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A. Zuevsky

Publications and source records attributed to A. Zuevsky.

31 records · Page 2Linked to original sources
arXiv2023

Relativity of reductive chain complexes of non-abelian simplexes

Chain total double complexes with reductive differentials for non-abelian simplexes with associated spaces are considered. It is conjectured that corresponding relative cohomology is equivalent to the coset space of vanishing over non-vanishing functionals related to differentials of complexes. The conjecture is supported by the theorem for the case of spaces of correlation functions and generalized connections on vertex operator algebra bundles.

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arXiv2023

Cosimplicial cohomology of restricted meromorphic functions on foliated manifolds

Starting from the axiomatic description of meromorphic functions with prescribed analytic properties, we introduce the cosimplicial cohomology of restricted meromorphic functions defined on foliations of smooth complex manifolds. Spaces for double chain-cochain complexes and coboundary operators are constructed. Multiplications of several restricted meromorphic functions with non-commutative parameters, as well as for elements of double complex spaces are introduced and their properties are discussed. In particular, we prove that the construction of invariants of cosimplicial cohomology of restricted meromorphic functions is non-vanishing, independent of the choice of the transversal basis for a foliation, and invariant with respect to changes of coordinates on a smooth manifold and on transversal sections. As an application, we provide an example of general cohomological invariants, in particular, generalizing the Godbillon--Vay invariant for codimension one foliations.

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arXiv2022

Product-type classes for vertex algebra cohomology of foliations on complex curves

We define a product of pairs of double complex spaces $C^n_m(V, \mathcal{W}, \mathcal{F})$ for gradig-restricted vertex algebra cohomology of codimension one foliation on a complex curve. We introduce a vertex algebra counterpart of the classical %product-type cohomological class using the orthogonality conditions on elements of double complex spaces with respect to the product we introduced.

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arXiv2022

K-theory cohomology of associative algebra twisted bundles

We introduce and study a $K$-theory of twisted bundles for associative algebras $A(\mathfrak g)$ of formal series with an infinite-Lie algebra coefficients over arbitrary compact topological spaces. Fibers of such bundles are given by elements of algebraic completion of the space of all formal series in complex parameters, sections are provided by rational functions with prescribed analytic properties. In this paper we introduce and study K-groups $K(A(\mathfrak g), X)$ of twisted $A(\mathfrak g)$-bundles as equivalence classes $[\mathcal{E}]$ of $A(\mathfrak g)$-bundle $\mathcal{E}$. We show that for any twisted $A(\mathfrak g)$-bundle $\mathcal{E}$ there exist another bundle $\widetilde{\mathcal{E}}$ such that an element of $K(A(\mathfrak g), X)$ for $\mathcal{E}$ can be represented in the form $[\mathcal{E}]/[\widetilde{\mathcal{E}}]$. The group $K(A(\mathfrak g), X)$ homomorphism properties with respect to tensor product, and splitting properties with respect to reductions of $X$ into base points. We determine also cohomology of cells of K-groups for the factor $X/Y$ of two compact spaces $X$ and $Y$.

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arXiv2022

Canonical torsor bundles of prescribed rational functions on complex curves

Prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. We define and construct explicitly prescribed rational functions-valued bundle $\mathcal{W}_M$ over a smooth complex curve $M$. An intrinsic coordinate-independent formulation for such bundle is is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for purposed of conformal field theory, deformation theory, and the theory of foliations.

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arXiv2021

Reduction cohomology of Riemann surfaces

We study the algebraic conditions leading to the chain property of complexes for vertex operator algebra $n$-point functions with differential being defined through reduction formulas. The notion of the reduction cohomology of Riemann surfaces is introduced. Algebraic, geometrical, and cohomological meanings of reduction formulas is clarified. A counterpart of the Bott-Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven. It is shown that the reduction cohomology is given by the cohomology of $n$-point connections over the vertex operator algebra bundle defined on a genus $g$ Riemann surface $Σ^{(g)}$. The reduction cohomology for a vertex operator algebra with formal parameters identified with local coordinates around marked points on $Σ^{(g)}$ is found in terms of the space of analytical continuations of solutions to Knizhnik-Zamolodchikov equations. For the reduction cohomology, the Euler-Poincare formula is derived. Examples for various genera and vertex operator cluster algebras are provided.

