D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kahler-Einstein surfaces

D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, H(y, y ($) over bar) multiplying at power -1/2 the first summand, i.e., the Minkowski metric of the D3 brane world-sheet, and at power 1/2 the second summand, i.e., the Ricci-flat metric on a six-dimensional transverse space M-6, whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where M6 = tot[ K [(MB)] is the total space of the canonical bundle over a complex Kahler surfaceMB. This situation emerges in many cases while considering the resolution a la Kronheimer of singular manifolds of type M-6 = C-3/ Gamma, where Gamma subset of SU(3) is a discrete subgroup. When Gamma = Z(4), the surface M-B Phi is the second Hirzebruch surface endowed with a Kahler metric having SU(2) x U(1) isometry. There is an entire class Met(FV) of such cohomogeneity one Kahler metrics parameterized by a single function FK( v) that are best described in the Abreu-Martelli-Sparks-Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass Met(FV) (KE). Met(FV) of Kahler-Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to S(2)xS(2), but are singular as complex manifolds. Actually, Met(FV) (KE) subset of Met(FV) (ext). Met(FV) is a subset of a four parameter subclass Met(FV) (ext) of cohomogeneity one extremal Kahler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism. Met(FV) (ext) contains also a two-parameter subclass Met(FV) (extF2) disjoint from Met(FV) (KE) of extremal smooth metrics on the second Hirzebruch surface that we rederive using constraints on period integrals of the Ricci 2-form. The Kahler-Einstein nature of the metrics inMet(FV) (KE) 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. The metrics

Automated summary

D3 brane solutions of type IIB supergravity can be obtained using a classical Ansatz with a harmonic warp factor. Of particular interest is the case where the transverse space is a complex Kahler surface. There is a subclass of Kahler-Einstein metrics on these surfaces called Met(FV) (KE).