Commun. Math. Phys. 140,149 157 (1991) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP Communications inzyxwvutsrq Mathematical Physics © Springer Verlag 1991 Anomalies and Curvature of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM W M anifolds C. N. Pope 1 *, L. J. Romans 2 *, E. Sezgin 1 and X. Shen 1 1 Center for Theoretical Physics, Texas A&M University, College Station, TX 77843 4242, USA 2 Department of Physics, University of Southern California, Los Angeles, CA 90089 0484, USA Received December 4, 1990 Abstract. We study the holomorphic structure of certain complex manifolds associated withzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED W^ algebras, namely, the flag manifolds WJT^ and ^i + JT ι + oo9 and the spaces WJSL{oo,R) and PF 1 + oo /GL(oo,K), where Γ *, and T 1+ o0 are the maximal tori in W^ and W 1 + oo . We compute their Ricci curvature and show how the results are related to the anomaly freedom conditions for W^ and l^n oo. We discuss the relation of these manifolds with extensions of universal Teichmϋ ller space. 1. Introduction An important problem in string theory is the search for a better understanding of its geometrical underpinnings, in the spirit of the beautiful interpretation of general relativity in terms of Riemannian geometry. It has been argued that a natural arena for addressing such geometrical issues is provided by the study of the manifold Jt = diff (S 1 )/^ 1 of complex structures on loop space related by reparametrisations [1]. This remarkable manifold proves to possess a natural Kahler structure [1], and it has been found that many statements concerning the consistency of string theory can be reformulated in terms of geometric data for Jt or related structures [1 5]. For example, the condition of nilpotency for the BRST charge Q (required for quantisation in the BRST formalism) is replaced by the requirement that a certain vector bundle over J( have vanishing Ricci curvature [1,2]. In this paper, we show that this geometrical formalism admits very natural extensions when one replaces the algebra difϊ XS 1 ) (essentially the centreless Virasoro algebra) by certain higher spin extended algebras which have been * Supported in part by the U.S. Department of Energy, under grant DE AS05 81ER40039 ** Supported in part by the U.S. Department of Energy, under grant DE FG03 84ER40168