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

TL;DR

The paper develops and extends the combined fiber diagram (CFD) framework to connect 5d SCFTs obtained from circle reductions of 6d theories with their weakly coupled gauge/quiver descriptions. By introducing flavor-equivalence classes and BG-CFDs, the authors constrain admissible quivers from CFDs and systematically uncover UV-dualities among gauge theories that flow to the same UV fixed point. They bootstrap marginal CFDs in cases lacking complete geometric data and derive rich networks of descendants for minimal conformal matter theories (e.g., (E6,E6), (E7,SO7), (E8, E8)) as well as rank-two E-string and D-type matter theories. The work also ties box-graph phases to explicit Calabi–Yau fiber structures, enabling a geometric verification of CFD-driven predictions and offering a path toward a gluing program to classify all 5d SCFTs descended from 6d theories.

Abstract

In arXiv:1906.11820 and arXiv:1907.05404 we proposed an approach based on graphs to characterize 5d superconformal field theories (SCFTs), which arise as compactifications of 6d $\mathcal{N}= (1,0)$ 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.

Figures (42)

• Figure 1: The representation graph (or undecorated box graph) for the $({\bf 2,16})$ representation of $SU(2)_\text{gauge} \times SO(16)$. The simple roots of $SO(16)$ are $\alpha_i$, and for $SU(2)_\text{gauge}$ the simple root is $\alpha^{SU(2)}$. The arrows indicate how weights are mapped into each other under the addition of the roots. The weights are $L_{i,j}= L_i^{\bf 2} +L_{j}^{\bf 16}$, where $L^{\bm R}_i$ are the fundamental weights of the representation ${\bm R}$. The action of the roots is indicated by the arrows. Note that $L^{\bf 16}_{i+8}= - L^{\bf 16}_{9-i}$ for $i= 1, \ldots, 8$.
• Figure 2: 5d rank one theories: Box graphs for $(\bf{2}, \bm{16})$ of $SU(2)\times SO(16)$, for the marginal theory $SU(2)+ 8 \bm{F}$, which is shown at the top of the tree. Connections indicate 'flop transitions', which in the gauge theory correspond to different phases of the extended Coulomb branch, and from the point of view of the $SU(2)$ gauge theory, corresponds to decoupling fundamental hypermultiplets.
• Figure 3: Representation graph for $(\bm{3}, \bm{9})$ of $SU(3)_{\text{gauge}} \times U(9)_\text{BG}$. $L_{i,j}= L^{SU(3)}_i + L^{SU(9)}_j$, where $L_k$ indicates the fundamental weights of the respective groups. $\alpha_i$ are the simple roots of the classical flavor symmetry $U(9)$, and $\alpha_i^{\text{gauge}}$ those of the gauge group $SU(3)$.
• Figure 4: An example of a flavor-equivalence class for $SU(3) + 9 \bm{F}$. The equivalence class is shown on the left-hand side --- there we indicate in green the key characteristics of the associated box graph: the splitting of both roots of the gauge group $\alpha_i^{\text{gauge}}$, $i=1,2$ combined will contain all the roots $\alpha_3, \alpha_4, \alpha_5, \alpha_6$. On the right-hand side, we show the complete set of box graphs that comprise the flavor-equivalence class. E.g., the top diagram corresponds to the case where the splitting of $\alpha_1^{\text{gauge}}$ contains all $\alpha_i$, $i=3,4,5,6$. In the second one, which is related by a flop to the top one, $\alpha_1^{\text{gauge}}$ does not contain $\alpha_6$, which is now part of the splitting of $\alpha_{2}^{\text{gauge}}$, etc.
• Figure 5: Flavor-equivalence class and BG-CFD: In this figure we show an example for a flavor-equivalence class of Coulomb branch phases for $SU(3)$ with $9{\bm{F}}$. The top graph is the box graph reduced to its flavor-equivalence class by omitting the sign assignments in the middle row: all sign assignments consistent with the standard box graph rules would correspond to the same combined gauge root splitting. This is encoded in the CFD-subgraph that we refer to as BG-CFD: the roots $\alpha_i$, $i=3,4,5,6$ of $G_{\text{BG}}= U(9)$ that participate in the splitting of the roots $\alpha_i^{\text{gauge}}$ of $G_{\text{gauge}}= SU(3)$ are $(-2)$-vertices in the CFD --- shown at the bottom. The $(-1)$-vertices of the CFD correspond to the F-extremal weights $(-L_{3,3})$ and $L_{1,7}$.

Theorems & Definitions (5)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5