Dynamical supersymmetry enhancement of black hole horizons

This thesis is devoted to the study of dynamical symmetry enhancement of black hole horizons in string theory. In particular, we consider supersymmetric horizons in the low energy limit of string theory known as supergravity and we prove the $\textit{horizon conjecture}$ for a number of supergravity theories. We first give important examples of symmetry enhancement in $D=4$ and the mathematical preliminaries required for the analysis. Type IIA supergravity is the low energy limit of $D=10$ IIA string theory, but also the dimensional reduction of $D=11$ supergravity which itself the low energy limit of M-theory. We prove that Killing horizons in IIA supergravity with compact spatial sections preserve an even number of supersymmetries. By analyzing the global properties of the Killing spinors, we prove that the near-horizon geometries undergo a supersymmetry enhancement. This follows from a set of generalized Lichnerowicz-type theorems we establish, together with an index theory argument. We also show that the symmetry algebra of horizons with non-trivial fluxes includes an $\mathfrak{sl}(2, \mathbb{R})$ subalgebra. As an intermediate step in the proof, we also demonstrate new Lichnerowicz type theorems for spin bundle connections whose holonomy is contained in a general linear group. We prove the same result for Roman's Massive IIA supergravity. We also consider the near-horizon geometry of supersymmetric extremal black holes in un-gauged and gauged 5-dimensional supergravity, coupled to abelian vector multiplets. We consider important examples in $D=5$ such as the BMPV and supersymmetric black ring solution, and investigate the near-horizon geometry to show the enhancement of the symmetry algebra of the Killing vectors. We repeat a similar analysis as above to prove the horizon conjecture. We also investigate the conditions on the geometry of the spatial horizon section $\cal{S}$.