New non-supersymmetric flux vacua from generalised calibrations
We construct a new class of non-supersymmetric ten-dimensional type II flux vacua, by studying first order differential equations which are deformations of the $\mathcal{N}=1$ supersymmetry conditions. We do so within the context of Generalised Complex Geometry, where there is a natural interpretation of the $\mathcal{N}=1$ supersymmetry conditions in terms of calibration conditions for probe D-branes, called D-string, domain-wall or space-filling branes, depending on them wrapping 2, 3 or 4 non-compact dimensions. We focus on the class of non-supersymmetric vacua violating the D-string calibration condition, and write down their general equations of motion in the language of pure spinors. We solve them for a subclass of vacua, where the deformation of the calibration condition is dictated by the foliated geometry of the internal space. We also construct backgrounds violating both the domain-wall and D-string calibration conditions, generalising the one-parameter DWSB class of backgrounds introduced in Lust et al. We present several explicit solutions with $SU(2)$ and $SU(3)$ structures, and we investigate briefly their associated low energy effective theories.