ALE MANIFOLDS AND CONFORMAL FIELD-THEORIES

We address the problem of constructing the family of (4,4) theories as sociated with the sigma model on a parametrized family M(zeta) of asym ptotically locally Euclidean (ALE) manifolds. We rely on the ADE class ification of these manifolds and on their construction as hyper-Kahler quotients, due to Kronheimer. By so doing we are able to define the f amily of (4,4) theories corresponding to a M(zeta) family of ALE manif olds as the deformation of a solvable orbifold C2/GAMMA conformal fiel d theory, GAMMA being a Kleinian group. We discuss the relation betwee n the algebraic structure underlying the topological and metric proper ties of self-dual four-manifolds and the algebraic properties of nonra tional (4,4) theories admitting an infinite spectrum of primary fields . In particular, we identify the Hirzebruch signature tau with the dim ension of the local polynomial ring R = C[x,y,z]/partial derivative W associated with the ADE singularity, with the number of nontrivial con jugacy classes in the corresponding Kleinian group and with the number of short representations of the (4,4) theory minus four.