5d SCFTs from Decoupling and Gluing

TL;DR

This work develops and extends the CFD framework to classify and construct 5d SCFTs arising from circle reductions of 6d (1,0) theories, including non-very-Higgsable cases and non-minimal conformal matter. It presents two routes to UV-fixed-point 5d theories: mass deformations and decoupling of gauge sectors, with decoupling linked to decompactification in the M-theory geometry. CFDs are shown to encode flavor symmetries, BPS spectra, and RG flows, and can be derived from geometry as flop-invariants; the authors further introduce a gluing program to assemble higher-rank 5d SCFTs from building blocks guided by the 6d tensor-branch structure. They compute CFDs for NHCs and NMCMs, discuss possible IR duals, and illustrate gluing with explicit examples, thereby providing a scalable framework for the landscape of 5d SCFTs descended from 6d. The results offer a geometrically transparent path to understand flavor symmetry enhancements and IR descriptions in 5d, with potential applications to broader classes of 6d-derived theories and their UV completions.

Abstract

We systematically analyse 5d superconformal field theories (SCFTs) obtained by dimensional reduction from 6d $\mathcal{N}=(1,0)$ SCFTs. Such theories have a realization as M-theory on a singular Calabi-Yau threefold, from which we determine the so-called combined fiber diagrams (CFD) introduced in arXiv:1906.11820, arXiv:1907.05404, arXiv:1909.09128. The CFDs are graphs that encode the superconformal flavor symmetry, BPS states, low energy descriptions, as well as descendants upon flavor matter decoupling. To obtain a 5d SCFT from 6d, there are two approaches: the first is to consider a circle-reduction combined with mass deformations. The second is to circle-reduce and decouple an entire gauge sector from the theory. The former is applicable e.g. for very Higgsable theories, whereas the latter is required to obtain a 5d SCFT from a non-very Higgsable 6d theory. In the M-theory realization the latter case corresponds to decompactification of a set of compact surfaces in the Calabi-Yau threefold. To exemplify this we consider the 5d SCFTs that descend from non-Higgsable clusters and non-minimal conformal matter theories. Finally, inspired by the quiver structure of 6d theories, we propose a gluing construction for 5d SCFTs from building blocks and their CFDs.

Figures (13)

• Figure 1: The flop operation on an example with $D_1^2\cdot S_3=-1$, $D_1\cdot S_1\cdot S_3=1$ and $D_1\cdot S_2\cdot S_3=1$. In the picture, each line segment denotes an intersection curve $S\cdot S'$, and the integer label on the side of $S$ (or $S'$) is the triple intersection number $S\cdot (S')^2$ (or $S'\cdot S^2$). After shrinking the curve $D_1\cdot S_3$, the surface geometry of $\tilde{D}_1$ and $\tilde{S}_i$ has a conifold singularity. Then after blowing up $\tilde{S}_1$ and $\tilde{S}_2$, the geometry will become the $D_1'$ and $S_i'$, which is the flopped geometry of the original one.
• Figure 2: The necessary flop operations to determine the multiplicity factor $\xi_{i,1}$ of the non-compact surface $D_1$ on $S_i$. The procedure includes two conifold transitions, where we do not draw the singular geometry explicitly.
• Figure 3: The flop operation on the resolution geometry of $(E_8,SU(2))$ conformal matter (rank-two E-string). On each surface component $S_i$ (labeled by the letter in the box), each node $D_\alpha$ corresponds to the intersection curve $S_i\cdot D_\alpha$. The number besides the node is the intersection number $(D_\alpha)^2\cdot S_i$. The genus of such a curve is by default zero unless otherwise labeled. In this geometry, the intersection curve $S_1\cdot S_2$ has genus-zero. After the flop, the intersection curve $S_1'\cdot S_2'$ has genus-one.
• Figure 4: The configuration of curves on the seven compact surfaces ($U,u_1,u_2,u_3,u_4,S_1,S_2$) in the resolution geometry of $(SO(8),SO(8))$$N=2$ non-minimal conformal matter theory. Here $u_i$ denotes $u_1$, $u_2$ and $u_3$, which has the topology of Hirzebruch surface $\mathbb{F}_2$.
• Figure 5: CFDs (on the LHS) of tables \ref{['tab:nm-CM-CFD']} and \ref{['tab:nm-E-CM-CFD']} and the embeddable BG-CFDs. Below the BG-CFDS we note the classical flavor symmetry.