arXiv subjects

A. Zuevsky

Publications and source records attributed to A. Zuevsky.

At least 19 recordsLinked to original sources

arXiv2026

Modular properties of affine $\SLA{sl}{2}$ torus $n$-point functions

Krauel, Shafiq and Wood realized torus $1$-point functions for the simple affine \voa{}s $L(k,0)$ built from $\alg{sl}(2)$ as vector-valued modular forms attached to a cyclic $R$-module structure and to the modular tensor category $\rep{L(k,0)}$. We extend this to torus $n$-point functions, defined as traces of chains of $n$ intertwining operators winding once around the $\tau$-cycle. Every coefficient of their local (Laurent, or Puiseux) expansion about a diagonal stratum is again a \vvmf{}, lying, whenever a mild lowest-weight hypothesis holds, in the very $R$-modules of \cite{KSW}. Granted a torus-primarity hypothesis verified explicitly below for $\alg{sl}(2)$, the operator product expansion then reduces leading short-distance behaviour to the classified $1$-point theory. For $\alg{sl}(2)$ we classify the fusion-chain conformal block spaces, identify torus $n$-point primary vectors, and treat $n=2$ in detail, obtaining an explicit $(k+1)$-dimensional family of vector-valued Jacobi-type forms generalizing the level-$k$ forms $\eta^{3k/2}$ of \cite{KSW}. We also extend \KSW's{} categorical picture, representations of $\Bthree=\PMod(\surf{1}{1})$ built from a modular tensor category, to representations, on fusion-chain block spaces, of a mapping-class subgroup of the $n$-punctured torus generated by $S$, $T$, and adjacent braidings, with a Verlinde-type dimension formula and categorical $S$-, $T$-operators. The explicit $\alg{sl}(2)$ matrix form of $S$ beyond this abstract construction remains open.

math.FA↗

arXiv2026

Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension

We study the exponential growth of mean multiplicities (EGMM) in the geodesic length spectrum of a semi-arithmetic Fuchsian group $\Gamma$ of finite covolume and arithmetic dimension $r\geq 1$ admitting a generalized modular embedding into $(\pm\bH)^{r-1}$. We introduce two new ingredients. First, a {multi-dimensional Schwarz-Pick contraction lemma}: the generalized modular embedding $F:\bH\to\bH^{r-1}$, being holomorphic and strictly contracting with respect to the product Kobayashi metric, satisfies $\norm{DF_z}_{\mathrm{op}}\leq\sqrt{r-1}\,(1-\delta)$ for a uniform $\delta=\delta(\Gamma)>0$ and all $z\in\bH$. Second, a geometry-of-numbers norm-form estimate: Minkowski's theorem applied to the lattice of algebraic integers in the invariant trace field $K$ gives $\#(\cL(\Gamma)\cap[N-1,N])\leq CN^{(r-1)^{3/2}(1-\delta)}$ for all $N$; unlike the case $r\leq 2$, this exponent depends on $r$. Combining the two ingredients shows that $\Gamma$ has EGMM whenever $r\leq 2$, or $r\geq 3$ and $\delta$ satisfies the {strong contraction condition} $\delta>1-\bigl(\sqrt2\,(r-1)\bigr)^{-1}$, the sharpest threshold our method gives, obtained by a refined geometry-of-numbers argument (Proposition \ref{propsharp}) that improves on the cruder exponent $(r-1)^{3/2}$ obtained directly from the norm form (the two coincide exactly at $r=3$).

math.FA↗

arXiv2026

Trace maps on chiral Clifford algebras for the rank two fermionic vertex operator superalgebra

