Explicit Toric Metric on Resolved Calabi-Yau Cone

We present an explicit non-singular complete toric Calabi-Yau metric using the local solution recently found by Chen, Lu and Pope. This metric gives a new supergravity solution representing D3-branes.

D3-branes on the tip of toric Calabi-Yau cones have been extensively studied in connection with the AdS/CFT correspondence [1]. It is natural to consider the deformations of cone metrics in order to explore non-conformal theories [2,3,4,5,6,7].
In this letter, we study a Calabi-Yau metric, i.e. Ricci-flat Kähler metric, constructed as the BPS limit of the six dimensional Euclideanised Kerr-NUT-AdS black hole metric [8]. In the black hole with equal angular momenta, the corresponding Calabi-Yau metric is of the form with two parameters a and b. We assume that the roots x i (i = 1, 2, 3) of f (x) = 0 are all distinct and real, and further they are ordered as z 1 < x 1 < x 2 < x 3 for the smallest real root z 1 of h(z) = 0. In order to avoid a curvature singularity we take the coordinates x and z to lie in the region z ≤ z 1 < x 1 ≤ x ≤ x 2 < 1. Indeed, it is easy to see that such a singularity appears at z = x.
For r → ∞ (z = −r 2 /2) , the metric tends to a cone metric dr 2 + r 2ḡ , wherē This metric yields the Sasaki-Einstein metric Y p,q when we impose a suitable condition for the parameter a [9].
Next let us look at the geometry near z = z 1 . We introduce new coordinates given by Then the metric behaves as where . Therefore, the periodicity of ϕ 1 should be 2π in order to avoid an orbifold singularity. The four dimensional Kähler metric g (4) is given by We now argue that by taking the special parameters a = (1/2) − (1/32) √ 13 which corresponds to Y 2,1 , and b = (1/4)(137 + 37 √ 13), the four dimensional space with metric g (4) is a non-trivial S 2 -bundle over S 2 , i.e. the first del Pezzo surface dP 1 . To see this, introduce a radial coordinate space defined by fixing the S 2 coordinates θ and ϕ 3 in (6). Then, the fibre metric near boundary x = x i (i = 1, 2) is written as Using the values of a and b, we have and then x i (1−z 1 )/(x i −z 1 ) = ±1/2. The apparent singularities at x = x i can be avoided by choosing the periodicity of ϕ 2 to be 4π. Thus, the (x, ϕ 2 )-fibre space is topologically On the other hand, fixing the coordinate x in (6), we obtain a principal U(1)-bundle over S 2 with the Chern number The metric g (4) can be regarded as a metric on the associated S 2 -bundle of the principal U(1)-bundle. The associated bundle is non-trivial since the Chern number is odd, and hence the total space is the dP 1 .
Let us describe the Calabi-Yau metric (1) from the point of view of toric geometry.
The metric has an isometry T 3 , locally generated by the Killing vector fields, ∂/∂ψ, ∂/∂φ and ∂/∂β. The symplectic (Kähler) form ω is given by Using the following generators of the T 3 action [10], one has Darboux coordinates (ξ 1 , ξ 2 , ξ 3 ) on which the symplectic form takes the standard form ω = dξ i ∧ dφ i : with ℓ −1 = −5 + 2 √ 13. For the range of variables: 0 ≤ θ ≤ π, where x 1 , x 2 and z 1 are given by (8), we find the Delzant polytope (see Fig.1) Here, each v a is a primitive element of the lattice Z 3 ⊂ R 3 and an inward-pointing normal vector to the two dimensional face of P . Explicitly, the set of five vectors v a = (1, w a ) can be chosen as (see Fig. 2) The constants λ a are given by Thus, we see that the five faces F a = {ξ ∈ R 3 | (ξ, v a ) = λ a } correspond to degeneration surfaces at x = x 1 , θ = 0, x = x 2 , θ = π and z = z 1 , respectively.
Finally, we note that a D3-brane solution can be constructed from the Calabi-Yau metric given by g (1) with the special parameters a, b: We find the warp factor H as a harmonic function △ g H = 0 ; where the constant L is given by For large −z = r 2 /2, the warp factor behaves as while near z = z 1 H ≃ 2(7 − 2 √ 13)L 4 27 log(z 1 − z).