A wide class of Seiberg–Witten models constructed by M -theory techniques and described by non-hyperelliptic Riemann surfaces are shown to possess an associative algebra of holomorphic differentials. This is a first step towards proving that also these models satisfy the Witten–Dijkgraaf–Verlinde–Verlinde equation. In this way, similar results known for simpler Seiberg–Witten models (described ...

The quantum mechanics of a system of charged particles interacting with a magnetic field on Riemann surfaces is studied. We explicitly construct the wave functions of ground states in the case of a metric proportional to the Chern form of the θ-bundle, and the wave functions of the Landau levels in the case of the the Poincaré metric. The degeneracy of the the Landau levels is obtained by using...

In this paper, we give both lower and upper bound estimates of the Bergman kernel for a degeneration of Riemann surfaces with constant curvature −1. As a result, we give a geometric proof of the Riemann–Rock theorem for a singular Riemann surface. Mathematics Subject Classification (2010). Primary 53C55; Secondary 32S30.

For singular metrics, Ray and Singer’s analytic torsion formalism cannot be applied. Hence we do not have the so-called Quillen metric on determinant of cohomology with respect to a singular metric. In this paper, we introduce a new metric on determinant of cohomology by adapting a totally different approach. More precisely, by strengthening results in the first paper of this series, we develop...

We use the theory of arithmetic surfaces to show that the Riemann-Roch theorem for Q-graphs is a direct consequence of the usual Riemann-Roch theorem for curves in algebraic geometry.

These notes summarize four expository lectures delivered at the Advanced School of the ICTS Program Groups, Geometry and Dynamics, December, 2012, Almora, India. The target audience was a group of students at or near the end of a traditional undergraduate math major. My purpose was to expose the types of discrete groups that arise in connection with Riemann surfaces. I have not hesitated to sho...

For a generic set in the Teichmüller space, we construct a covariant functor with the range in a category of the operator AF -algebras. The functor maps isomorphic Riemann surfaces to the stably isomorphic AF algebras. Our construction is based on a Hodge theory for measured foliations, elaborated by Hubbard, Masur and Thurston. As an application, we consider the following two problems: (i) a c...