Adjoint operators, gauge invariant perturbations, and covariant   symplectic structure for black holes in string theory
  Expressions for the general and complete perturbations in terms of Debye potentials of static charged black holes in string theory, valid for curvature below the Planck scale, are derived starting from a decoupled set of equations and using Wald's method of adjoint operators. Our results cover both extremal and nonextremal black holes and are valid for arbitrary values of the dilaton coupling parameter. The decoupled set is obtained using the Newman-Penrose formulation of the Einstein-Maxwell-dilaton theory and involves naturally field quantities invariant under both ordinary gauge transformations of the electromagnetic potential perturbations and infinitesimal rotations of the perturbed tetrad. Furthermore, using the recent pointed out relationship between adjoint operators and conserved currents, a local continuity law for the field perturbations in terms of the potentials is also obtained. It is shown that such continuity equation implies the existence of conserved quantities and of a covariant symplectic structure on the phase space. Future extensions of the present results are discussed. 
R. Cartas-Fuentevilla (Universidad Autonoma de Puebla, Mexico)