Global structure of five-dimensional fuzzballs

Abstract

We describe and study families of BPS microstate geometries, namely, smooth, horizonless asymptotically-flat solutions to supergravity. We examine these solutions from the perspective of earlier attempts to find solitonic solutions in gravity and show how the microstate geometries circumvent the earlier "No-Go" theorems. In particular, we re-analyse the Smarr formula and show how it must be modified in the presence of non-trivial second homology. This, combined with the supergravity Chern-Simons terms, allows the existence of rich classes of BPS, globally hyperbolic, asymptotically flat, microstate geometries whose spatial topology is the connected sum of N copies of S 2 × S 2 with a "point at infinity" removed. These solutions also exhibit "evanescent ergo-regions," that is, the non-space-like Killing vector guaranteed by supersymmetry is time-like everywhere except on time-like hypersurfaces (ergo-surfaces) where the Killing vector becomes null. As a by-product of our work, we are able to resolve the puzzle of why some regular soliton solutions violate the BPS bound: their spactimes do not admit a spin structure.

Non-BPS multi-bubble microstate geometries

“…We thus observe the familiar story that these solitonic solutions are supported by fluxes, as discussed in [40] and also by [54][55][56]. It would be interesting to verify explicitly that the Komar-type integral defining the mass of our solution can be decomposed using the intersection form of the Euclidean base space, as discussed in [54].…”

“…Unfortunately, it turns out that these solutions violate the BPS bound and do not admit a spin structure [39,40], and hence they are not good candidates for describing black hole microstates. There is also a proposal for constructing microstates of neutral black holes [41], which can be very long-lived, but do not appear to be described by stationary supergravity solutions.…”

Floating JMaRT

“…Within the floating brane ansatz, we had W = V 2 and w 0 = σ, such that the Euclidean base was the solution to the Maxwell-Einstein equations, but this is not the case in this system. These functions are determined in terms of the four Ernst potentials and the six additional functions L I and 19) and 20) whereas the vector fields are determined by the first order equations…”

“…Applying the procedure above to the vector fields dv 2 and dw 1 leads to the following expressions for the functions K 2 and L 1 , and respectively to the vector fields dv 1 and dw 2 for K 1 and L 2 , i.e. 20) where the constants p a and q a are related to asymptotic charges, whereas the additional integration constants l a parametrise the asymptotic value of the dilatons. Given these, one may proceed in analysing the vector fields dw 3 , dw 0 and dω, leading to the final two functions …”

“…This is a system of equations that is based on an Euclidean Maxwell-Einstein subsystem, which governs the spatial section of the metric and generalises the hyper-Kähler metrics appearing in BPS solutions. However, it has been noted in [19,20] that solutions to this system generically lack a spin structure and therefore seem unattractive as microstate geometries. This is not so for the JMaRT solution of [12], which is not described by the floating brane ansatz and is known to have many desirable features for a microstate geometry, including a dual CFT description, despite the fact that it violates the over-rotating regularity bound for the associated asymptotically Minkowski black hole solution in five dimensions.…”