ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA $\mathcal{W}(p)$: A-SERIES

Motivated by [On the triplet vertex algebra [Formula: see text], Adv. Math.217 (2008) 2664–2699], for every finite subgroup Γ ⊂ PSL(2, ℂ) we investigate the fixed point subalgebra [Formula: see text] of the triplet vertex [Formula: see text], of central charge [Formula: see text], p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, ℂ). First, we prove the C2-cofiniteness of the Am-fixed subalgebra [Formula: see text]. Then we construct a family of [Formula: see text]-modules, which are expected to form a complete set of irreducible representations. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for m = 2. We also present a rigorous proof of the fact that the full automorphism group of [Formula: see text] is PSL(2, ℂ).

STRUCTURE CONSTANTS OF THE D5, CHIRAL, MINIMAL MODEL

The operator product algebra of the simplest chiral, minimal model, D5, in the ADE classification, is determined. The construction uses the Coulomb gas technique that was already employed in the nonchiral A series. This technique is supplemented by conformal blocks special to the D series. Explicit results for the structure constants of three-point correlations are given. They demonstrate that the ground state has a discrete symmetry, Z2.

ALE MANIFOLDS AND CONFORMAL FIELD THEORIES

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 [Formula: see text] 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.