BPS D3-branes on smooth ALE manifolds

TL;DR

This paper reviews new classical BPS solutions in type IIB supergravity describing D3-branes on ALE manifolds, characterized by fluxes through 2-cycles and preserving $ ext{N}=2$ supersymmetry, with solutions involving a quasi harmonic warp factor.

Contribution

It introduces a new family of solutions for D3-branes on smooth ALE spaces, analyzing the warp factor's properties and its relation to the confluent Heun equation.

Findings

Solutions preserve $ ext{N}=2$ supersymmetry.

Warp factor satisfies a confluent Heun equation.

Explicit analysis for Eguchi-Hanson ALE manifold.

Abstract

In this talk I review the recent construction of a new family of classical BPS solutions of type IIB supergravity describing 3-branes transverse to a 6-dimensional space with topology $\mathbb{R}^{2}\times$ALE. They are characterized by a non-trivial flux of the supergravity 2-forms through the homology 2-cycles of a generic smooth ALE manifold. These solutions have two Killing spinors and thus preserve $\mathcal{N}=2$ supersymmetry. They are expressed in terms of a quasi harmonic function $H$ (the ``warp factor''), whose properties was studied in detail in the case of the simplest ALE, namely the Eguchi-Hanson manifold. The equation for $H$ was identified as an instance of the confluent Heun equation.