Bulk Singularities and the Effective Cosmological Constant for Higher Co-dimension Branes

We study a general configuration of parallel branes having co-dimension >2 situated inside a compact d-dimensional bulk space within the framework of a scalar and flux field coupled to gravity in D dimensions, such as arises in the bosonic part of some D-dimensional supergravities. A general relation is derived which relates the induced curvature of the observable noncompact n dimensions to the asymptotic behaviour of the bulk fields near the brane positions. For compactifications down to n = D-d dimensions we explicitly solve the bulk field equations to obtain the near-brane asymptotics, and by so doing relate the n-dimensional induced curvature to physical near-brane properties. In the special case where the bulk geometry remains nonsingular (or only conically singular) at the brane positions our analysis shows that the resulting n dimensions must be flat. As an application of these results we specialize to n=4 and D=6 and derive a new class of solutions to chiral 6D supergravity for which the noncompact 4 dimensions have de Sitter or anti-de Sitter geometry.