Moduli Space Of Riemann Surfaces Research Articles

The Virasoro minimal string

We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge c≥≥ 25 coupled to timelike Liouville CFT with central charge 26-c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, thus motivating the name Virasoro minimal string. The duality holds for any value of the continuous parameter cc and reduces to the JT gravity/matrix integral duality in the large central charge limit. It thus provides a precise stringy realization of JT gravity. The main observables of the Virasoro minimal string are quantum analogues of the Weil-Petersson volumes, which are computed as absolutely convergent integrals of worldsheet CFT correlators over the moduli space of Riemann surfaces. By exploiting a relation of the Virasoro minimal string to three-dimensional gravity and intersection theory on the moduli space of Riemann surfaces, we are able to give a direct derivation of the duality. We provide many checks, such as explicit numerical - and in special cases, analytic - integration of string diagrams, the identification of the CFT boundary conditions with asymptotic boundaries of the two-dimensional spacetime, and the matching between the leading non-perturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.

Singular spin structures and superstrings

There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙg−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric.

Liouville theory and the Weil-Petersson geometry of moduli space

Liouville theory describes the dynamics of surfaces with constant negative curvature and can be used to study the Weil-Petersson geometry of the moduli space of Riemann surfaces. This leads to an efficient algorithm to compute the Weil-Petersson metric to arbitrary accuracy using Zamolodchikov’s recursion relation for conformal blocks. For example, we compute the metric on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}0,4 numerically to high accuracy by considering Liouville theory on a sphere with four punctures. We numerically compute the eigenvalues of the Weil-Petersson Laplacian, and find evidence that the obey the statistics of a random matrix in the Gaussian Orthogonal Ensemble.

One dimensional equisymmetric strata in moduli space with genus 1 quotient surfaces

The complex orbifold structure of the moduli space of Riemann surfaces of genus g (g≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ge 2$$\\end{document}) produces a stratification into complex subvarieties named equisymmetric strata. Each equisymmetric stratum is formed by the surfaces where the group of automorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata of dimension one are of two structurally different types. Type 1 equisymmetric strata correspond to Riemann surfaces where the group of automorphisms produces a quotient surface of genus zero, while those of Type 2 appear when such a quotient is a surface of genus one. Type 1 equisymmetric strata have been extensively studied by the authors of the present work in a previous recent paper, we now focus on Type 2 strata. We first establish the existence of such strata and their frequency of occurrence in moduli spaces. As a main result we obtain a complete description of Type 2 strata as coverings of the sphere branched over three points (Belyi curves) and where certain isolated points (punctures) have to be eliminated. Finally, we study in detail the doubly infinite family of Type 2 strata whose automorphism groups have order the product of two primes.

Asymptotic behavior of the Zhang–Kawazumi's φ-invariants

The φ-invariant was introduced by Shou-Wu Zhang and Nariya Kawazumi. We study the continuity property of the φ-invariants for degenerating graphs, show that this continuity property induces an adelic divisor on the moduli space of Riemann surfaces, and then give an asymptotic expression of the Zhang–Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space. We mainly use Yuan–Zhang's adelic divisors, and follow Yuan's idea of globalization of φ-invariants.

Moduli spaces of Riemann surfaces as Hurwitz spaces

We consider the moduli space Mg,n of Riemann surfaces of genus g≥0 with n≥1 ordered and directed marked points. For d≥2g+n−1 we show that Mg,n is homotopy equivalent to a component of the simplicial Hurwitz space HurΔ(Sdgeo) associated with the partially multiplicative quandle Sdgeo. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space Ω∞−2MTSO(2) of Hurwitz flavour.

On generators and relations of the rational cohomology of Hilbert schemes

We consider for $$d\ge 1$$ the graded commutative $${\mathbb {Q}}$$ -algebra $${\mathcal {A}}(d):=H^*({\text {Hilb}}^d({\mathbb {C}}^2);{\mathbb {Q}})$$ , which is also connected to the study of generalised Hurwitz spaces by work of the first author. These Hurwitz spaces are in turn related to the moduli spaces of Riemann surfaces with boundary. We determine two distinct, minimal sets of $${\lfloor }d/2{\rfloor }$$ multiplicative generators of $${\mathcal {A}}(d)$$ . Additionally, we prove when the lowest degree generating relations occur. For small values of d, we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for $${\mathcal {A}}(d)$$ .

Equivariant Picard groups of the moduli spaces of some finite Abelian covers of the Riemann sphere

In this note, following Kordek's work we will compute the equivariant Picard groups of the moduli spaces of Riemann surfaces with certain finite abelian symmetries.

Billiards and Teichmüller curves

A Teichmüller curve V ⊂ M g V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems.

