We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi-Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d $$ \mathcal{N} $$N = (1, 0) SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi-Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations. The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, how- ever, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.

In [1, 2] we proposed an approach based on graphs to characterize 5d superconformal field theories (SCFTs), which arise as compactifications of 6d $$ \mathcal{N} $$N = (1, 0) SCFTs. The graphs, so-called combined fiber diagrams (CFDs), are derived using the realization of 5d SCFTs via M-theory on a non-compact Calabi-Yau threefold with a canonical singularity. In this paper we complement this geometric approach by connecting the CFD of an SCFT to its weakly coupled gauge theory or quiver descriptions and demonstrate that the CFD as recovered from the gauge theory approach is consistent with that as determined by geometry. To each quiver description we also associate a graph, and the embedding of this graph into the CFD that is associated to an SCFT provides a systematic way to enumerate all possible consistent weakly coupled gauge theory descriptions of this SCFT. Furthermore, different embeddings of gauge theory graphs into a fixed CFD can give rise to new UV-dualities for which we provide evidence through an analysis of the prepotential, and which, for some examples, we substantiate by constructing the M-theory geometry in which the dual quiver descriptions are manifest.

Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of ‘trinion’ singularities which exhibit these properties. In Type IIB, they give rise to 4d $$ \mathcal{N} $$N = 2 SCFTs that we call $$ {D}_p^b $$D p b (G)-trinions, which are marginal gaugings of three SCFTs with G flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d $$ \mathcal{N} $$N = 4 theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as ‘ugly’ components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver ‘bad’. We propose a way to redeem the badness of these quivers using a class $$ \mathcal{S} $$S realization. We also discover new S-dualities between different $$ {D}_p^b $$D p b (G)-trinions. For instance, a certain E₈ gauging of the E₈ Minahan-Nemeschansky theory is S-dual to an E₈-shaped Lagrangian quiver SCFT.

Applying the Ashok-Denef-Douglas estimation method to elliptic Calabi-Yau fourfolds suggests that a single elliptic fourfold \( {\mathrm{\mathcal{M}}}_{\max } \) gives rise to \( \mathcal{O}\left({10}^{272,000}\right) \) F-theory flux vacua, and that the sum total of the numbers of flux vacua from all other F-theory geometries is suppressed by a relative factor of \( \mathcal{O}\left({10}^{-3000}\right) \). The fourfold \( {\mathrm{\mathcal{M}}}_{\max } \) arises from a generic elliptic fibration over a specific toric threefold base B_max, and gives a geometrically non-Higgsable gauge group of E₈⁹ × F₄⁸ × (G₂ × SU(2))¹⁶, of which we expect some factors to be broken by G-flux to smaller groups. It is not possible to tune an SU(5) GUT group on any further divisors in \( {\mathrm{\mathcal{M}}}_{\max } \), or even an SU(2) or SU(3), so the standard model gauge group appears to arise in this context only from a broken E₈ factor. The results of this paper can either be interpreted as providing a framework for predicting how the standard model arises most naturally in F-theory and the types of dark matter to be found in a typical F-theory compactification, or as a challenge to string theorists to explain why other choices of vacua are not exponentially unlikely compared to F-theory compactifications on \( {\mathrm{\mathcal{M}}}_{\max } \).