Abstract
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 supersymmetric 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.
Highlights
Were still linear it would make the construction of explicit solutions much more straightforward but it would enable a much more precise analysis of the moduli space of such solutions
We have found that the system of equations describing supersymmetric solutions of sixdimensional, minimal, ungauged supergravity coupled to an anti-symmetric tensor multiplet can be solved in a linear process once the base metric and the vector field, β, have been determined
When the solutions to this theory do not depend on the “common D1-D5” direction, v, they can be reduced to solutions of yields N = 2 ungauged five-dimensional supergravity coupled to two vector multiplets, and the complicated linear system found here collapses to the simpler linear system found in [3]
Summary
Minimal ungauged N = 1 supergravity in six dimensions has a bosonic field content consisting of a graviton and a two-index tensor gauge field, Bμν, whose field strength, G, is required to be self-dual If one reduces this to five dimensions one obtains N = 2 supergravity coupled to one vector multiplet. The supersymmetry conditions imply that the base is almost hyper-Kahler in that there are three anti-self-dual 2-forms,. These forms are required to satisfy the differential identity: dJ (A) = ∂v β ∧ J (A)) ,. We note that one can, simplify things by taking the base to be hyper-Kahler and restricting β to be independent of v This means that ψ = 0 and that the equation (2.14) is a simple, linear self-duality condition on the base
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