Enhanced Euclidean supersymmetry, 11D supergravity and $SU(\infty )$ Toda equation

Abstract

We show how to lift solutions of Euclidean Einstein-Maxwell equations with non-zero cosmological constant to solutions of eleven-dimensional supergravity theory with non-zero fluxes. This yields a class of 11D metrics given in terms of solutions to $SU(\infty )$ Toda equation. We give one example of a regular solution and analyse its supersymmetry. We also analyse the integrability conditions of the Killing spinor equations of N=2 minimal gauged supergravity in four Euclidean dimensions. We obtain necessary conditions for the existence of additional Killing spinors, corresponding to enhancement of supersymmetry. If the Weyl tensor is anti-self-dual then the supersymmetric metrics satisfying these conditions are given by separable solutions to the $SU(\infty )$ Toda equation. Otherwise they are ambi-Kähler and are conformally equivalent to Kähler metrics of Calabi type or to product metrics on two Riemann surfaces.

19 pages. The relationship between our solutions, and those found by Martelli, Passias and Sparks clarified. A discussion of sufficient conditions for supersymmetry added. Final version, to appear in JHEP

References in corpus (4)

• The supersymmetric NUTs and bolts of holography
• N=31 is not IIB
• Cosmological Einstein-Maxwell Instantons and Euclidean Supersymmetry: Beyond Self-Duality
• Self-Dual Conformal Gravity

Cited by in corpus (2)

• On 2-dimensional Kaehler metrics with one holomorphic isometry
• Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors