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In arXiv:1906.11820 and arXiv:1907.05404 we proposed an approach based ongraphs to characterize 5d superconformal field theories (SCFTs), which arise ascompactifications of 6d $\mathcal{N}= (1,0)$ SCFTs. The graphs, so-calledcombined fiber diagrams (CFDs), are derived using the realization of 5d SCFTsvia M-theory on a non-compact Calabi--Yau threefold with a canonicalsingularity. In this paper we complement this geometric approach by connectingthe CFD of an SCFT to its weakly coupled gauge theory or quiver descriptionsand demonstrate that the CFD as recovered from the gauge theory approach isconsistent with that as determined by geometry. To each quiver description wealso associate a graph, and the embedding of this graph into the CFD that isassociated to an SCFT provides a systematic way to enumerate all possibleconsistent weakly coupled gauge theory descriptions of this SCFT. Furthermore,different embeddings of gauge theory graphs into a fixed CFD can give rise tonew UV-dualities for which we provide evidence through an analysis of theprepotential, and which, for some examples, we substantiate by constructing theM-theory geometry in which the dual quiver descriptions are manifest.

Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities