D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler–Einstein surfaces
Authors: Bruzzo Ugo , Fré Pietro , Shahzad Umar , Trigiante Mario 
Published: June 2023

D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, $$H(\textbf{y},\bar{\textbf{y}})$$

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.