Higher-dimensional field theories from type II supergravity

In this thesis we give a pedagogical introduction to the construction of NS5-D6(-O6)-D8(-O8) brane setups in massive type IIA string theory and the six-dimensional $(1,0)$ SCFT's they engineer. After describing how to introduce the various ingredients (and their interplay) in type IIA, we explain how to read off an effective linear quiver description of the SCFT's from the brane setups. We then explain in detail how to construct all AdS$_7$ solutions of massive IIA supergravity which are dual to the field theories. We do this by exploiting the power of generalized complex geometry. We then show how to establish a holographic dictionary between four-dimensional $\mathcal{N}=1$ compactifications on a Riemann surface (without punctures) of the $(1,0)$ SCFT's and a certain class of massive IIA AdS$_5$ solutions, which are in one-to-one correspondence with the AdS$_7$ vacua. We provide evidence for the advocated AdS$_7$/CFT$_6$ and AdS$_5$/CFT$_4$ dualities by computing the (leading order of the) field theory $a$ conformal anomaly from supergravity. Finally, we briefly touch upon the construction of five-dimensional $\mathcal{N}=1$ SCFT's through fivebrane webs, and reduce the classification problem of their type IIB AdS$_6$ duals to a system of two coupled PDEs.