Narrow Results

In this paper, we describe Zhu recursion for a vertex operator algebra (VOA) and its modules on a genus $g$ Riemann surface in the Schottky uniformization. We show that $n$-point (intertwiner) correlation functions can be written as linear combinations of $(n-1)$-point functions with universal coefficients given by derivatives of the differential of the third kind, the Bers quasiform and certain holomorphic forms. We use this formula to describe conformal Ward identities framed in terms of a canonical differential operator which acts with respect to the Schottky moduli and to the insertion points of the $n$-point function. We consider the generalized Heisenberg VOA and determine all its correlation functions by Zhu recursion. We also use Zhu recursion to derive linear partial differential equations for the Heisenberg VOA partition function and various structures such as the bidifferential of the second kind, holomorphic $1$-forms, the prime form and the period matrix. Finally, we compute the genus $g$ partition function for any rational Euclidean lattice generalized VOA.

A way to characterize the space of leaves of a foliation in terms of connections is proposed. A particular example of vertex algebra cohomology of codimension one foliations on complex curves is considered. Multiple applications in Mathematical Physics are revealed.

We study the algebraic conditions leading to the chain property of complexes for vertex operator algebra n-point functions (with their convergence assumed) 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 are 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 ${\mathrm{\Sigma }}^{(g)}$. The reduction cohomology for a vertex operator algebra with formal parameters identified with local coordinates around marked points on ${\mathrm{\Sigma }}^{(g)}$ is found in terms of the space of analytical continuations of solutions to Knizhnik–Zamolodchikov equations. For the reduction cohomology, the Euler–Poincaré formula is derived. Examples for various genera and vertex operator cluster algebras are provided.

We discuss the problem of determining the de Rham, Dolbeault and algebraic cohomological dimension of ${{ℳ}}_g$, focusing on possible strategies of attack and then concentrating on exhaustion functions. In the final section, we explain how these techniques can be employed to provide a nontrivial upper bound for the Dolbeault cohomological dimension of ${{ℳ}}_g$.

We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ω_Λ of the second kind on the Riemann surface. Using Λ and ω_Λ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

In this paper, we construct the foliation of a space associated to correlation functions of vertex operator algebras, 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.

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.

Starting from Zhu recursion formulas for correlation functions for vertex operator algebras with formal parameters associated to local coordinates around marked points on a Riemann surfaces, we introduce a cluster algebra structure over a noncommutative set of variables. Cluster elements and mutation rules are explicitly defined. In particular, we propose an elliptic version of vertex operator cluster algebras arising from correlation functions and Zhu reduction procedure for vertex operators on the torus.

In this work we give pairs of generators (x, y) for the alternating groups A_n, 5 ≤ n ≤ 19, such that they determine the minimal genus of a Riemann surface on which A_n acts as the automorphism group. Using these results we prove that A₁₅ is the unique of these groups that is an H*-group, i.e., the groups achieving the upper bound of the order of an automorphism group acting on non-orientable unbordered surfaces.

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S, f) to be conformally equivalent to (S′, f′) where S′ is a bordered orientable Klein surface embedded in the Euclidean space and f′ is the restriction to S′ of a prime order rotation. Our results can allow the visualization of some Riemann surfaces automorphisms by representing them as restrictions of isometries of 𝕊⁴ or ℝ⁴. We illustrate this method with the order seven automorphisms of two famous Riemann surfaces: the Klein quartic and the Wiman surface.

Let $m\in \mathbb{N}$, $P(t)\in ℂ[t]$. Then we have the algebraic curve $u^m=P(t)$, and its coordinate algebras (the Riemann surfaces) $R_m(P)=ℂ[t^{\pm 1},u|u^m=P(t)]$ and $S_m(P)=ℂ[t,u|u^m=P(t)].$ The Lie algebras ${{ℛ}}_m(P)=Der(R_m(P))$ and ${{𝒮}}_m(P)=Der(S_m(P))$ are called the $m$th superelliptic Lie algebras associated to $P(t)$. In this paper, we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras, which will help to understand the birational equivalence of some algebraic curves of the form $u^m=P(t)$.

We show that for any fixed point P₀ on a Riemann surface Σ the distinct realizations of cocycles in correspond to the natural appearances of the standard Heisenberg vertex operator algebra Π(P₀) and to the commutative Heisenberg vertex operator algebra Π⁰(P₀), respectively.

A theorem of Jentzsch–Szegő describes the limit measure of a sequence of discrete measures associated to zeroes of a sequence of polynomials in one variable. Following the presentation by Andrievskii and Blatt in [Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] we extend this theorem to compact Riemann surfaces and to analytic curves in the sense of Berkovich over ultrametric fields, using classical potential theory in the former case, and Baker/Rumely, Thuillier's potential theory on analytic curves in the latter case. We then apply this equidistribution theorem to the question of irreducibility of truncations of power series with coefficients in ultrametric fields.

Résumé français: Le théorème de Jentzsch–Szegő décrit la mesure limite d'une suite de mesures discrètes associée aux zéros d'une suite convenable de polynômes en une variable. Suivant la présentation que font Andrievskii et Blatt dans [Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] on étend ici ce résultat aux surfaces de Riemann compactes, puis aux courbes analytiques sur un corps ultramétrique. On donne pour finir quelques corollaires du cas particulier de la droite projective sur un corps ultramétrique à l'irréductibilité des polynômes-sections d'une série entière en une variable.

The main goal of this paper is to present a proof of Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main result states that any hyperbolic sphere with n cusps has a pants decomposition with all of its geodesics of length bounded by . Other results include lower and upper bounds for Bers' constants for hyperelliptic surfaces and spheres with boundary geodesics.