Aspects of type IIB theory on asymptotically locally Euclidean spaces

Abstract

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 ${R}^{6}\ifmmode\times\else\texttimes\fi{}\mathcal{M},$ 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 ``hyper-K\"ahler 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 \ensuremath{\sigma} models to conformal field theories.