Surface charge algebra in gauge theories and thermodynamic integrability

Abstract

ABSTRACT. Surface charges and their algebra in interacting Lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus. In the case of exact solutions and symmetries, the surface charges are interpreted as a Pfaff system. Integrability is governed by Frobenius' theorem and the charges associated with the derived symmetry algebra are shown to vanish. Surfaces charges reproduce well-known Hamiltonian and covariant phase space expressions. In the asymptotic context, we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended. Comparision with Hamiltonian and covariant phase space methods are done in appendix.

Free boundary conditions and the AdS3/CFT2 correspondence

“…In particular, having identified L with the generator of Virasoro transformations of the matter and ghost fields, their central charge can be obtained in the bulk from the anomalous transformation of L and is given by c M = 3l/2G in the semiclassical approximation. The Polyakov action is invariant under Kac-Moody transformations where 20) and Λ satisfies ∂ 3 − Λ − n 2 ∂ − Λ = 0. Therefore the parameter Λ may be written as in eq.…”

“…The charges associated with these symmetries may be computed by standard methods [17][18][19][20]. For three-dimensional gravity the infinitesimal charge corresponding to the asymptotic Killing vector ξ is given by…”

$ \mathcal{W} $ symmetry and integrability of higher spin black holes

“…Here, the Poisson bracket can be computed using the infinitesimal charge formula given in (5.12) even though we do not have at hand the conserved charge Q Λ at all times (see [72] for a general proof). The infinitesimal charge is linear in the variations of the fields, but it might depend non-linearly on the fields L and W and their φ derivatives.…”

Warped AdS3 black holes in higher derivative gravity theories

“…This ambiguity generalizes the one in the symplectic potential (n − 1)-form under Θ → Θ + dY, and hence in the symplectic structure. One proposal to fix this ambiguity [78,79] is by acting on the weakly vanishing Noether current with a contracting homotopy operator, yielding an (n − 2)-form denoted k BB ξ (δΦ, Φ). In essence, this operator is the inverse of the exterior derivative d (see e.g.…”

“…2 Then, exploiting again the symmetries and using the Covariant Phase Formalism [26,27,75,[77][78][79][80] we will be able to compute the charges and central charges appearing in (1.4) and (1.5). In particular, we will find that…”