B. Enriquez, S. Pakuliak, V. Rubtsov
We construct level 1 basic representations of the quantized current algebras associated to higher genus algebraic curves using one free field. We also clarify the relation between the elliptic current algebras of the papers [EF] and [JKOS].
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We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the $t=q$ specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.
We consider all genus two correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of a sequence of globally defined differential operators on which the genus two Siegel modular group $Sp(4,\mathbb{Z})$ has a natural action.
We give a presentation in terms of generators and relations of the cohomology in degree zero of the Campos-Willwacher graph complexes associated to compact orientable surfaces of genus $g$. The results carry a natural Lie algebra structure, and for $g=1$ we recover Enriquez' elliptic Grothendieck-Teichmüller Lie algebra. In analogy to Willwacher's theorem relating Kontsevich's graph complex to Drinfeld's Grothendieck-Teichmüller Lie algebra, we call the results higher genus Grothendieck-Teichmüller Lie algebras. Moreover, we find that the graph cohomology vanishes in negative degrees.
We continue our program to define and study $n$-point correlation functions for a vertex operator algebra $V$ on a higher genus compact Riemann surface obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas for the genus two partition function for free bosonic theories and lattice vertex operator algebras $V_L$. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. We also compute the genus two Heisenberg v…
We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.
This is a survey of our construction of current algebras, associated with complex curves and rational differentials. We also study in detail two classes of examples. The first is the case of a rational curve with differentials $z^n dz$; these algebras are ``building blocks'' for the quantum current algebras introduced in our earlier work. The second is the case of a genus $>1$ curve $X$, endowed with a regular differential having only double zeroes.
Presentations of skein algebras of genus 2
该研究给出亏格2的闭或单洞定向曲面的考夫曼括号纽结代数的表示，为亏格大于1的相关问题提供了首个此类结果。
We describe the construction of the genus-zero parts of conformal field theories in the sense of G. Segal from representations of vertex operator algebras satisfying certain conditions. The construction is divided into four steps and each step gives a mathematical structure of independent interest. These mathematical structures are intertwining operator algebras, genus-zero modular functors, genus-zero holomorphic weakly conformal field theories, and genus-zero conformal field theories.
Quasi-lisse vertex (super)algebras
本文研究拟光滑顶点（超）代数，给出保证零亏格和一亏格 $n$ 点相关函数收敛的新有限性条件。
In this paper, we study representations for three-point Lie algebras of genus zero based on the Cox-Jurisich's presentations. We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into simple modules over the three-point affine algebras of genus zero. As a corollary, vertex representations are constructed for the three-point affine algebra of genus zero using vertex operators. Moreover, we construct a Fock module for certain quotient of three-point Virasoro algebra of genus zero.
We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.
For higher genus multi-point current algebras of Krichever-Novikov type associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles are studied. They yield as central extensions almost-graded higher genus affine Lie algebras. In case that the Lie algebra is reductive a complete classification is given. For a simple Lie algebra, like in the classical situation, there is up to equivalence and rescaling only one non-trivial almost-graded central extension. The classification is extended to the algebras of meromorphic differential operators of order less or equal one on the curr…
We define the partition and $n$-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two H…
Classical affine Lie algebras appear e.g. as symmetries of infinite dimensional integrable systems and are related to certain differential equations. They are central extensions of current algebras associated to finite-dimensional Lie algebras g. In geometric terms these current algebras might be described as Lie algebra valued meromorphic functions on the Riemann sphere with two possible poles. They carry a natural grading. In this talk the generalization to higher genus compact Riemann surfaces and more poles is reviewed. In case that the Lie algebra g is reductive (e.g. g is simple, semi-si…
It is well known that any triangulation of a marked surface produces a quiver. In this paper we will provide a triangulation for orientable surfaces of genus $n$ with an arbitrary number interior marked points (called punctures) whose corresponding quiver has a maximal green sequence.
