The Wheeler-De Witt equation for brane gravity

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We consider the gravity in the system consisting of the BPS D3-brane embedded in the flat background geometry, produced by the solutions of the supergravity. The effective action for this system is represented by the sum of the Hilbert-Einstein and DBI actions. We derive the Wheeler-De Witt equation for this system and obtain analytical solutions in some special cases. We also calculate tunneling probability from Planckian size of D3-brane to the classical regime. This paper appeared in Phys. Rev. D 77, 066017 (2008)