For a holomorphic vector bundle $F$ of rank $r$ on a smooth Riemann surface $X$ we construct a trace map on the chiral homology of the chiral Clifford algebra $\CE$ attached to the purely odd bundle $E=\Pi(F\oplus F^\vee\otimes\omega_X)$. It is the chiral-algebraic realization of the rank two fermionic vertex operator superalgebra. The free-fermion (bc-type) conformal field theory built from a dual pair of odd fields $ \beta_i\in F$, $\gamma^j\in F^\vee\otimes\omega_X$. We give a complete construction of this vertex operator superalgebra, its associated vertex superalgebra bundle, and the isomorphism between the latter's chiral algebra and the chiral envelope $\CE$. Using the Batalin-Vilkovisky (BV) formalism together with Feynman diagrams we prove that the resulting trace map \[ \Trch : \bigl(\widetilde\sC^{\ch}(X,\CE)_{\sQ},\, \dch_{\CE}\bigr)\longrightarrow (\OBV,-\DBV) \] is a chain map satisfying a generalized quantum master equation and is a quasi-isomorphism, generalizing to the odd/Clifford setting the trace map on chiral Weyl algebras constructed by Gui for symplectic bosons. We establish existence, homotopy uniqueness, and functoriality (including explicit metric-independence up to chain homotopy) of the trace map, prove cyclicity of the relevant supertrace, and verify nilpotency of $\DBV$, the graded Leibniz rule, $d^2=0$ for every differential introduced. As an application we compute the trace map on a modified affine current and on a modified energy-momentum tensor, recovering, purely algebraically from the chiral chain complex, Fay's classical formulas for the variation of the fermionic (Ray-Singer) analytic torsion along the moduli of the bundle $F$ and along the moduli of the curve $X$.

math.FA↗

arXiv2026

The first chiral homology group in higher genus

We extend the theory of the first chiral homology group of vertex algebras, developed by van Ekeren and Heluani for elliptic curves, to compact Riemann surfaces of arbitrary genus. Our approach realizes a genus $g$ surface by iterated self-sewing of $g$ handles onto the Riemann sphere, each governed by a sewing parameter $\rho_i$ in a punctured disc, so that the construction of van Ekeren-Heluani is recovered. We construct an explicit complex computing the chiral homology groups $H^{\mathrm{ch}}_0$ and $H^{\mathrm{ch}}_1$ of a vertex algebra $V$ on a genus $g$ surface with $n$ marked points, equip it with a projectively flat connection, with an explicit central-charge anomaly, over the $g$-dimensional space of sewing parameters, and prove a genus $g$ Fourier-space Borcherds identity for the associated modified vertex operators. We show that the same two finiteness hypotheses isolated by van Ekeren and Heluani in genus $1$ - finite dimensionality of the first Poisson homology $\HP_1(R_V)$ of the Zhu $C_2$- algebra, and finite generation of a certain Koszul homology of the associated graded algebra - imply finite dimensionality of $H^{\mathrm{ch}}_1(X,V)$ for every genus $g$ and every vertex algebra $V$, answering a question left open in their work. Using the degeneration $\rho_i \to 0$ together with the factorization theorem of Damiolini- Gibney-Tarasca, we relate the totally degenerate limit of $H_1^{\mathrm{ch}}$ to the Hochschild homology of an iterated construction on the Zhu algebra, and deduce vanishing of the first chiral homology group in every genus for the same classically free, rational vertex algebras treated in genus one.

math.FA↗

arXiv2026

Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups

We generalize the geometric construction of vertex operator algebra (VOA) bundles and their associated automorphic forms from the elliptic modular curve to arbitrary Fuchsian groups $ \Gamma \subset \mathrm{PSL}_2(\mathbb{R})$. A sharp topological dichotomy emerges regarding the existence of a holomorphic weight-$2$ quasi-automorphic generator $E_2^\Gamma$. When $\Gamma$ has a cusp, we construct $E_2^\Gamma$ via the analytic continuation of parabolic Eisenstein series and prove that the space of quasi-automorphic forms is a free polynomial extension, allowing the algebraic setup of the genus one theory, including the quasi-VOA structure and the characterization of strict automorphic forms via a lowering operator. Conversely, when $\Gamma$ is cocompact of genus $g\ge 2$, Atiyah's theorem on holomorphic connections rigorously obstructs the existence of $E_2^\Gamma$. For this obstructed case, we provide exact dimension formulas that link the shortage in lifting quasi-automorphic forms directly to the failure of quasi-primarity within the VOA, fully resolving the torsion-free case and conjecturing the extension to groups with elliptic points.

