Quantum Effective Action in Spacetimes with Branes and Boundaries:   Diffeomorphism Invariance
  We construct a gauge-fixing procedure in the path integral for gravitational models with branes and boundaries. This procedure incorporates a set of gauge conditions which gauge away effectively decoupled diffeomorphisms acting in the $(d+1)$-dimensional bulk and on the $d$-dimensional brane. The corresponding gauge-fixing factor in the path integral factorizes as a product of the bulk and brane (surface-theory) factors. This factorization underlies a special bulk wavefunction representation of the brane effective action. We develop the semiclassical expansion for this action and explicitly derive it in the one-loop approximation. The one-loop brane effective action can be decomposed into the sum of the gauge-fixed bulk contribution and the contribution of the pseudodifferential operator of the brane-to-brane propagation of quantum gravitational perturbations. The gauge dependence of these contributions is analyzed by the method of Ward identities. By the recently suggested method of the Neumann-Dirichlet reduction the bulk propagator in the semiclassical expansion is converted to the Dirichlet boundary conditions preferable from the calculational viewpoint. 
A.O.Barvinsky