[Paper Review] On 2-dimensional Kaehler metrics with one holomorphic isometry

Samuele Chimento, Tomás Ortı́n|arXiv (Cornell University)|Oct 6, 2016

Black Holes and Theoretical Physics9 references3 citations

TL;DR

This paper presents a general ansatz for 2-dimensional Kähler metrics with one holomorphic isometry, expressing them as a deformation of Gibbons-Hawking metrics via two real functions $H$ and $W$. The key result is a closed-form metric structure that unifies hyper-Kähler metrics with tri-holomorphic isometries and scalar-flat Kähler metrics with mono-holomorphic isometries, providing a powerful ansatz for supersymmetric solutions in 5D and 6D gauged supergravity.

ABSTRACT

We show how to write any Kaehler metric of complex dimension 2 admitting a holomorphic isometry as a simple 1-real-function deformation of a Gibbons-Hawking metric. Hyper-Kaehler metrics with a tri-holomorphic isometry (Gibbons-Hawking metrics) or with a mono-holomorphic isometry are recovered for particular values of the additional function. The new general metric can be used as an Ansatz in several interesting physical problems.

Motivation & Objective

• To close the gap in the literature by providing a closed-form expression for 2-dimensional Kähler metrics admitting one holomorphic isometry.
• To generalize Gibbons-Hawking metrics, which describe hyper-Kähler spaces with tri-holomorphic isometries, to include metrics with only one holomorphic isometry.
• To enable the construction of supersymmetric solutions in 5D and 6D Fayet-Iliopoulos U(1)-gauged supergravity by providing a tractable metric ansatz.
• To unify known special cases—such as scalar-flat Kähler metrics and hyper-Kähler metrics with mono-holomorphic isometries—within a single, coherent framework.

Proposed method

• The metric is constructed in the form $ds^2 = H^{-1}(dz + ilde\chi)^2 + H\left[(dx^1)^2 + W^2((dx^2)^2 + (dx^3)^2)\right]$, where $H$ and $W$ are real functions of $x^1, x^2, x^3$, and $\tilde\chi$ is a 1-form on the base space.
• The 1-form $\tilde\chi$ is determined by the constraints $(d\tilde\chi)_{ij} = \partial_k H$ for appropriate index combinations, ensuring the metric is Kähler and admits a holomorphic isometry.
• The integrability condition $\mathfrak{D}^2 H = \partial_1^2 H + \partial_2^2 (W^2 H) + \partial_3^2 H = 0$ ensures the existence of a globally defined $\tilde\chi$ up to closed 1-forms.
• The ansatz reduces to Gibbons-Hawking metrics when $W$ is constant and $H$ is harmonic, and to scalar-flat Kähler metrics when $H = \partial_2 \log W^2$ and $W$ satisfies the Toda equation.
• The construction uses a vierbein formalism and spin connection to derive the integrability conditions and verify the Kähler and holomorphic isometry properties.
• The method is applied to the non-compact symmetric space $\overline{\mathbb{CP}}^2$, showing that its metric fits the general ansatz with explicit expressions for $H$, $W$, and $\tilde\chi$.

Experimental results

Research questions

• RQ1 Can a general closed-form expression be derived for 2-dimensional Kähler metrics admitting a single holomorphic isometry, beyond the known Gibbons-Hawking class?
• RQ2 How do known special cases—such as hyper-Kähler metrics with mono-holomorphic isometries and scalar-flat Kähler metrics—fit into a unified framework?
• RQ3 What are the necessary and sufficient conditions on the functions $H$ and $W$ for the metric to be Kähler and admit a holomorphic isometry?
• RQ4 How does the conformal factor $W$ act as an obstruction to the metric being hyper-Kähler with tri-holomorphic isometry?
• RQ5 Can this ansatz be used to systematically construct supersymmetric solutions in 5D and 6D gauged supergravity?

Key findings

• The general Kähler metric with one holomorphic isometry is given by $ds^2 = H^{-1}(dz + \tilde\chi)^2 + H\left[(dx^1)^2 + W^2((dx^2)^2 + (dx^3)^2)\right]$, with $H$, $W$, and $\tilde\chi$ depending only on $x^1, x^2, x^3$.
• The integrability condition $\mathfrak{D}^2 H = \partial_1^2 H + \partial_2^2 (W^2 H) + \partial_3^2 H = 0$ ensures the existence of a globally defined 1-form $\tilde\chi$ satisfying the required constraints.
• When $W$ is constant, the metric reduces to a Gibbons-Hawking metric, recovering hyper-Kähler metrics with tri-holomorphic isometries.
• When $H = \partial_2 \log W^2$, the metric becomes scalar-flat Kähler and admits a hyper-Kähler structure with a mono-holomorphic isometry.
• The example of $\overline{\mathbb{CP}}^2$ is shown to fit the general ansatz, with explicit expressions for $H$, $W^2$, and $\tilde\chi$ derived from its standard metric.
• The conformal factor $W$ acts as a measure of the obstruction to the metric being hyper-Kähler with tri-holomorphic isometry, as the integrability condition reduces to the Laplace equation only when $W$ is constant.