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

Abstract

D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, H(textbf{y},bar{textbf{y}}) 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 mathcal {M}_6, whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where mathcal {M}_6={text {tot}}[ Kleft[ left( mathcal {M}_Bright) right] is the total space of the canonical bundle over a complex Kähler surface mathcal {M}_B. This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type mathcal {M}_6=mathbb {C}^3/Gamma , where Gamma subset mathrm {SU(3)} is a discrete subgroup. When Gamma = mathbb {Z}_4, the surface mathcal {M}_B is the second Hirzebruch surface endowed with a Kähler metric having mathrm {SU(2)times U(1)} isometry. There is an entire class {text {Met}}(mathcal{F}mathcal{V}) of such cohomogeneity one Kähler metrics parameterized by a single function mathcal{F}mathcal{K}(mathfrak {v}) that are best described in the Abreu–Martelli–Sparks–Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}}subset {text {Met}}(mathcal{F}mathcal{V}) of Kähler–Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to S^2times S^2, but are singular as complex manifolds. Actually, {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}}subset {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}}subset {text {Met}}(mathcal{F}mathcal{V}) is a subset of a four parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}} of cohomogeneity one extremal Kähler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism.{text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}} contains also a two-parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}mathbb {F}_2} disjoint from {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} of extremal smooth metrics on the second Hirzebruch surface that we rederive using constraints on period integrals of the Ricci 2-form. The Kähler–Einstein nature of the metrics in {text {Met}}(mathcal{F}mathcal{V})_{textrm{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 in {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} are defined on the base manifolds of U(1) fibrations supporting the family of Sasaki–Einstein metrics textrm{SEmet}_5 introduced by Gauntlett et al. (Adv Theor Math Phys 8:711–734, 2004), and already appeared in Gibbons and Pope (Commun Math Phys 66:267–290, 1979). However, as we show in detail using Weyl tensor polynomial invariants, the six-dimensional Ricci-flat metric on the metric cone of {mathcal M}_5 in {text {Met}}(textrm{SE})_5 is different from the Ricci-flat metric on {text {tot}}[ Kleft[ left( mathcal {M}_{textrm{KE}}right) right] constructed via Calabi Ansatz. This opens new research perspectives. We also show the full integrability of the differential system of geodesics equations on mathcal {M}_B thanks to a certain conserved quantity which is similar to the Carter constant in the case of the Kerr metric.