The Gibbons-Hawking ansatz in generalized K\"ahler geometry

TL;DR

This paper generalizes the Gibbons-Hawking ansatz to generalized K"ahler surfaces with symmetry, deriving a local construction that includes an arbitrary function and classifying solutions under certain equations.

Contribution

It introduces a new local ansatz for generalized K"ahler surfaces with symmetry, extending the classic Gibbons-Hawking approach and classifies solutions satisfying specific geometric and physical equations.

Findings

Derived a local ansatz for generalized K"ahler surfaces with symmetry.

Allowed for an arbitrary function in the ansatz, generalizing previous models.

Classified all complete solutions with minimal symmetry group under certain equations.

Abstract

We derive a local ansatz for generalized K\"ahler surfaces with nondegenerate Poisson structure and a biholomorphic $S^1$ action which generalizes the classic Gibbons-Hawking ansatz for invariant hyperK\"ahler manifolds, and allows for the choice of one arbitrary function. By imposing the generalized K\"ahler-Ricci soliton equation, or equivalently the equations of type IIB string theory, the construction becomes rigid, and we classify all complete solutions with the smallest possible symmetry group.