69,137 research outputs found
Random Construction of Riemann Surfaces
In this paper, we address the following question: What does a typical compact
Riemann surface of large genus look like geometrically? We do so by
constructing compact Riemann surfaces from oriented 3-regular graphs. The set
for such Riemann surfaces is dense in the space of all compact Riemann
surfaces, namely Belyi surfaces. And in this construction we can control the
geometry of the compact Riemann surface by the geometry of the graph. We show
that almost all such surfaces have large first eigenvalue and large Cheeger
constant
Selberg Supertrace Formula for Super Riemann Surfaces III: Bordered Super Riemann Surfaces
This paper is the third in a sequel to develop a super-analogue of the
classical Selberg trace formula, the Selberg supertrace formula. It deals with
bordered super Riemann surfaces. The theory of bordered super Riemann surfaces
is outlined, and the corresponding Selberg supertrace formula is developed. The
analytic properties of the Selberg super zeta-functions on bordered super
Riemann surfaces are discussed, and super-determinants of Dirac-Laplace
operators on bordered super Riemann surfaces are calculated in terms of Selberg
super zeta-functions.Comment: 43 pages, amste
Some differential equations for the Riemann -function on Jacobians
We prove some differential equations for the Riemann theta function
associated to the Jacobian of a Riemann surface. The proof is based on some
variants of a formula by Fay for the theta function, which are motivated by
their analogues in Arakelov theory of Riemann surfaces. Moreover, we give a
generalization of Rosenhain's formula to hyperelliptic Riemann surfaces as
conjectured by Gu\`ardia.Comment: Comments are welcom
Noncommutative Riemann Surfaces
We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1.
Following [1], we construct a projective unitary representation of pi_1(Sigma)
realized on L^2(H), with H the upper half-plane. As a first step we introduce a
suitably gauged sl_2(R) algebra. Then a uniquely determined gauge connection
provides the central extension which is a 2-cocycle of the 2nd Hochschild
cohomology group. Our construction is the double-scaling limit N\to\infty,
k\to-\infty of the representation considered in the Narasimhan-Seshadri
theorem, which represents the higher-genus analog of 't Hooft's clock and shift
matrices of QCD. The concept of a noncommutative Riemann surface Sigma_\theta
is introduced as a certain C^\star-algebra. Finally we investigate the Morita
equivalence.Comment: LaTeX, 1+14 pages. Contribution to the TMR meeting ``Quantum aspects
of gauge theories, supersymmetry and unification'', Paris 1-7 September 199
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