of moment map. We describe this duality in terms of the spectral curve of non-reduced system in comparison to the Ruijsenaars duality.

math-ph↗
arXiv2019

Noncommutative cross-ratio and Schwarz derivative

We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to "noncommutative elementary geometry" and relate it to noncommutative integrable systems. We also provide a noncommutative version of the celebrated "pentagramma mirificum".

math.RA↗
arXiv2017

Lie algebroid cohomology and Lie algebroid extensions

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.

math.RA↗
arXiv2017

Algebras of quantum monodromy data and decorated character varieties

The Riemann-Hilbert correspondence is an isomorphism between the de Rham moduli space and the Betti moduli space, defined by associating to each Fuchsian system its monodromy representation class. In 1997 Hitchin proved that this map is a symplectomorphism. In this paper, we address the question of what happens to this theory if we extend the de Rham moduli space by allowing connections with higher order poles. In our previous paper arXiv:1511.03851, based on the idea of interpreting higher order poles in the connection as boundary components with bordered cusps (vertices of ideal triangles in the Poincaré metric) on the Riemann surface, we introduced the notion of decorated character variety to generalize the Betti moduli space. This decorated character variety is the quotient of the space of representations of the fundamental groupid of arcs by a product of unipotent Borel sub-groups (one per bordered cusp). Here we prove that this representation space is endowed with a Poisson structure induced by the Fock--Rosly bracket and show that the quotient by unipotent Borel subgroups giving rise to the decorated character variety is a Poisson reduction. We deal with the Poisson bracket and its quantization simultaneously, thus providing a quantisation of the decorated character variety. In the case of dimension 2, we also endow the representation space with explicit Darboux coordinates. We conclude with a conjecture on the extended Riemann-Hilbert correspondence in the case of higher order poles.

math.AG↗
arXiv2016

Painlevé monodromy manifolds, decorated character varieties and cluster algebras

In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlevé differential equations. Since all Painlevé differential equations (apart from the sixth one) exhibit Stokes phenomenon, it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character is defined as complexification of the bordered cusped Teichmüller space introduced in arXiv:1509.07044. We show that the decorated character variety of a Riemann sphere with s holes and n>1 bordered cusps is a Poisson manifold of dimension 3 s+ 2 n-6 and we explicitly compute the Poisson brackets which are naturally of cluster type. We also show how to obtain the confluence procedure of the Painlevé differential equations in geometric terms.

math-ph↗
arXiv2014

Nonabelian holomorphic Lie algebroid extensions

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild-Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.

math.SG↗
arXiv2014

Quantization and dynamisation of Trace-Poisson brackets

The quantization problem for the trace-bracket algebra, derived from double Poisson brackets, is discussed. We obtain a generalization of the boundary YBE (or so-called ABCD-algebra) for the quantization of quadratic trace-brackets. A dynamical deformation is proposed on the lines of Gervais-Neveu-Felder dynamical quantum algebras.

math-ph↗