We show that the non-gravitational sectors of certain 6d and 5d supergravity theories can be decomposed into superconformal field theories (SCFTs) which are coupled together by pairwise identifying and gauging mutual global symmetries. In the case of 6d supergravity, we consider F-theory on compact elliptic Calabi-Yau 3-folds with base $B=T^4/\mathbb{Z}_n\times \mathbb{Z}_m$ and we show in many examples that the non-gravitational field theory sectors can be described as configurations of coupled 6d $(1,0)$ SCFTs. We also conjecture that the effective 2d $(0,4)$ SCFTs living on the self-dual strings of the 6d theories lead to holographically dual descriptions of type IIB string theory on $AdS_3\times S^3\times B$ and moreover that their elliptic genera can be used to compute the degeneracies of 5d spinning BPS black holes along with all-genus topological string amplitudes on the corresponding compact 3-fold. In the case of 5d supergravity, we consider M-theory on compact Calabi-Yau 3-folds and using similar ideas as in the 6d case we show the complete non-gravitational sector of 5d supergravity theories can be decomposed into coupled 5d $\mathcal{N}=1$ SCFTs. Furthermore, using this picture we propose a generalized topological vertex formalism which, excluding some curve classes, seems to capture all-genus topological string amplitudes for the mirror quintic.

115 pages, 36 figures