Supersymmetric Solutions in Six Dimensions: A Linear Structure

TL;DR

This paper reveals a linear structure in six-dimensional supersymmetric equations, enabling systematic construction of solutions, exemplified by new geometries with multiple charges, which could advance understanding of black hole microstates.

Contribution

It introduces a re-parameterization that transforms complex non-linear equations into linear ones, facilitating the construction of supersymmetric solutions in six dimensions.

Findings

Linear structure allows systematic solution construction

New geometries with D1, D5, and P charges constructed

Potential applications to black hole microstate geometries

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.