Finite-dimensionality of the Space of Conformal Blocks

Without using Gabber’s theorem, the finite-dimensionality of the space of conformal blocks in the Wess-Zumino-Novikov-Witten models is proved. §0 Introduction. Conformal field theory with non-abelian gauge symmetry, called the Wess-ZuminoNovikov-Witten (WZNW) model, has been studied by many physicists and mathematicians. A mathematical formulation of this model over the projective line is given in [TK], and it is generalized in [TUY] over algebraic curves of arbitrary genus. In (chiral) conformal field theory, the main objects are N -point functions, and in [TUY] they are regarded as sections of a certain vector bundle over the moduli space of N -pointed stable curves. The fiber of this bundle at a stable curve X is called the space of conformal blocks (or the space of vacua) attached to X. The finite-dimensionality of this space is essential to the mathematical treatment of conformal field theory, and is proved in [TUY] as a consequence of Gabber’s theorem [Ga] stating the involutivity of characteristic varieties. The aim of the present paper is to give a proof of the finite-dimensionality without using Gabber’s theorem. In §1 we summarize some basic facts on affine Lie algebras following [Ka]. In §2 and §3, we recall the definition of pointed stable curves and the space of conformal blocks. The essence in our proof is explained in §4 and the complete proof is given in §5. §1. Integrable highest weight modules of Affine Lie algebras. By C[[t]] and C((t)), we mean the ring of formal power series in t and the field of formal Laurent series in t, respectively. Let g be a simple Lie algebra over C, h its Cartan subalgebra and h the dual space of h over C. By △, we denote the root system of (g, h), and for α ∈ △ we denote the root vector corresponding to α by Xα. Let ( , ) : g × g → C be the Cartan-Killing form normalised by the condition (Hθ, Hθ) = 2, Typeset by AMS-TEX 1