Supersymmetric Solutions in Six Dimensions: A Linear Structure

TL;DR

This work shows that the equations governing all supersymmetric solutions in six-dimensional minimal ungauged supergravity coupled to an anti-self-dual tensor multiplet admit a linear structure after fixing a base geometry and a self-dual vector field. By introducing new variables $Z_i$, $\\Theta_i$, and reexpressing the fluxes, the authors reduce the problem to decoupled linear systems for these potentials, with the remaining metric data encoded linearly in ${\\cal F}$ and $\\omega$. They illustrate the method by constructing explicit D1-D5-P spirals and their generalizations, highlighting an action-at-a-distance property for parallel strands and the absence of bubble equations in these cases. The results point toward a broader class of six-dimensional microstate geometries, including the anticipated superstratum, and set the stage for systematic, superposable solution-building in higher dimensions.

Abstract

The equations underlying all supersymmetric solutions of six-dimensional minimal ungauged supergravity coupled to an anti-self-dual tensor multiplet have been known for quite a while, and their complicated non-linear form has hindered all attempts to systematically understand and construct BPS solutions. In this paper we show that, by suitably re-parameterizing these equations, one can find a structure that allows one to construct supersymmetric solutions by solving a sequence of linear equations. We then illustrate this method by constructing a new class of geometries describing several parallel spirals carrying D1, D5 and P charge and parameterized by four arbitrary functions of one variable. A similar linear structure is known to exist in five dimensions, where it underlies the black hole, black ring and corresponding microstate geometries. The unexpected generalization of this to six dimensions will have important applications to the construction of new, more general such geometries.