Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-04-27
Int.J.Mod.Phys.A9:3007-3058,1994
Physics
High Energy Physics
High Energy Physics - Theory
48 pages, Latex, SISSA/44/92/EP, IFUM/443/FT
Scientific paper
10.1142/S0217751X94001199
We address the problem of constructing the family of (4,4) theories associated with the sigma-model on a parametrized family ${\cal M}_{\zeta}$ of Asymptotically Locally Euclidean (ALE) manifolds. We rely on the ADE classification of these manifolds and on their construction as HyperK\"ahler quotients, due to Kronheimer. So doing we are able to define the family of (4,4) theories corresponding to a ${\cal M}_{\zeta}$ family of ALE manifolds as the deformation of a solvable orbifold ${\bf C}^2 \, / \, \Gamma$ conformal field-theory, $\Gamma$ being a Kleinian group. We discuss the relation among the algebraic structure underlying the topological and metric properties of self-dual 4-manifolds and the algebraic properties of non-rational (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 ${\cal R}=\o {{\bf C}[x,y,z]}{\partial W}$ associated with the ADE singularity, with the number of non-trivial conjugacy classes in the corresponding Kleinian group and with the number of short representations of the (4,4)-theory minus four.
Anselmi Damiano
Billo' Marco
Fre' Pietro
Girardello Luciano
Zaffaroni Alberto
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