math.FA↗

arXiv2026

Hardy spaces on Riemann surfaces under ramified coverings

We extend the theory of indefinite Hardy spaces on finite bordered Riemann surfaces to the setting of ramified analytic coverings. Given a finite $n$-sheeted ramified covering $F\colon S_1\to S_2$ of finite bordered Riemann surfaces satisfying a spin-compatibility hypothesis, we construct (i) the direct image of a unitary flat vector bundle $\VxX{1}\otimes \Del{1}$ on the double $X_1$ under $F$, taking full account of the ramification divisor $R_F$ and establishing the extension across the branch locus via a local analysis; (ii) a canonical matrix function $G_2$ encoding the parahermitian structure on $X_2$, together with the induced representation $\chi_2$ of $\piX{X_2}{p_0}$; (iii) an explicit isometric isomorphism $\phi_F\colon H^{2,J_1(p)}(S_1,\VxS{1}\otimes\Del{1}) \xrightarrow{\;\sim\;} H^{2,J_2(p)}(S_2,\VxS{2}\otimes\Del{2})$ between the associated Hardy-Kre\u{\i}n spaces, provided that $h^0(X_1,\VxX{1}\otimes\Del{1})=0$ and that the branch locus is disjoint from $\partial S_2$. We then develop the resulting operator theory in terms of vessels and Bezoutian operators. To each object in the category $\mathcal{RH}$ of finite bordered surfaces with unitary flat bundles we attach a triangular vessel whose input and output spaces are the Hardy-Kre\u{\i}n spaces on the two surfaces. The Bezoutian of the vessel is expressed as a finite-rank operator on $\mathcal{H}_2$ whose kernel is built from bounded holomorphic point-evaluation functionals in $\mathcal{H}_2$ evaluated at the interior ramification images $F(r_\nu)\in S_2$, consistently with the boundary-transversality hypothesis $\partial S_2\cap B_F=\left\{\varnothing\right\}$. We prove that the assignment $(S,\Vx{},J)\mapsto H^{2,J(p)}(S,\Vx{}\otimes\Delta)$ extends to a covariant functor from $\mathcal{RH}$ (with ramified morphisms) to the category of Kre\u{\i}n spaces.

math.FA↗

arXiv2026

Topological invariant responsible for the integer QHE and non-commutative geometry

We consider a wide class of $2D$ tight - binding models of solid state physics. These models are, in the most general case, non - homogeneous. The topological invariant ${\cal N}_3$ responsible for the quantization of the Hall conductivity, for the specific case of the integer quantum Hall effect in $2D$, is expressed through the Wigner transformation of the two-point electron Matsubara Green function. We express this invariant as a pairing of the element of the $K^{-1}$ group (generated by the Green function) with the specific element of the cyclic cohomology group $HC^3$. According to a set of local index theorems the values of ${\cal N}_3$ can be shown to be integer for a limited class of tight - binding models.

cond-mat.mes-hall↗

arXiv2025

Determinant representations for Garvan formulas

In this note, we demonstrate how determinant representations for correlation functions in conformal field theory can be used to derive explicit determinant formulas for powers of the classical $\eta$-function, expressed via deformed elliptic functions with parameters. In particular, we obtain counterparts of Garvan's formulas for the modular discriminant corresponding to the genus two Riemann surface case.

