  Experience with Randall-Sundrum models teaches the importance of following
how branes back-react onto the bulk geometry, since this can dramatically
affect the system's low-energy properties. Yet the practical use of this
observation for model building is so far mostly restricted to branes having
only one transverse dimension (codimension-1) in the bulk space, since this is
where tools for following back-reaction are well-developed. This is likely a
serious limitation since experience also tells us that one dimension is rarely
representative of what happens in higher dimensions. We here summarize recent
progress on developing the matching conditions that describe how codimension-2
branes couple to bulk metric, gauge and scalar fields. These matching
conditions are then applied to three situations: D7-branes in F-theory
compactifications of 10D Type IIB string vacua; 3-branes coupled to bulk axions
in unwarped and non-supersymmetric 6D systems; and 3-branes coupled to chiral,
gauged 6D supergravity. For each it is shown how the resulting brane-bulk
dynamics is reproduced by the scalar potential for the low-energy moduli in the
dimensionally reduced, on-brane effective theory. For 6D supergravity we show
that the only 4D-maximally symmetric bulk geometries supported by
positive-tension branes are flat.
