Conformal field theory and the cohomology of the moduli space of stable bundles

Abstract

Let Σg be a compact Riemann surface of genus g ≥ 2, and let Λ be a line bundle over Σg of degree 1. Then the moduli space of rank 2 stable bundles V over Σg such that ΛV ∼= Λ was shown by Seshadri [19] to be a nonsingular projective variety Ng . Its rational cohomology ring H(Ng ) is exceedingly rich and despite years of study has never quite been computed in full. The object of this paper is to give an essentially complete characterization of this ring, or at least to reduce the problem to a matter of linear algebra. In particular, we find a proof of Newstead’s conjecture that pg1(Ng ) = 0. We also obtain a formula for the volume of Ng , which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by Witten. The approach we shall take is not from algebraic geometry but from mathematical physics: it relies on the SU(2) Wess-Zumino-Witten model, which is a functor Zk associating a finite-dimensional vector space to each Riemann surface with marked points. The relationship with the moduli space is that when the “level” k of the functor is even, the vector space associated to Σg with no marked points can be identified with H(Ng ;L), where L is a fixed line bundle over Ng . Now the work of Verlinde provides us with a means of calculating the dimension of any vector space arising from our functor, and in particular dimH(Ng ;L), which we shall denote D(g, k). On the other hand Newstead [14] found explicit generators for H(Ng ), and we can also express D(g, k) in terms of them using a Riemann-Roch theorem. Equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of Ng , and by Poincare duality this is sufficient, at least in principle, to determine the ring structure of H(Ng ). In the discussion above one important point has been skated over. The SU(2) WZW model is of course associated to bundles of degree 0, not degree 1, so in order to make use of Verlinde’s work it is necessary to formulate a “twisted” version of the field theory. This is carried out in §2, but the crucial properties of the twisted theory, analogous to those which make the ordinary theory a modular functor, are not proved in this paper. Rather, we will confine ourselves to exploring the consequences of these claims, and hope to return to justify them in a later paper. An outline of the remaining sections goes as follows. In §3 we review those parts of Verlinde’s work we shall need, show how they must be modified in the twisted case, and work out some explicit formulas for D(g, k). In §4 we study the cohomology of Ng , which we regard throughout as a space of representations via the theorem of Narasimhan and Seshadri. We define Newstead’s generators α , β , and ψi of H (Ng )