math.FA↗
arXiv2021

Cosimplicial meromorphic functions cohomology on complex manifolds

Developing ideas of \cite{Fei}, we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold $M$. Graded differential cohomology of a sheaf of Lie algebras $\mathcal G$ via the cosimplicial cohomology of $\mathcal G$-formal series for any covering by Stein spaces on $M$ is computed. A relation between cosimplicial cohomology (on a special set of open domains of $M$) of formal series of an infinite-dimensional Lie algebra $\mathcal G$ and singular cohomology of auxiliary manifold associated to a $\mathcal G$-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.

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arXiv2021

Associative algebra twisted bundles over compact topological spaces

For the associative algebra $A(\mathfrak g)$ of an infinite-dimensional Lie algebra $\mathfrak g$, we introduce twisted fiber bundles over arbitrary compact topological spaces. Fibers of such bundles are given by elements of algebraic completion of the space of formal series in complex parameters, sections are provided by rational functions with prescribed analytic properties. Homotopical invariance as well as covariance in terms of trivial bundles of twisted $A(\mathfrak g)$-bundles is proven. Further applications of the paper's results useful for studies of the cohomology of infinite-dimensional Lie algebras on smooth manifolds, $K$-theory, as well as for purposes of conformal field theory, deformation theory, and the theory of foliations are mentioned.

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arXiv2021

Foliation of a space associated to vertex operator algebra

We construct the foliation of aspace associated to correlation functions of vertex operator algebras on considered on Riemann surfaces. We prove that the computation of general genus $g$ correlation functions determines a foliation on the space associated to these correlation functions a sewn Riemann surface. Certain further applications of the definition are proposed.

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arXiv2020

Schottky vertex operator cluster algebras

Using recursion formulas for vertex operator algebra higher genus characters with formal parameters identified with local coordinates around marked points on a Riemann surface of arbitrary genus, we introduce the notion of a vertex operator cluster algebra structure. Cluster elements and mutation rules are explicitly defined, and the simplest example of a vertex operator cluster algebra is presented.

math.FA↗
arXiv2020

The simplest cohomological invariants for vertex algebras

For the double complex structure of grading-restricted vertex algebra cohomology defined in \cite{Huang}, we introduce a multiplication of elements of double complex spaces. We show that the orthogonality and bi-grading conditions applied on double complex spaces, provide in relation among mappings and actions of co-boundary operators. Thus, we endow the double complex spaces with structure of bi-graded differential algebra. We then introduce the simples cohomology classes for a grading-restricted vertex algebra, and show their independence on the choice of mappings from double complex spaces. We prove that its cohomology class does not depend on mappings representing of the double complex spaces. Finally, we show that the orthogonality relations together with the bi-grading condition bring about generators and commutation relations for a continual Lie algebra.

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arXiv2009

Hardy Spaces on Compact Riemann Surfaces with Boundary

We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles $X_1$ and $X_2$ of Riemann surfaces we define the vector bundle on the second double as a direct image of the vector bundle on first double. %% We choose line bundles of half-order differentials $Δ_1$ and $Δ_2$ so that the vector bundle $V^{X_2}_{χ_\ad} \otimes Δ_2$ on $X_2$ would be the direct image of the vector bundle $V^{X_1}_{χ_\ao} \otimes Δ_1$. We then show that the Hardy spaces $H_{2, J_1(p)} (S_1,V_{χ_\ao} \otimes Δ_1)$ and $H_{2,J_2(p)} (S_2,V_{χ_\ad} \otimes Δ_2)$ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation $χ_\ad$ of the fundamental group $\piod(X_2, p_0)$ given a matrix representation $χ_\ao$ of the fundamental group $\piod(X_1, p'_0)$. %% On the basis of the results of \cite{vin} and Theorem \ref{theorem_1} proven in the present work we then conjecture that there exists a covariant functor from the category ${\cal RH}$ of finite bordered surfaces with vector bundle and signature matrices to the category of Kre\uın spaces and isomorphisms which are ramified covering of Riemann surfaces.

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