A new uniform lower bound on Weil–Petersson distance

In this paper we study the injectivity radius based at a fixed point along Weil-Petersson geodesics. We show that the square root of the injectivity radius based at a fixed point is $ 0.3884$-Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is uniformly Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric, where the Lipschitz constant can be chosen to be $0.5492$. Applications to asymptotic geometry of moduli space of Riemann surfaces for large genus will also be discussed.

The moduli space of the tropicalizations of Riemann surfaces

In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by its pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. This, in the covering level, is analogous to adding the frontier set, subordinate to a pants decomposition, to the Teichmüller space. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratifications of these two moduli spaces.

Small eigenvalues of closed Riemann surfaces for large genus

In this article we study the asymptotic behavior of small eigenvalues of hyperbolic surfaces for large genus. We show that for any positive integer k k , as the genus g g goes to infinity, the minimum of k k -th eigenvalues of hyperbolic surfaces over any thick part of moduli space of Riemann surfaces of genus g g is uniformly comparable to 1 g 2 \frac {1}{g^2} in g g . And the minimum of a g ag -th eigenvalues of hyperbolic surfaces in any thick part of moduli space is bounded above by a uniform constant only depending on ε \varepsilon and a a . In the proof of the upper bound, for any constant ε > 0 \varepsilon >0 , we will construct a closed hyperbolic surface of genus g g in any ε \varepsilon -thick part of moduli space such that it admits a pants decomposition whose curves all have length equal to ε \varepsilon , and the number of separating systole curves in this surface is uniformly comparable to g g .

The vanishing rate of Weil–Petersson sectional curvatures

The Weil–Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil–Petersson metric is approximately a product metric. An almost product metric has sections with almost vanishing curvature. We bound the sectional curvature away from zero in terms of the product of lengths of short geodesics on Riemann surfaces. We give examples and an expectation for the actual vanishing rate.

Real-normalized differentials with a single order 2 pole

A meromorphic differential on a Riemann surface is said to be {\it real-normalized} if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. Our goal is to develop tools to study this topology. We propose a combinatorial model for the real normalized differentials with a single order $2$ pole and use it to analyze the corresponding absolute period foliation.

Real classical geometry with arbitrary deficit parameter(s) alpha (_{I}) in deformed Jackiw\u2013Teitelboim gravity

An interesting deformation of Jackiw–Teitelboim (JT) gravity has been proposed by Witten by adding a potential term U(phi ) as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over phi as R(x)+2=2alpha delta (vec {x}-vec {x}'). The resulting Euclidean metric suffered from a conical singularity at vec {x}=vec {x}'. A possible geometry is modeled locally in polar coordinates (r,varphi ) by mathrm{d}s^2=mathrm{d}r^2+r^2mathrm{d}varphi ^2,varphi cong varphi +2pi -alpha . In this letter we show that there exists another family of ”exact” geometries for arbitrary values of the alpha . A pair of exact solutions are found for the case of alpha =0. One represents the static patch of the AdS and the other one is the non-static patch of the AdS metric. These solutions were used to construct the Green function for the inhomogeneous model with alpha ne 0. We address a type of phase transition between different patches of the AdS in theory because of the discontinuity in the first derivative of the metric at x=x'. We extended the study to the exact space of metrics satisfying the constraint R(x)+2=2sum _{i=1}^{k}alpha _idelta ^{(2)}(x-x'_i) as a modulus diffeomorphisms for an arbitrary set of deficit parameters (alpha _1,alpha _2,ldots ,alpha _k). The space is the moduli space of Riemann surfaces of genus g with k conical singularities located at x'_k, denoted by mathcal {M}_{g,k}.

Volumes And Random Matrices

AbstractThis article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020)

D-instanton perturbation theory

D-instanton world-volume theory has open string zero modes describing collective coordinates of the instanton. The usual perturbative amplitudes in the D-instanton background suffer from infra-red divergences due to the presence of these zero modes, and the usual approach of analytic continuation in momenta does not work since all open string states on a D-instanton carry strictly zero momentum. String field theory is well-suited for tackling these issues. However we find a new subtlety due to the existence of additional zero modes in the ghost sector. This causes a breakdown of the Siegel gauge, but a different gauge fixing consistent with the BV formalism renders the perturbation theory finite and unambiguous. At each order, this produces extra contribution to the amplitude besides what is obtained from integration over the moduli space of Riemann surfaces.

GL2ℝ–invariant measures in marked strata : generic marked points, Earle–Kra for strata and illumination

We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmuller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2,R) orbit.

Open KdV hierarchy of 2d minimal gravity of Lee-Yang series

We present the open KdV hierarchy of 2d minimal gravity of Lee-Yang series which uses the boundary cosmological constant as a flow parameter. The boundary cosmological constant is a conjugate variable to the boundary flow parameter used in the open KdV hierarchy of the intersection numbers on the moduli space of Riemann surfaces with boundaries. The two generating functions are related through the Laplace transform.

A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

We survey Mirzakhani’s work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of nonexperts.