ALE MANIFOLDS AND CONFORMAL FIELD THEORIES

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Abstract
We address the problem of constructing the family of (4,4) theories associated with the σ model on a parametrized family ℳζ of asymptotically locally Euclidean (ALE) manifolds. We rely on the ADE classification of these manifolds and on their construction as hyper-Kähler quotients, due to Kronheimer. By so doing we are able to define the family of (4,4) theories corresponding to a ℳζ family of ALE manifolds as the deformation of a solvable orbifold C2/Γ conformal field theory, Γ 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 τ with the dimension of the local polynomial ring 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.
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