ALE MANIFOLDS AND CONFORMAL FIELD-THEORIES

We address the problem of constructing the family of (4,4) theories associated with the sigma model on a parametrized family M(zeta) of asymptotically locally Euclidean (ALE) manifolds. We rely on the ADE classification of these manifolds and on their construction as hyper-Kahler quotients, due to Kronheimer. By so doing we are able to define the family of (4,4) theories corresponding to a M(zeta) family of ALE manifolds as the deformation of a solvable orbifold C2/GAMMA conformal field theory, GAMMA being a Kleinian group. We discuss the relation between the algebraic structure underlying the topological and metric properties of self-dual four-manifolds and the algebraic properties of nonrational (4,4) theories admitting an infinite spectrum of primary fields. In particular, we identify the Hirzebruch signature tau with the dimension of the local polynomial ring R = C[x,y,z]/partial derivative W associated with the ADE singularity, with the number of nontrivial conjugacy classes in the corresponding Kleinian group and with the number of short representations of the (4,4) theory minus four.