Localized four-dimensional gravity in the D-brane background with NS $B$ field

We calculate small correction terms to gravitational potential near $p$-branes embedded in a constant NS $B$ field background in the context of M-theory or string theory. The normalizable wave functions of gravity fluctuations around the brane describe only massive modes. We compute such wave functions analytically. We estimate the correction to gravitational potential for small and long distances, and show that there is an intermediate range of distances in which we can identify $4D$ gravity on the brane below a crossover scale given in terms of components of the $B$ field. The $4D$ gravity is metastable and for distances much larger than the crossover scale the $5D$ gravity is recovered.

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