Fine Tuning and Six-Dimensional Gauged $N=(1,0)$ Supergravity Vacua

TL;DR

This paper demonstrates that six-dimensional chiral gauged N=(1,0) supergravity harbors a richer set of supersymmetric vacua than previously recognized. It constructs a one-parameter family of AdS$_3 \times S^3$ solutions with a squashed $S^3$ along Hopf fibres, where the squashing is controlled by the 3-form flux $P$ and supersymmetry is preserved for all squashings; in the limit $P\to0$ (with a fibre rescaling) the solution reduces to the familiar $(Minkowski)_4 \times S^2$ vacuum, reframing the cosmological-constant issue as a Hopf-fibration fine-tuning. The work also identifies dyonic string solutions whose near-horizon geometry approaches the AdS$_3$ times squashed $S^3$ vacua, and proves their consistency via Killing-spinor integrability. Altogether, the results broaden the landscape of SUSY vacua in this theory and reveal a deep connection between fluxes, fibre geometry, and the emergence of four-dimensional spacetime structures, with implications for braneworld scenarios and potential higher-dimensional origins.

Abstract

We find a new family of supersymmetric vacuum solutions in the six-dimensional chiral gauged N=(1,0) supergravity theory. They are generically of the form AdS_3 x S^3, where the 3-sphere is squashed homogeneously along its Hopf fibres. The squashing is freely adjustable, corresponding to changing the 3-form charge, and the solution is supersymmetric for all squashings. In a limit where the length of the Hopf fibres goes to zero, one recovers, after a compensating rescaling of the fibre coordinate, a solution that is locally the same as the well-known (Minkowski)_4 x S^2 vacuum of this theory. It can now be viewed as a fine tuning of the new more general family. The traditional "Cosmological Constant Problem" is replaced in this theory by the problem of why the four-dimensional (Minkowski)_4 x S^2 vacuum should be selected over other members of the equally supersymmetric AdS_3 x S^3 family. We also obtain a family of dyonic string solutions in the gauged N=(1,0) theory, whose near-horizon limits approach the AdS_3 times squashed S^3 solutions.