Einstein-Hilbert action on the brane for the bulk graviton

TL;DR

The paper addresses the problem of next-to-leading corrections to the Dp-brane effective action in the bosonic string by computing tree-level disk amplitudes for scattering of massless closed-string states off a brane and matching them to a brane-bulk field theory. The main approach is to identify the necessary $O(\alpha')$ kinetic terms for pullbacks of bulk fields on the brane, including an induced Einstein-Hilbert term and second fundamental- form contributions, and to fix their coefficients via S-matrix comparisons, with careful treatment of field-redefinition ambiguities. The key findings are that induced gravitational and related kinetic terms are indeed required on the brane, with explicit coefficients: $1/\kappa_p^{2}=2\tau_p$, $\beta_1=-1$, $\beta_2=1$, $\beta_3=\beta_4=0$, and analogous fixed values for dilaton and B-field couplings; the analysis also discusses scheme dependence and tachyon couplings. The results illuminate how brane-localized gravity can arise from stringy corrections and provide a framework for assessing brane-world gravity scenarios, including potential transitions between bulk and brane gravity regimes and the role of tachyons in the open-string sector.

Abstract

We find new closed string couplings on Dp-branes for the bosonic string. These couplings are quadratic in derivatives and therefore take the form of induced kinetic terms on the brane. For the graviton in particular we find the induced Einstein-Hilbert term as well as terms quadratic in the second fundamental tensor. We comment on tachyon dependences of these brane-localized couplings.