| 總結: | <p>The aim of this thesis is to explore new constructions in supergravity to gain insights into the AdS$_4$/CFT$_3$ and AdS$_5$/CFT$_4$ correspondences.</p>
<p>We start by deriving a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS$_4$/CFT$_3$ correspondence. We find that the holographically renormalized action localizes to the fixed points of a supersymmetric Killing vector and may then be expressed entirely in terms of topological data. In this sense, the gravitational partition function localizes in the bulk. Our general formula reproduces all known examples in the literature and makes predictions for new solutions, assuming they exist.</p>
<p>Also in $D=4$, $\mathcal{N}=2$ minimal gauged supergravity, we sudy a class of accelerating, rotating and dyonic black holes in AdS$_4$. %When the acceleration is non-vanishing,
These $D=4$ black hole metrics have conical singularities but we find a new way to desingularize them by uplifting to solutions of $D=11$ supergravity using a regular Sasaki-Einstein $SE_7$ manifold. These $D=11$ solutions are associated to M2-branes wrapped around a two-dimensional weighted projective space, $\mathbb{WCP}^1_{[n_-,n_+]}$, also known as a spindle, and we find that supersymmetry is not preserved through a usual topological twist. We also discuss the supersymmetric and extremal limits of these solutions and compute the entropy of these black holes, arguing that it should be possible to derive it from certain $\mathcal{N}=2$, $d=3$ quiver gauge theories compactified on a spinning spindle with magnetic flux.</p>
<p>In the AdS$_5$/CFT$_4$ correspondence we continue exploring these ideas by constructing AdS$_3 \times \mathbb{WCP}^1_{[n_-,n_+]}$ solutions of different $D=5$ supergravity theories. Again we find that these solutions can be uplifted to find smooth solutions in $D=10/11$ and that supersymmetry is preserved in a different way than through the topological twist. Our construction suggests a path to the dual $d=2$, $\mathcal{N}=(0,2)$ SCFTs in the IR, associated to D3 or M5 branes wrapped on a spindle. We make a precision test of this conjecture by computing the central charge and R-symmetry of the field theories and find agreement with the gravity computations in all cases.</p>
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