Instanton Strings and HyperKaehler Geometry

TL;DR

This paper explores the relationship between instanton moduli spaces on K3 surfaces, hyperkaehler geometry, and their realization in superconformal field theories, revealing how background fields deform these geometries.

Contribution

It establishes a detailed correspondence between string moduli, conformal field theories, and hyperkaehler geometries, including the deformation from symmetric products to Hilbert schemes.

Findings

The instanton moduli space metric degenerates to the symmetric product of K3.

A self-dual B-field deforms the symmetric product into a Hilbert scheme.

The work links string duality with hyperkaehler deformation theory.

Abstract

We discuss two-dimensional sigma models on moduli spaces of instantons on K3 surfaces. These N=(4,4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS_3. We derive a precise map relating the moduli of the K3 type IIB string compactification to the moduli of these conformal field theories and the corresponding classical hyperkaehler geometry. We conclude that, in the absense of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K3. Turning on a self-dual NS B-field deforms this symmetric product to a manifold that is diffeomorphic to the Hilbert scheme. We also comment on the mathematical applications of string duality to the global issues of deformations of hyperkaehler manifolds.