Torsion Constraints and Super Riemann Surfaces

We define super Riemann surfaces as smooth 2|2-dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2×2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the torsion constraints of 2d supergravity. We show that they are both necessary and sufficient for a frame to admit local superconformal coordinates. The other torsion constraints are merely conditions to fix some of the gauge freedom in this description, or to specify a particular connection on such a manifold, analogous to the Levi-Civita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface cannot be regarded as having only one complex dimension. Nevertheless, in certain important aspects super Riemann surfaces behave as nicely as if they had only one dimension. In particular they posses an analog $$\widehat\partial $$ of the Cauchy-Riemann operator on ordinary Riemann surfaces, a differential operator taking values in the bundle of half-volume forms. This operator furnishes a short resolution of the structure sheaf, making possible a Quillen theory of determinant line bundles. Finally we show that the moduli space of super Riemann surfaces is embedded in the larger space of complex curves of dimension 1|1.

Contents Introduction § 1. Fuchsian groups and their sequential generators § 2. Geometry of Fuchsian groups § 3. Free Fuchsian groups of rank two § 4. Spaces of Fricke-Klein-Teichmüller type § 5. Moduli of Riemann surfaces § 6. The space of holomorphic morphisms of a Riemann surface § 7. Lifting Fuchsian groups to § 8. Topological classification of Arf functions and of pairs of Arf functions § 9. Topological classification of independent sets of Arf functions on compact surfaces § 10. The moduli space of spinor bundles of rank one § 11. Super Fuchsian groups, super Riemann surfaces, and their topological types § 12. Moduli of super Riemann surfaces § 13. super Fuchsian groups, super Riemann surfaces, and their topological invariants § 14. Moduli of super Riemann surfaces § 15. Superholomorphic morphisms of super Riemann surfacesBibliography

Line bundles and divisors are defined on a super Riemann surface. The isomorphism between them is shown.

The free motion of a nonrelativistic superparticle on the super Riemann surface (SRS) of genus h≥2 is investigated. Geodesics or classical paths are given explicitly on the super Poincaré upper half-plane SH, a universal covering space of the SRS, and the paths with some suitable initial conditions yield periodic orbits on the SRS. The periodic orbits are unstable and the system is chaotic. Quantum mechanics is solved on the universal covering space SH and the heat kernel is given on the SRS. This leads to a superanalog of the Selberg trace formula. The Selberg super zeta function is introduced whose zero points and poles determine the energy spectrum on the SRS.

A new (1,1)-dimensional super vector bundle which exists on any super Riemann surface is described. Cross sections of this bundle provide a new class of fields on a super Riemann surface which closely resemble holomorphic functions on a super Riemann surface, but which (in contrast to the case with holomorphic functions) form spaces which have a well defined dimension which does not change as odd moduli become non-zero.

The super Mumford form and Sato Grassmannian

Resolutions of non-regular Ricci-flat Kähler cones

Line bundles over families of (super) Riemann surfaces. I: The non-graded case

Following the work of Krichever and Novikov, Bonora, Martellini, Rinaldi and Russo defined a superalgebra associated to each compact Riemann surface with spin structure. Noting that this data determines a graded Riemann surface, we find a natural interpretation of the BMRR-algebra in terms of the geometry of graded Riemann surfaces. We also discuss the central extensions of these algebras (correcting the form of the central extension given by Bonoraet al.). It is hoped that this work will be the first step towards defining Krichever-Novikov algebras for (the more general) super-Riemann surfaces; in particular we emphasise the importance ofgraded conformal vectorfields.

Two different approaches to (Kostant-Leites-) super Riemann surfaces are investigated. In the local approach, i.e. glueing open superdomains by superconformal transition functions, deformations of the superconformal structure are discussed. On the other hand, the representation of compact super Riemann surfaces of genus greater than one as a fundamental domain in the Poincare upper half-plane provides a simple description of super Laplace operators acting on automorphicp-forms.

In type II superstring theory, the vacuum amplitude at a given loop order g can receive contributions from the boundary of the compactified, genus g supermoduli space of curves M̄g. These contributions capture the long distance or infrared behavior of the amplitude. The boundary parameterizes degenerations of genus g super Riemann surfaces. A holomorphic projection of the supermoduli space onto its reduced space would then provide a way to integrate the holomorphic, superstring measure and thereby give the superstring vacuum amplitude at g-loop order. However, such a projection does not generally exist over the bulk of the supermoduli spaces in higher genera. Nevertheless, certain boundary divisors in ∂M̄g may holomorphically map onto a bosonic space upon composition with universal morphisms, thereby enabling an integration of the holomorphic, superstring measure here. Making use of ansatz factorizations of the superstring measure near the boundary, our analysis shows that the boundary contributions to the three loop vacuum amplitude will vanish in closed oriented type II superstring theory with unbroken spacetime supersymmetry.

The super period matrix with Ramond punctures

I review the description of super Riemann surfaces as algebraic curves in projective superspace, as opposed to the usual descriptions as 2d supergravity geometries or (1,1) supermanifolds with superconformai structure. Some applications and several open problems in this approach are described.

Conformal Lie superalgebras and moduli spaces

Super Riemann Surface, superprojective structures