Abstract
Abstract We construct the first eleven-dimensional supergravity solutions, which are regular, have no smearing and possess only SO(2, 4) × SO(3) × U(1) R isometry. They are dual to four-dimensional field theories with $ \mathcal{N} $ = 2 superconformal symmetry. We utilise the Toda frame of self-dual four-dimensional Euclidean metrics with SU(2) rotational symmetry. They are obtained by transforming the Atiyah-Hitchin instanton under SL(2, $ \mathbb{R} $ ) and are expressed in terms of theta functions. The absence of any extra U(1) symmetry, even asymptotically, renders inapplicable the electrostatic description of our solution.
Highlights
Where the AdS5 and S2 have unit radii and γijdxidxj = dz2 + eΨ(dx2 + dy2)
The main result of this paper is the construction of the first in the literature solution of eleven-dimensional supergravity as dual of field theories with N = 2 superconformal symmetry which has only SO(2, 4) × SO(3) × U(1)R isometry
Our construction was made possible by making contact with solutions of the continual Toda equation corresponding to the four-dimensional Atiyah-Hitchin gravitational instanton and subsequent use of modular transformations in order to satisfy the appropriate boundary conditions
Summary
If ∂φ is a translational Killing vector we can always choose a coordinate system such that dV −1 = ± γ dω , γij = δij For this metric the self-duality (or anti-self duality) condition can be written as. Regular solutions of four-dimensional Euclidean self-dual metrics require that ∂zΨ = 0. This is not in conflict with the boundary condition (1.4) since the latter refers to eleven-dimensional metrics. This implies that we cannot take over solutions to Toda equations appropriate for the four-dimensional metrics (2.2) and use them in (1.1) since the resulting background will be singular
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