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

Abstract

AbstractD3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, $$H(\textbf{y},\bar{\textbf{y}})$$ H ( y , y ¯ ) multiplying at power $$-1/2$$ - 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 $$\mathcal {M}_6$$ M 6 , whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where $$\mathcal {M}_6={\text {tot}}[ K\left[ \left( \mathcal {M}_B\right) \right] $$ M 6 = tot [ K M B is the total space of the canonical bundle over a complex Kähler surface $$\mathcal {M}_B$$ M B . This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type $$\mathcal {M}_6=\mathbb {C}^3/\Gamma $$ M 6 = C 3 / Γ , where $$\Gamma \subset \mathrm {SU(3)} $$ Γ ⊂ SU ( 3 ) is a discrete subgroup. When $$\Gamma = \mathbb {Z}_4$$ Γ = Z 4 , the surface $$\mathcal {M}_B$$ M B is the second Hirzebruch surface endowed with a Kähler metric having $$\mathrm {SU(2)\times U(1)}$$ SU ( 2 ) × U ( 1 ) isometry. There is an entire class $${\text {Met}}(\mathcal{F}\mathcal{V})$$ Met ( F V ) of such cohomogeneity one Kähler metrics parameterized by a single function $$\mathcal{F}\mathcal{K}(\mathfrak {v})$$ F K ( v ) that are best described in the Abreu–Martelli–Sparks–Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})$$ Met ( F V ) KE ⊂ Met ( F V ) of Kähler–Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to $$S^2\times S^2$$ S 2 × S 2 , but are singular as complex manifolds. Actually, $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})$$ Met ( F V ) KE ⊂ Met ( F V ) ext ⊂ Met ( F V ) is a subset of a four parameter subclass $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}$$ Met ( F V ) ext of cohomogeneity one extremal Kähler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism.$${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}$$ Met ( F V ) ext contains also a two-parameter subclass $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}\mathbb {F}_2}$$ Met ( F V ) ext F 2 disjoint from $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$$ Met ( F V ) 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 $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$$ Met ( F V ) 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 $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$$ Met ( F V ) KE are defined on the base manifolds of U(1) fibrations supporting the family of Sasaki–Einstein metrics $$\textrm{SEmet}_5$$ 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 $${\mathcal M}_5 \in {\text {Met}}(\textrm{SE})_5$$ M 5 ∈ Met ( SE ) 5 is different from the Ricci-flat metric on $${\text {tot}}[ K\left[ \left( \mathcal {M}_{\textrm{KE}}\right) \right] $$ tot [ K M KE constructed via Calabi Ansatz. This opens new research perspectives. We also show the full integrability of the differential system of geodesics equations on $$\mathcal {M}_B$$ M B thanks to a certain conserved quantity which is similar to the Carter constant in the case of the Kerr metric.