Spheres, deficit angles and the cosmological constant

We consider compactifications of six-dimensional gravity in four-dimensional Minkowski or de Sitter space times a twodimensional sphere, S2. As has been pointed out recently, it is possible to introduce 3-branes in these backgrounds with arbitrary tension without affecting the effective four-dimensional cosmological constant, since its only effect is to induce a deficit angle in the sphere. We show that if a monopole-like configuration of a 6D U(1) gauge field is used to produce the spontaneous compactification of the two extra dimensions in a sphere, a fine-tuning between brane and bulk parameters is reintroduced once the quantization condition for the gauge field is taken into account, so the 4D cosmological constant depends on the brane tension. This problem is absent if instead of the monopole we consider a 4-form field strength in the bulk to obtain the required energy–momentum tensor. Also, making use of the 4-form field, we generalize the solution to an arbitrary number of dimensions (>6), keeping always four noncompact dimensions and compactifying the rest in an n-dimensional sphere. We show that an (n + 1)-brane with arbitrary tension can be introduced in this background without affecting the effective 4D cosmological constant.