New fields on super Riemann surfaces

Super Riemann surfaces are the arena for two-dimensional superconformal field theory. They can be regarded as smooth 212-dimensional supermanifolds equipped with a reduction of their structure group to the group of upper triangular 2 × 2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the famous torsion constraints of 2d supergravity. 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-Ci vita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface \(\hat{X}\) cannot be regarded as having only one complex dimension; instead it has one commuting and one anticommuting complex coordinate. Nevertheless, in certain important aspects SRS behave as nicely as if they had only one dimension. In particular they possess an analog of the Cauchy-Riemann operator \(\bar{\partial }\) on an ordinary Riemann surface X. The latter is a first-order differential operator with values in a canonical line bundle ω º Ω1X over the surface; this bundle in turn is in a sense a square root of the bundle of volume forms. Similarly super Riemann surfaces have a canonically-defined first-order operator \(\hat{\bar{\partial }}\) taking values in the bundle \(\hat{\omega } \equiv BER{{\Omega }^{1}}\hat{X}\) of half-volume forms. This observation allows us to write a superstring world-sheet action depending only on a “superconformal structure” analogous to a conformal structure on an ordinary 2-surface. Furthermore, the operator \(\hat{\bar{\partial }}\) furnishes a short resolution of the structure sheaf of the super Riemann surface, making possible a Quillen theory of determinant line bundles. The material in this talk, and further discussion, can be found in the following papers: 1. S. Giddings and P. Nelson, “Torsion constraints and super Riemann surfaces,” Phys. Rev. Lett. 59, 2619 (1987). 2. P. Nelson, “Holomorphic coordinates for supermoduli space,” Comm. Math. Phys. 115, 167 (1988). 3. S. Giddings and P. Nelson, “The geometry of super Riemann surfaces,” Comm. Math. Phys. (1988)

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.

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.

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 Mumford form and Sato Grassmannian

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.

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

Line integral on the super Riemann surface is discussed. A “super differential operator” which possesses both properties of differential and of differential operator is proposed. With this “super-differential operator” a new theory of differential form on the super Riemann surface is constructed. We call “the new differentials on the super Riemann surface” “the super differentials”. As the applications of our theory, the existency theorems of singular “super differentials” such as “super abelian differentials of the 3rd kind” and of a super projective connection are examined.

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.

In a previous paper, the authors together with L. Vrancken initiated the study of $3$-dimensional CR submanifolds of the nearly K\" ahler homogeneous $\mathbb S^3\times \mathbb S^3$. As is shown by Butruille this is one of only four homogeneous $6$-dimensional nearly K\"ahler manifolds. Besides its almost complex structure $J$ it also admits a canonical almost product structure $P$. Along a $3$-dimensional CR submanifold the tangent space of $\mathbb S^3\times\mathbb S^3$ can be naturally split as the orthogonal sum of three $2$-dimensional vector bundles $\mathcal D_1$, $\mathcal D_2$ and $\mathcal D_3$. We study the CR submanifolds in relation to the behavior of the almost product structure on these vector bundles.