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A continuation of supergravity solutions on warped spacetimes


In this dissertation, we investigate the existence of solutions with sixteen supersymmetries to Type IIB supergravity on two sets of spacetimes that each contain an internal factor of two-dimensional Anti-de Sitter space ($AdS_{2}$).

The first case is $AdS_{2}\times S^{6}$ warped over a Riemann surface $\Sigma$. We construct the general Ansatz for the bosonic supergravity fields and supersymmetry generators compatible with the $SO(2, 1) \oplus SO(7)$ isometry algebra of the spacetime, which extends to the corresponding real form of the exceptional Lie superalgebra $F(4)$. We reduce the BPS equations to this Ansatz, obtain their general local solutions, and show that these local solutions solve the full Type IIB supergravity field equations and Bianchi identities. We contrast the $AdS_{2} \times S^{6}$ solution with the closely related $AdS_{6} \times S^{2}$ case and present the results for both in parallel.

In the second part of this work, we seek global half-BPS $AdS_{2}\times S^{6}$ solutions corresponding to the near-horizon behavior of $(p, q)$-string junctions. The general local solution was obtained in terms of two holomorphic functions $\mathcal{A}_{\pm}$ on $\Sigma$, which are constrained by a set of positivity and regularity conditions. We identify the type of singularity in $\mathcal{A}_{\pm}$ needed at the boundary of $\Sigma$ to match the solutions locally onto the classic $(p, q)$-string solution. We then construct and discuss solutions with multiple $(p, q)$-strings, however the existence of geodesically complete solutions remains unsettled.

The other case we consider is warped $AdS_{2}\times S^{5}\times S^{1}$. The existence of the Lie superalgebra $SU(1, 1|4) \subset PSU(2, 2|4)$, whose maximal bosonic subalgebra is $SO(2, 1) \oplus SO(6) \oplus SO(2)$, motivates the search for half-BPS solutions with this same isometry that are asymptotic to $AdS_{5}\times S^{5}$. We reduce the BPS equations to the Ansatz for the bosonic fields and supersymmetry generators compatible with these symmetries, then show that the only non-trivial solution is the maximally supersymmetric solution $AdS_{5}\times S^{5}$. We argue that this implies that no solutions exist for fully back-reacted D7 probe or D7/D3 intersecting branes whose near-horizon limit is of the form $AdS_{2}\times S^{5}\times S^{1}$.

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