A Note on Conifolds

We present the Ricci-flat metric and its Kähler potential on the conifold with theO(N) isometry, whose conical singularity is repaired by the complex quadric surfaceQ = SO(N)/SO(N− 2)× U(1). ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] Introduction. Conformally invariant nonlinear sigma models withN = 2 supersymmetry in twodimensions can describe the superstring in curved space. The target space is a Ricci-flat Kähler manifold by the requirement of finiteness [1, 2, 3]. In the previous letter [4], we presented the simple derivation of the Ricci-flat metric on the deformed conifold with the O(N) isometry, whose conical singularity is removed by SN−1. It coincides with the Stenzel metric on the cotangent bundle over SN−1 [5], and includes the Eguchi-Hanson gravitational instanton [6] and the six-dimensional deformed conifold [7, 8] in the cases of N = 3 and N = 4, respectively. The metric contains the deformation parameter, and the manifold becomes a conifold when the parameter vanishes. In this letter, we present the explicit form of the Ricci-flat metric and its Kähler potential on the conifold, whose conical singularity is repaired by the complex quadric surface QN−2 ≡ SO(N)/SO(N − 2)×U(1). It contains a resolution parameter b as an integration constant, which controls the size of QN−2. The limit of b → 0 corresponds to the conifold, which coincides with the singular limit of the deformed conifold. Our manifold can be interpreted as the line bundle over QN−2. The four-dimensional manifold of N = 3 is again the Eguchi-Hanson space, in which the conical singularity is removed by Q1 ≃ S2. In the case of the six-dimensional manifold of N = 4, the conical singularity is repaired by Q2 ≃ S2 × S2, and it gives a way to repair the singularity different from the deformation by S3 [7, 8] or the so-called small resolution by S2 [7, 9]. Definition of the model. N = 2 supersymmetric nonlinear sigma models in two dimensions are described by the chiral superfields φα(x, θ, θ̄) and the Kähler potential K(φ,φ∗) [10]. The Lagrangian is given by L = ∫ d4θK = gαβ∗(φ,φ)∂μφ∂φ + · · · , where the Kähler metric is defined by gαβ∗ = ∂α∂β∗K with ∂α = ∂/∂φα and ∂α∗ = ∂/∂φ∗α. (Here we have used the same letters for chiral superfields and their components.) First, we prepare chiral superfields φA(x, θ, θ̄) (A = 1, 2, · · · , N ; N ≥ 3), constituting the vector ~ φ(x, θ, θ̄) of O(N). We define the O(N) symmetric target space by imposing the constraint N