This article is the first report of an ongoing project aimed at finding a geometric interpretation of the Witten genus and other tmf classes. Section 2 reviews the sheaves of chiral differential operators (CDOs) over a complex manifold, including their construction, obstructions and relation with the Witten genus. In section 3, the structure of each sheaf of CDOs is reorganized in terms of modules over the sheaf of holomorphic functions. This invokes the notion of a differential graded vertex algebroid. The construction of sheaves of CDOs is due to Gorbounov, Malikov and Schechtman, and so is …
By the classical genus zero Sugawara construction one obtains from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type) representations of the Virasoro algebra. In this lecture first the classical construction is recalled. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in joint work with O.K. Sheinman. We were able to show that also in the higher genus, multi-point situa…
We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion where we express an arbitrary genus two $n$--point correlation function in terms of $(n-1)$--point functions. We consider several applications including the correlation functions for the Heisenberg vertex operator algebra and its modules, Virasoro correlation functions and genus two Ward identities. We derive novel differential equations in terms of a differential operator on the genus two Siegel upper half pl…
We define the $n$-point function for a vertex operator algebra on a genus two Riemann surface in two separate sewing schemes where either two tori are sewn together or a handle is sewn to one torus. We explicitly obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and lattice vertex operator algebras in both sewing schemes. We prove that the partition functions are holomorphic in the sewing parameters on given suitable domains and describe their modular properties. Finally, we show that the partition functions cannot be equal in the neighborhood o…
Equivalence Relations on Vertex Operator Algebras, I: Genus
本文研究顶点算子代数的体亏格与双曲亏格两种等价关系，证明双曲亏格更精细，提出c=32全纯顶点算子代数分类计划与数量下界，并猜想Siegel-Weil恒等式。
In [H5] (q-alg/9512024) and [H7] (q-alg/9704008), the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras were solved implicitly earlier in [H3] (q-alg/9505019). In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [H1], [H2], [HL1] (hep-th/9301009), [HL2] and [H6] to the notion of intertwining operator…
Monoidal categorification of genus zero skein algebras
本研究证明了零亏格带边曲面Kauffman括号扭结代数与量子化$K$-理论库仑分支相关的猜想，得到扭结代数的幺半群范畴化构造，部分解答了D. Thurston的相关问题。
Lyubashenko's construction associates representations of mapping class groups Map_{g,n} of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H. For any such Hopf algebra we find an invariant of Map_{g,n} for all values of g and n. More generally, we obtain such invariants for any pair (H,omega), where omega is a ribbon automorphism of H. Our results are motivated by the quest to understand correlation functions of bulk fiel…
By twisted quantum invariants we mean polynomial invariants of knots in the three-sphere endowed with a representation of the fundamental group into the automorphism group of a Hopf algebra $H$. These are obtained by the Reshetikhin-Turaev construction extended to the $\mathrm{Aut}(H)$-twisted Drinfeld double of $H$, provided $H$ is finite dimensional and $\mathbb{N}^m$-graded. We show that the degree of these polynomials is bounded above by $2g(K)\cdot d(H)$ where $g(K)$ is the Seifert genus of a knot $K$ and $d(H)$ is the top degree of the Hopf algebra. When $H$ is an exterior algebra, our t…
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r…
This paper explains the conjectured algebraic duality between genus zero Gromov-Witten theory and genus zero "Closed String topology". This duality in another perspective is discussed on page 87 of the book "Frobenius manifold, quantum cohomology, and moduli spaces" (by Yuri Manin). This paper also discusses Fulton MacPherson strata.
Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras
本文研究简单自对偶强CFT型顶点算子代数在亏格g黎曼面上的Virasoro关联函数，证明其由微分算子作用于归一化配分函数决定，并给出该算子在Virasoro图上的显式权重表达及模性质。
We consider a system of three commuting difference operators in three variables $x_{12},x_{13},x_{23}$ with two generic complex parameters $q,t$. This system and its eigenfunctions generalize the trigonometric $A_1$ Ruijsenaars-Schneider model and $A_1$ Macdonald polynomials, respectively. The principal object of study in this paper is the algebra generated by these difference operators together with operators of multiplication by $x_{ij} + x_{ij}^{-1}$. We represent the Dehn twists by outer automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping c…
We describe Zhu recursion for a vertex operator algebra (VOA) on a general genus Riemann surface in the Schottky uniformization where $n$-point correlation functions are written as linear combinations of $(n-1)$-point functions with universal coefficients. These coefficients are identified with specific geometric structures on the Riemann surface. We apply Zhu recursion to the Heisenberg VOA and determine all its correlation functions. For a general VOA, Zhu recursion with respect to the Virasoro vector is shown to lead to conformal Ward identities expressed in terms of derivatives with respec…
According to seminal work of Kontsevich, the unstable homology of the mapping class group of a surface can be computed via the homology of a certain lie algebra. In a recent paper, S. Morita analyzed the abelianization of this lie algebra, thereby constructing a series of candidates for unstable classes in the homology of the mapping class group. In the current paper, we show that these cycles are all nontrivial, representing degree 4k+1 homology classes in the homology of the mapping class group of a genus one surface with 4k+1 punctures.
We present a free field realisation for the vertex operator algebra associated to the genus-two, class $\mathcal{S}$ superconformal field theory of type $\mathfrak{a}_1$. The free field realisation is in the style of recent work by the authors, and is formulated in terms of a one-dimensional isotropic lattice vertex algebra along with two pairs of symplectic fermions. Our realisation makes manifest an enhanced ${\rm USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an ${\rm SU(2)}$ outer automorphism that has been observed in recent…
On the gauge invariance of the Kuperberg invariant of certain high genus framed 3-manifolds
本文研究了某些高亏格带框架的3-流形的Kuperberg不变量的规范不变性，并通过双曲3-流形提供了有限维Hopf代数规范不变量的第一个例子。
We give a functorial construction of the genus zero chiral algebras of class $\mathcal{S}$, that is, the vertex algebras corresponding to the theory of class $\mathcal{S}$ associated with genus zero pointed Riemann surfaces via the 4d/2d duality discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees in physics. We show that there is a unique family of vertex algebras satisfying the required conditions and show that they are all simple and conformal. In fact, our construction works for any complex semisimple group G that is not necessarily simply laced. Furthermore, we show that the…
Classical Limit of genus two DAHA
本文证明了亏格二曲面skein代数的单参数形变是平坦的，求解了字问题并描述单项式基，计算了其经典极限为平坦Poisson形变，并给出了特征簇坐标环的简洁表示。
Askey-Wilson polynomial and genus-two skein module
本文利用亏格二 skein 代数的 q-差分算子，建立了约化 Askey-Wilson 多项式与亏格二环柄体 skein 模之间的对应关系。
Genus $g$ Zhu Recursion for Vertex Operator Algebras and Their Modules
本文在 Schottky 单值化下建立 VOA 及其模的亏格 $g$ Zhu 递归，用于关联函数、Ward 恒等式、Heisenberg VOA 结构及有理格广义 VOA 配分函数。
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