We compute how bulk loops renormalize both bulk and brane effective
interactions for codimension-two branes in 6D gauged chiral supergravity, as
functions of the brane tension and brane-localized flux. We do so by explicitly
integrating out hyper- and gauge-multiplets in 6D gauged chiral supergravity
compactified to 4D on a flux-stabilized 2D rugby-ball geometry, specializing
the results of a companion paper, arXiv:1210.3753, to the supersymmetric case.
While the brane back-reaction generically breaks supersymmetry, we show that
the bulk supersymmetry can be preserved if the amount of brane-localized flux
is related in a specific BPS-like way to the brane tension, and verify that the
loop corrections to the brane curvature vanish in this special case. In these
systems it is the brane-bulk couplings that fix the size of the extra
dimensions, and we show that in some circumstances the bulk geometry
dynamically adjusts to ensure the supersymmetric BPS-like condition is
automatically satisfied. We investigate the robustness of this residual
supersymmetry to loops of non-supersymmetric matter on the branes, and show
that supersymmetry-breaking effects can enter only through effective brane-bulk
interactions involving at least two derivatives. We comment on the relevance of
this calculation to proposed applications of codimension-two 6D models to
solutions of the hierarchy and cosmological constant problems.Comment: 49 pages + appendices. This is the final version to appear in JHE
