Letters in Mathematical Physics 14 (,1987) 47-53. 9 1987 by D. Reidel Publishing Company. 47 A Direct Calculation of Super-Teichmtiller Space LUKE HODGKIN Department of Mathematics, King's College, University of London. The Strand, London WC2R 2LS, U.K. (Recewed: 9 March 1987) Abstract. A method of finding the moduli space of de Witt super-Riemann surfaces in any genus is described. The approach is to use cohomology to compute the homomorphisms from the surface group into the superconformal group; it turns out that this is a simpler problem, with a smaller class of solutions than had been suspected. The description of super-Telchmtiller space as a complex supermamfold is confirmed. 1. Results It is generally agreed that the moduli spaces of super-Riemann surfaces, or 'super- Teichmiiller spaces' (STS) are basic objects of superstring theory [1]. The aim here is to give a simple description of how to get to them, independent of metrics and other considerations. The framework is the de Witt theory of supermanifolds [2] in its complex (dimension 1) version. The main results are as follows. Let Fg be the fundamental group for (compact) surfaces of genus g: F~= ( al, b I..... ag, bg:a~blallb~'...ag'bg 1 = 1 ) [3] (1) When g = 0, the STS is trivial [4], so we restrict attention to g > 0. The 'genus g superconformal group', SCf(g), is the group of all superconformal automorphisms of X, where X is (i) the super-complex plane sC if g = 1, (ii) the super-half plane SU if g > 1. THEOREM 1. (i) The super-Teichmiiller space in genus g, or STS(g), can be identified with the set of all homomorphisms f: Fg ---,SCf(g) up to conjugation in SCf(g); subject to the condition that the body o f f must be a properly discontinuous monomorphism. (ii) S TS (g) is a complex analytic supermanifold of dimension (3g - 3 [2g - 2) ifg > 1, (1 [ 1) if g = 1 and the underlying spin structure is trivial, (1 IO) if g = 1 and the spin structure is nontrivial. (iii) The bo@ of STS(g) is the manifold of pairs (marked Riemann surface, spin structure) in genus g. The ideas underlying the theorem are simple, but they require some technicalities. Full details will be given in [5]; a description of the main points follows below.