Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States

TL;DR

This work develops a graph-based CFD framework to classify 5d SCFTs arising from circle reductions of 6d SCFTs. By encoding partially resolved elliptic CY3 geometries into Combined Fiber Diagrams, the authors read off strongly coupled flavor symmetries, mass deformations, and BPS spectra directly from geometry, avoiding reliance on particular weakly coupled descriptions. The approach yields complete rank-one and rank-two classifications and provides systematic predictions for higher-rank conformal matter theories, including symmetry enhancements and new SCFTs. The CFD method offers a scalable, combinatorial route to map the UV fixed-point structure of 5d theories and their interconnections via RG flows and UV dualities, with clear geometric interpretations through non-flat resolutions and Mori cone data.

Abstract

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 $\mathcal{N}=(1,0)$ 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, however, 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.

Figures (23)

• Figure 1: (a) Field-Theory overview : Starting from a 6d SCFT, the circle-reduction yields the 5d marginal theory. Mass deforming the marginal theory gives rise to 5d gauge theories that flows to 5d SCFTs in the UV. Alternatively, one can reduce the 6d theory on a circle with holonomies and then flow to said SCFTs. A third alternative is to take the 6d SCFT onto the tensor branch and reduce to 5d. (b) Geometry overview: The geometric realization in F/M-theory complements the field theoretic approach. A 6d SCFT is constructed from F-theory on a non-compact elliptically fibered Calabi--Yau threefold CY$_3$ with non-minimal singularities. These have crepant resolutions either by blow-ups in the fiber (an approach useful e.g. for conformal matter theories) or by blow-ups that modify the base of the fibration. Each of these approaches introduces compact surfaces, $S_i$, and the 5d strongly coupled flavor symmetry is encoded in the geometry of certain curves associated to the 6d flavor symmetry inside the surfaces $S_i$. We propose a succinct way of tracking these, so-called flavor curves, in terms of a graphical tool, making $G_\text{F}^{(\text{5d})}$ that is encoded in the geometry manifest --- see figure \ref{['fig:OverviewOfTools']}.
• Figure 2: This diagram shows schematically the approaches taken in the present paper and in the companion paper Apruzzi:2019enx, where box graphs and Coulomb branch phases will be discussed. The CFDs were initially introduced in Apruzzi:2019vpe and encode (among other things) the strongly-coupled flavor symmetry $G_\text{F}^{(5d)}$ of the 5d SCFT. The present paper provides the geometric foundation for and a derivation of CFDs using the structure of "flavor curves" within compact surfaces in the resolution of Calabi--Yau threefold singularities. In Apruzzi:2019enx the focus will be on a derivation using weakly-coupled gauge theory descriptions, whenever these exist.
• Figure 3: Rank one E-string: Depicted is the collision of two codimension one singularities of type $E_8$ and $I_1$, respectively, as well as the codimension two fiber including the non-flat component $S$. In codimension one, the fiber above $u=0$ is a collection of $(-2)$-curves $F_i \cong \mathbb{P}^1$ that are in one-to-one correspondence with the affine roots of $E_8$. At the collision point $u=v=0$, one of these rational curves splits (here $F_1 \rightarrow C^+ + C^-$). The non-flat fiber component $S$ is a complex surface above the codimension two locus, which contains some of the $(-2)$-curves of the codimension one fiber -- the so-called flavor curves. In the situation depicted, $S$ contains $F_2, \cdots, F_8$, which intersect in the $E_7$ Dynkin diagram -- we will refer to these curves in the following as flavor curves. Indeed, we will argue that they encode the flavor symmetry of the associated strongly coupled SCFT to be $E_7$.
• Figure 4: Affine $E_8$ Dynkin diagram with labels corresponding to the exceptional sections $u_i$ and associated to the Cartan divisors $D_{k}$ that correspond to the simple roots $\alpha_k$. The labels $u_i$ refer to the exceptional sections of the blow-up.
• Figure 5: The negative curves on the vertical divisor gdP$_9$, where the numbers in the brackets denote the self-intersection numbers and the letter $u_i$ or $z$ indicates the intersection of that divisor in the resolved Calabi--Yau with $S=\hbox{gdP}_9$. We also denote the expression of each curve in terms of the standard basis of the Picard group $h$ and $e_i$ of a rational surface.