# 5D and 6D SCFTs from $\mathbb{C}^3$ orbifolds

### Submission summary

 Authors (as Contributors): Yinan Wang
Submission information
Date accepted: 2022-03-25
Date submitted: 2022-03-13 14:45
Submitted by: Wang, Yinan
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
Approach: Theoretical

### Abstract

We study the orbifold singularities $X=\mathbb{C}^3/\Gamma$ where $\Gamma$ is a finite subgroup of $SU(3)$. M-theory on this orbifold singularity gives rise to a 5d SCFT, which is investigated with two methods. The first approach is via 3d McKay correspondence which relates the group theoretic data of $\Gamma$ to the physical properties of the 5d SCFT. In particular, the 1-form symmetry of the 5d SCFT is read off from the McKay quiver of $\Gamma$ in an elegant way. The second method is to explicitly resolve the singularity $X$ and study the Coulomb branch information of the 5d SCFT, which is applied to toric, non-toric hypersurface and complete intersection cases. Many new theories are constructed, either with or without an IR quiver gauge theory description. We find that many resolved Calabi-Yau threefolds, $\widetilde{X}$, contain compact exceptional divisors that are singular by themselves. Moreover, for certain cases of $\Gamma$, the orbifold singularity $\mathbb{C}^3/\Gamma$ can be embedded in an elliptic model and gives rise to a 6d (1,0) SCFT in the F-theory construction. Such 6d theory is naturally related to the 5d SCFT defined on the same singularity. We find examples of rank-1 6d SCFTs without a gauge group, which are potentially different from the rank-1 E-string theory.

Published as SciPost Phys. 12, 127 (2022)

Dear Editor-in-charge,

We have made changes to the manuscript according to the referee's comments and requested changes. In particular, regarding the skepticism in the claim of new 6d (1,0) theories and the interpretations of certain 5d N=1 theories, we have made weaker statements in the abstract, introduction and main texts. We also fixed a number of typos and added some references. Please read the list of changes for full details.

Sincerely,
The authors

### List of changes

1. In abstract, the sentence "Nonetheless, some of these theories are interpreted as the known rank-1 Seiberg $E_N$ theory" is removed. In the last sentence, "which are different from" is replaced by "which are potentially different from".

2. On page 2, in the second and third paragraph of the introduction section, new references [9][32][41][43][46][47] are added.

3. In Table 1 (page 4), we fixed typos in the rows $G_m$ and $G_{p,q}$, to $8m$ and $8pq^2$.

4. The second to last paragraph on page 5, we replaced the sentence "We argue that it corresponds to a 6d SCFT that is not the rank-1 E-string" with "We argue that it corresponds to a 6d SCFT that is potentially different from the rank-1 E-string". We also replaced "Similarly, the tensor branch of $H_{168}$ takes the form of type $II$ singular fiber on a $(-1)$-curve, which also leads to a new 6d SCFT, see appendix \ref{sec:H168-6d}." with "Similar arguments can be applied to the case of $H_{168}$ as well, whose tensor branch takes the form of type $II$ singular fiber on a $(-1)$-curve, see appendix \ref{sec:H168-6d}."

5. On page 7, we removed "Finally, the flavor symmetry $SU(3)\times SU(2)^2\times U(1)$ is enhanced to $SO(10)$ and matches the flavor symmetry of Seiberg rank-1 $E_5$ theory which matches the result using a brane web construction in \cite{Acharya:2021jsp}."

6. In Table 3 (page 7), we removed the rows of $G_3$, $E^{(2)}$, $E^{(4)}$, $E^{(6)}$, $\Delta(48)$ and $H_{168}$.

7. On page 16, in the last paragraph before section 2.3.2, we replaced "the elements in $\Gamma_{1,f}$ along with the identity element $\mathbf{1}$ exactly form subgroups $\mb{Z}_N^3\subset\Gamma$." with "the elements in $\Gamma_{1,f}$ along with the identity element $\mathbf{1}$ exactly form three commuting $\mb{Z}_N$'s in $\Gamma$."

8. In Table 5 (page 26), we fixed a typo in the row labeled by $\frac{1}{10}(1,2,7)$, from $\mb{Z}_2$ to $\mb{Z}_5$.

9. On page 41, in (4.27) we replaced "I_{n,s}$with$I_{4,s}$, and$w^3-27z^n$with$w^3-27z^4$. In (4.28) we replaced "\subset" with "=". 10. On page 45, last paragraph before section 4.3, we replaced "We conclude that the rank-1 SCFT from$\Delta(48)$is equivalent to the Seiberg$E_5$theory, although$S_1$is not a smooth dP$_5$surface. Additional evidences are also provided via brane-web arguments in \cite{Acharya:2021jsp}." with "There are two possible physical interpretations: \begin{enumerate} \item{The rank-1 SCFT from$\mb{C}^3/\Delta(48)$is equivalent to the Seiberg$E_5$theory. Additional evidences are also provided via brane-web arguments in \cite{Acharya:2021jsp}. Nonetheless, it is unclear how to get the correct BPS states from M2 brane wrapping curves in (\ref{3n2-n=4-inter}).} \item{The geometry describes a new rank-1 theory, analogous to the cases with terminal singularities or singular divisors in \cite{Closset:2020afy,Closset:2021lwy}.} \end{enumerate} To solve the problem, more physical quantities should be computed, such as the Higgs branch dimensions or superconformal indices. Similar subtleties appear in the cases of$\Gamma=E^{(2)}, E^{(4)}, G_3$and$H_{168}$as well." 11. On page 50, we replaced "Hence the flavor symmetry factors from$C_i$is$SU(4)\times SU(2)^2\times U(1)$, which can be enhanced to the flavor symmetry of Seiberg rank-1$E_6$theory: \be G_F=E_6\,. \ee" with "Hence the flavor symmetry factors from$C_i$is$G_F=SU(4)\times SU(2)^2\times U(1)$." 12. On page 51, we added Hao Y. Zhang to the Acknowledgements. 13. On page 79, we replaced "The flavor symmetry factors read off from the geometry are$SU(2)^2\times U(1)$, which can be enhanced to the flavor symmetry$G_F=SU(3)\times SU(2)$of Seiberg$E_3$theory." with "The flavor symmetry factors read off from the geometry are$G_F=SU(2)^2\times U(1)$." 14. On page 80, we replaced "Hence from the$(-2)$-curves on$S_1$, we can read off flavor symmetry factors$SU(3)\times SU(2)\times U(1)$, which can be enhanced to the flavor symmetry of Seiberg rank-1$E_4$theory \be G_F=SU(5)\,. \ee The 5d SCFT from$\mb{C}^3/E^{(2)}$is then concluded to be the Seiberg rank-1$E_4$theory." with "Hence from the$(-2)$-curves on$S_1$, we can read off flavor symmetry factors$G_F=SU(3)\times SU(2)\times U(1)$." 15. On page 81, we replaced "From the curves on$S_1$, one can read off the flavor symmetry factors$SU(3)\times SU(2)\times U(1)^2$, which can be embedded into the flavor symmetry of Seiberg rank-1$E_5$theory: \be G_F=SO(10)\,. \ee The 5d SCFT from$\mb{C}^3/E^{(4)}$is then concluded to be the Seiberg rank-1$E_5$theory." with "From the curves on$S_1$, one can read off the flavor symmetry factors$G_F=SU(3)\times SU(2)\times U(1)^2\$."

16. We fixed the typos of "Mckay" to "McKay" in various places.