Exact half-BPS flux solutions in M-theory III: Existence and rigidity of global solutions asymptotic to AdS ₄ × S ⁷

The BPS equations in M-theory for solutions with 16 residual supersymmetries, SO(2,2) × SO(4) × SO(4) symmetry, and AdS ₄ × S ⁷ asymptotics, were reduced in [arXiv:0806.0605] to a linear first order partial differential equation on a Riemann surface with boundary, subject to a non-trivial quadratic constraint. In the present paper, suitable regularity and boundary conditions are imposed for the existence of global solutions. We seek regular solutions with multiple distinct asymptotic AdS ₄ × S ⁷ regions, but find that, remarkably, such solutions invariably reduce to multiple covers of the M-Janus solution found by the authors in [arXiv:0904.3313], suggesting rigidity of the half-BPS M-Janus solution. In particular, we prove analytically that no other smooth deformations away from the M-Janus solution exist, as such deformations invariably violate the quadratic constraint. These rigidity results are contrasted to the existence of half-BPS solutions with non-trivial 4-form fluxes and charges asymptotic to AdS ₇ × S ⁴. The results are related to the possibility of M2-branes to end on M5-branes, but the impossibility of M5-branes to end on M2-branes, and to the non-existence of half-BPS solutions with simultaneous AdS ₄ × S ⁷ and AdS ₇ × S ⁴ asymptotic regions.