ALE manifolds and Conformal Field Theory

TL;DR

The work develops a framework to assign (4,4) superconformal field theories to ALE manifolds ${\cal M}_{\zeta}$ by employing Kronheimer's HyperKähler quotient, interpreting these theories as deformations of solvable orbifold CFTs ${\bf C}^2/\Gamma$ where $\Gamma$ is Kleinian. It demonstrates a deep correspondence between topological data of self-dual 4-manifolds (notably the Hirzebruch signature $\tau$) and algebraic structures of non-rational (4,4) theories, identifying $\tau$ with the dimension of the local ring ${\cal R}= {\bf C}[x,y,z]/\partial W$, the number of nontrivial conjugacy classes of $\Gamma$, and the count of short representations minus four. Through the orbifold construction and N=4 character analysis, the paper shows how the moduli of ALE spaces map to CFT data via deformations governed by the chiral ring and twisted sectors, linking geometric resolution data to CFT marginal operators. The results illuminate how non-compact Calabi–Yau-like geometries can be exactly described by non-rational CFTs, with implications for gravitational instanton physics and the interface between geometry and conformal field theory. Open questions remain regarding the precise matching of Chiral rings ${\cal R}$ with CFT operator products and the full mapping between geometric moduli ($\zeta$, $t_i$, and $\xi$) across the HyperKähler and CFT formalisms.

Abstract

We address the problem of constructing the family of (4,4) theories associated with the sigma-model on a parametrized family ${\cal M}_ζ$ of Asymptotically Locally Euclidean (ALE) manifolds. We rely on the ADE classification of these manifolds and on their construction as HyperKähler quotients, due to Kronheimer. So doing we are able to define the family of (4,4) theories corresponding to a ${\cal M}_ζ$ family of ALE manifolds as the deformation of a solvable orbifold ${\bf C}^2 \, / \, Γ$ conformal field-theory, $Γ$ 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 $τ$ with the dimension of the local polynomial ring ${\cal R}=ø{{\bf C}[x,y,z]}{\partial W}$ 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.

Figures (4)

• Figure 1: Interpretation of the numbers $k_i$
• Figure 2: Extended Dynkin diagrams of the infinite series
• Figure 3: Exceptional extended Dynkin diagrams
• Figure 4: Long and short representations