Abelian actions on compact nonorientable Riemann surfaces

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe)

We study the genus expansion on compact Riemann surfaces of the gravitational path integral $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z grav m in two spacetime dimensions with cosmological constant Λ > 0 coupled to one of the non-unitary minimal models ℳ2m − 1, 2. In the semiclassical limit, corresponding to large m, $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z grav m admits a Euclidean saddle for genus h ≥ 2. Upon fixing the area of the metric, the path integral admits a round two-sphere saddle for h = 0. We show that the OPE coefficients for the minimal weight operators of ℳ2m − 1, 2 grow exponentially in m at large m. Employing the sewing formula, we use these OPE coefficients to obtain the large m limit of the partition function of ℳ2m − 1, 2 for genus h ≥ 2. Combining these results we arrive at a semiclassical expression for $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z grav m . Conjecturally, $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z grav m admits a completion in terms of an integral over large random Hermitian matrices, known as a multicritical matrix integral. This matrix integral is built from an even polynomial potential of order 2m. We obtain explicit expressions for the large m genus expansion of multicritical matrix integrals in the double scaling limit. We compute invariant quantities involving contributions at different genera, both from a matrix as well as a gravity perspective, and establish a link between the two pictures. Inspired by the proposal of Gibbons and Hawking relating the de Sitter entropy to a gravitational path integral, our setup paves a possible path toward a microscopic picture of a two-dimensional de Sitter universe.

Foliation of a space associated to vertex operator algebra

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.

Generalised Beauville groups

A Beauville surface is defined by the action of a finite group (Beauville group) on the product of two compact Riemann surfaces. In this paper, we consider higher products and the possibility of a similar action by a finite group, which we call a generalised Beauville group; we prove several results regarding the existence and construction of infinite families of generalised Beauville groups and provide a complete classification of the abelian ones; we list all generalised Beauville groups of orders from 1 to 1023.

Log-lightning computation of capacity and Green's function

See Video Abstract (click the "Video Abstract" button next to the "PDF" button) A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces.