| Summary: | <p>This thesis is devoted to the study of several geometric aspects, as well as their role in the holographic context, of certain supersymmetric AdS solutions that are expected to describe the near-horizon region of accelerating black holes in 4d and accelerating black strings in 5d. In particular, these are AdS3 × <em>Y</em>7 solutions of type IIB supergravity, sourced by D3-branes, and AdS2×<em>Y</em>9 solutions of 11d supergravity, sourced by M2-branes. Both <em>Y</em>7 and <em>Y</em>9 are total spaces of fibrations with codimension two Sasakian fibres over a two-dimensional orbifold known as spindle. The latter is topologically a two-sphere but has conical deficits angles parametrized by coprime integers m± ∈ N whose physical effect is indeed to turn on the acceleration.</p>
<p>We exploit the formalism of geometric extremization together with a series of geometric methods to derive block formulas for many holographic observables of interest, including central charges, free energies, quantized fluxes, and probe brane actions. These allow us to recover previously known results for numerous explicit supergravity solutions and to significantly generalize them to various other compactifications.</p>
<p>For the AdS3 solutions (with toric internal space) we give a general proof that our off-shell supersymmetric action agrees with an appropriate off-shell central charge in the dual field theory, establishing a very general exact result in holography. For the AdS2 solutions we provide a general geometric prescription for reproducing various physical quantities appearing in the matrix model arising from the large <em>N</em> limit of the localized partition function of the dual field theory. Here we focus on the non-accelerating case but we comment on how our results should be extended to the accelerating case too.</p>
<p>We illustrate our findings with many examples throughout, including an explicit solution obtained in 5d STU gauged supergravity and uplifted on a five-sphere.</p>
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