Aspects of Type IIB Theory on ALE Spaces

D–brane technology and strong/weak coupling duality supplement traditional orbifold techniques by making certain background geometries more accessible. In this spirit, we consider some of the geometric properties of the type IIB theory on IR×M where M is an ‘Asymptotically Locally Euclidean (ALE)’ gravitational instanton. Given the self– duality of the theory, we can extract the geometry (both singular and resolved) seen by the weakly coupled IIB string by studying the physics of a D1–brane probe. The construction is both amusing and instructive, as the physics of the probe completely captures the mathematics of the construction of ALE instantons via ‘HyperKähler Quotients’, as presented by Kronheimer. This relation has been noted by Douglas and Moore for the A–series. We extend the explicit construction to the case of the D– and E–series — uncovering a quite beautiful structure — and highlight how all of the elements of the mathematical construction find their counterparts in the physics of the type IIB D–string. We discuss the explicit ALE metrics which may be obtained using these techniques, and comment on the role duality plays in relating gauged linear sigma models to conformal field theories.