D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of K\"ahler-Einstein surfaces

TL;DR

This paper constructs explicit D3-brane supergravity solutions using Ricci-flat metrics on canonical bundles over Kähler-Einstein surfaces, revealing new classes of solutions with integrable geodesic equations.

Contribution

It introduces a new class of Ricci-flat metrics on canonical bundles over Kähler-Einstein surfaces, derived via the AMSY formalism, and demonstrates their application to supergravity solutions.

Findings

Existence of a two-parameter subclass of Kähler-Einstein metrics on $S^2\times S^2$

Derivation of new formulas for Ricci-flat metrics via the Calabi Ansatz

Proof of integrability of geodesic equations with a Carter-like constant

Abstract

D3-brane solutions of type IIB supergravity can be obtained by means a classical ansatz involving a harmonic warp factor and two summands, the first being the flat Minkowskian metric of the D3 brane world-sheet and the second a Ricci flat metric on a suitable 6-dimensional transverse space, both twisted by the warp factor. Of particular interest is the case of the total space of thecanonical bundle over a complex K\"ahler 2-fold. This situation emerges in many cases while considering the resolution of finite quotient singulaties. When the group is $\mathbb{Z}_4$, the complex 2-fold is the second Hirzebruch surface endowed with a K\"ahler metric having SU(2)xU(1) isometry. There is actually an entire class of such metrics parameterized by a single function, and best described in the AMSY symplectic formalism. We recover the existence of a two parameter subclass of K\"ahler-Einstein metrics on manifolds that are homeorphic to $S^2\times S^2$, and study in detail this class. The K"ahler-Einstein nature of these manifolds allows the construction of the Ricci flat metric on their canonical bundle via the Calabi Ansatz, which we recast in the AMSY formalism deriving some new elegant formulae. Furthermore we show the full integrability of the differential system of geodesics equations thanks to an additional conserved quantity that we unveil and which is similar to the Carter constant in the case of the Kerr metric.