Bubbling 1/2 BPS solutions of minimal six-dimensional supergravity

TL;DR

This paper resolves the inconsistency found in bubbling 1/2 BPS solutions of minimal $D=6$ supergravity with an $S^1\times S^1$ reduction by introducing a torus axion $\chi$ in a $T^2$ reduction. The resulting bosonic theory in four dimensions is consistent and contains a metric, a gauge field, two scalars $H,G$, and the axion $\chi$, with a self-dual 3-form relating two gauge fields. The bubbling AdS$_3$ solutions are structured by two harmonic functions, and the Killing spinors carry an $SL(2,\mathbb{Z})$ charge, relating different solutions via duality. Depending on the $(\eta,\tilde{\eta})$ charges, explicit families of bubbling AdS$_3$ geometries are obtained, including cases related by SL$(2,\mathbb{Z})$ transformations; one class connects to known D1-D5–type configurations. Overall, the work extends bubbling geometries from AdS$_5$ to AdS$_3\times S^3$ within minimal six-dimensional supergravity and clarifies the role of the axion and duality in enabling regular, supersymmetric solutions.

Abstract

We continue our previous analysis (hep-th/0412045) of 1/2 BPS solutions to minimal 6d supergravity of bubbling form. We show that, by turning on an axion field in the T^2 torus reduction, the constraint F \wedge F, present in the case of an S^1 x S^1 reduction, is relaxed. We prove that the four-dimensional reduction to a bosonic field theory, whose content is the metric, a gauge field, two scalars and a pseudo-scalar (the axion), is consistent. Moreover, these reductions when lifted to the six-dimensional minimal supergravity represent the sought-after family of 1/2 BPS bubbling solutions.