Torsion Constraints and Super Riemann Surfaces

Abstract

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)