Variational approach to infrared divergencies: Applications to black hole action integrals and holographic Wilson loops

Renormalizing the on-shell action variation offers two important advantages over renormalizing the action itself. First, the variation of the action has a universal form in which the bulk contribution vanishes identically. As a result, the renormalization problem is automatically localized at the boundary; no bulk subtractions are required. Second, the variation of the action admits a natural interpretation as a 1-form on functional space. The distinction between exact and non-exact 1-forms provides a clear renormalization prescription. We apply these ideas to the computation of on-shell black hole actions. After presenting the general algorithm, we discuss some examples, including Kerr-Newman, Kerr-Newman-AdS, Taub-NUT, Taub-Bolt, and STU black holes. We also apply the same techniques to the computation of holographic Wilson loops. All results agree with standard results, in particular with holographic renormalization.