Extra Dimensional Curvature Supression of the Effective Four Dimensional Vacuum Energy Density

Considering a very large number of extra dimensions, $N\rightarrow \infty$, we show that in the effective four dimensional picture, to leading order in $N$, both the cosmological constant in $N+4$ dimensions and the curvature of the extra dimensions (curved as spheres) give the same type of contributions. Furthermore in this limit, the extra dimensional curvature naturally supress the effect of a positive Cosmological Constant, so that the resulting effective potential governing the vacuum energy in the effective $4-D$ picture has a leading 1/N dependence (i.e. vanishing in the large $N$ limit). We can understand qualitatively this effect in a heuristic picture, by thinking that all dimensions, both visible and extra have an equal sharing of the curvature caused by $\Lambda$, in this case when increasing the overall number of dimensions by adding $N$ extra dimensions, then if $N$ is large, the visible dimensions do not have to curve too much, hence a small four dimensional vacuum energy follows. In the large $N$ picture the potential can be also stabilized by a small (i.e. vanishing at large $N$) expectaction value of a four index field strength.