math.FA↗

arXiv2024

Torsor structure of level-raising operators

We consider families of reductive complexes related by level-raising operators and originating from an associative algebra. In the main theorem it is shown that the multiple cohomology of that complexes is given by the factor space of products of reduction operators. In particular, we compute explicit torsor structure of the genus $g$ multiple cohomology of the families of horizontal complexes with spaces of of canonical converging reductive differential forms for a $C_2$-cofinite quasiconformal strong-conformal field theory-type vertex operator algebra associated to a complex curve. That provides an equivalence of multiple cohomology to factor spaces of products of sums of reduction functions with actions of the group of local coordinates automorphisms.

math.FA↗

arXiv2024

Sequences of multiple products and cohomology classes for foliations of complex curves

The idea of transversality is explored in the construction of cohomology theory associated to regularized sequences of multiple products of rational functions associated to vertex algebra cohomology of codimension one foliations on complex curves. Explicit formulas for cohomology invariants results from consideration transversality conditions applied to sequences of multiple products for elements of chain-cochain transversal complexes defined for codimension one foliations.

math.FA↗

arXiv2024

Invariant classes for families of complexes

We consider families of chain-cochain infinite complexes $\mathcal C$ of spaces with elements depending on a number of parameters, and endowed with a converging associative multiple product. The existence of left/right local/non-local square-vanishing ideals is assumed for subspaces of $\mathcal C$-spaces. We show that a set of differential and orthogonality relations together with coherence conditions on indices of a chain-cochain complex $\mathcal C$ elements generates families of graded differential algebras. With the appropriate orthogonality conditions on completions of $\mathcal C$ elements in the multiple product, we define the equivalence classes of cohomology invariants.

math.FA↗

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.

math.FA↗

arXiv2022

Multiple products of meromorphic functions

Let $\mathfrak g$ be an infinite-dimensional Lie algebra and let $G$ be the algebraic completion of a graded $\mathfrak g$-module $W$. Using the Schottky uniformization of the Riemann sphere as a geometric model for a genus $\kappa$ Riemann surface, we construct a $\kappa$-parameter family of extended coboundary operators $\widetilde\delta^n_m(\rho_1,\ldots,\rho_\kappa)$ acting on the double complex of predetermined meromorphic functions on the configuration space $F_n\C$ with values determined by $G$. The extension is realized as a graded trace, defined coordinate-freely as the trace of a canonically associated finite-rank endomorphism of each homogeneous component $W_{(k)}$ of $W$, of the classical coboundary operator, the $\kappa$ sewing loci being held disjoint from the free marked points at which the classical differential acts. The sewing operator is exhibited as a chain map between explicitly defined complexes. We give a complete proof of the resulting chain property $\widetilde\delta^{n+1}_{m-2\kappa-1}\circ\widetilde\delta^n_m=0$ and of the convergence of the defining power series in the sewing parameters $\rho_p$ under an explicit growth hypothesis, and we determine precisely how the construction depends on the auxiliary choice of local coordinates and sewing annuli. Applications of the resulting cohomology theory - to the sheaf of conformal blocks on the Deligne-Mumford moduli space of stable curves, to secondary characteristic classes of holomorphic foliations, to graded trace functions arising in the description of topological phases of matter, and to integrable hierarchies of Toda type - are proposed and discussed as motivation, without claiming these correspondences as theorems of the present paper.

math.FA↗

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.

math.FA↗

arXiv2022

On holonomy groupoid of vertex operator algebra bundles on foliations

For a foliation $\F$ defined on a smooth complex manifold $M$ we introduce the category of vertex operator algebra $V$ bundles with sections provided by vectors of elements of the space of algebraically extended $V$-module $W$-valued differentials. An intrinsic coordinate-independent formulation for such bundles is given. Finally, we identify the cohomology of the spaces of sections for a vertex operator algebra $V$ bundle with vertex operator algebra cohomology of the holonomy groupoid $Hol(M, \F)$.

math.FA↗

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 $Y$.

math.FA↗

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.

math.FA↗

arXiv2022

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.

math.FA↗