Hessian Geometry and Phase Change of Gibbons-Hawking Metrics

Abstract

We study the Hessian geometry of toric Gibbons-Hawking metrics and their phase change phenomena via the images of their moment maps.

Hessian Geometry and Phase Changes of Multi-Taub-NUT Metrics

“…A simple modification by changing V to V ǫ in the proof of Theorem 4.1 in [18] then proves the following: Theorem 3.1. The metrics g ǫ and Kähler forms ω ǫ are given in local complex coordinates z 1 , z 2 as follows:…”

“…Based on the computations in an early paper [18], we consider the Hessian geometry of toric multi-Taub-NUT metrics. This means to find the moment maps together with their images, and complex potential functions on the moment map images, and use them to introduce some local complex coordinates to express the metrics and the Kähler forms.…”

“…Symplectic coordinates and Hessian geometry for toric multi-Taub-NUT spaces. The following result can be obtained by a simple modification of the corresponding result in [18]: Proposition 3.1. In the above symplectic coordinates, the multi-Taub-NUT metric takes the following form:…”

“…This is a sequel to [17] and [18] in which the Kepler metrics and the toric Gibbons-Hawking metrics are studied from the point of view of Hessian geometry [13], respectively. These are techniques developed in the setting of toric canonical Kähler metrics on compact toric manifolds in Kähler geometry [8,1,2], and also in the the study of toric Sasaki-Einstein metrics [7,11,12] that arise in AdS/CFT correspondence in string theory [10,16].…”

“…Our motivation is to test the applicability of some techniques developed in string theory to problems in classical gravity and general relativity, and to gain some new insights to these problems in doing this. For the examples in our earlier papers mentioned above, by examining the images of the moment maps, one can easily visualize the degenerations of Kähler metrics, more importantly, one can study their phase changes [18,15], a concept borrowed from statistical physics and introduced in [3,4] in the context of Kähler geometry. In this paper we will show that the results in [18] can be generalized to toric multi-Taub-NUT spaces with mild modifications, and more importantly, we will report a new type of phase transition not known from our earlier papers [3,4,18,15].…”