| \((r_+,r_-)\) | Charge lattice | \( \mathcal{G}\) | Singularity | Theory | Gauge groups |
| (17,1) | \( \Gamma_{1,1}\oplus 2\text{E}_8 \) | Heterotic string on \(S^1\) | Table | ||
| \(\text{(9,1)}\) | \(\text{E}_8 \oplus \Gamma_{\text{1,1}}\) | \(\mathbb{Z}_2\) | \([\text{E}_8]\) | \(\text{CHL string}\) | Table |
| \(\text{(1,1)}\) | \(\Gamma_{\text{1,1}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([\text{D}_{16}]\) | \(\text{IIB on DP,}\ C_0 = \frac{1}{2}\) | Table |
| \(\text{(1,1)}\) | \(\Gamma_{\text{1,1}}\) | \(\mathbb{Z}_2\) | \([\text{D}_{16} \mathbb{Z}_2]\) | \(\text{IIB on DP,}\ C_0 = 0\) | Table |
| \(\text{(1,1)}\) | \(\Gamma_{\text{1,1}}\) | \(\mathbb{Z}_2\) | \([2 \text{E}_8]\) | \(\text{M on KB}\) | Table |
| \((r_+,r_-)\) | Charge lattice | \( \mathcal{G}\) | Singularity | Theory | Gauge groups |
| (18,2) | \( \Gamma_{2,2}\oplus 2\text{E}_8 \) | Heterotic string on \(T^2\) | Table | ||
| \(\text{(10,2)}\) | \(\text{D}_8 \oplus \Gamma_{\text{2,2}}\) | \(\mathbb{Z}_2\) | \([\text{D}_8]\) | \(\text{CHL string ($\mathbb{Z}_2$-triple)}\) | Table |
| \(\text{(2,2)}\) | \(\Gamma_{\text{2,2}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([2 \text{D}_8]\) | \(\text{IIB on }S^1 \times \text{DP,}\ C_0 = \frac{1}{2}\) | Table |
| \(\text{(2,2)}\) | \(\Gamma_{\text{1,1}} \oplus \Gamma_{\text{1,1}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([2 \text{D}_8 \mathbb{Z}_2]\) | \(\text{IIB on }S^1 \times \text{DP,}\ C_0 = 0\) | Table |
| \((r_+,r_-)\) | Charge lattice | \( \mathcal{G}\) | Singularity | Theory | Gauge groups |
| (19,3) | \( \Gamma_{3,3}\oplus 2\text{E}_8 \) | Heterotic string on \(T^3\) | Table | ||
| \(\text{(11,3)}\) | \(2 \text{D}_4 \oplus \Gamma_{\text{3,3}}\) | \(\mathbb{Z}_2\) | \([2 \text{D}_4]\) | \(\text{F on }\frac{\text{K3} \times S^1}{\mathbb{Z}_2}\text{ ($\mathbb{Z}_2$-triple)}\) | Table |
| \(\text{(3,3)}\) | \(\Gamma_{\text{3,3}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([4 \text{D}_4 \mathbb{Z}_2]\) | \(\text{F on }\frac{(T^4 \times S^1)'}{\mathbb{Z}_2}\) | Table |
| \(\text{(3,3)}\) | \(\Gamma_{\text{1,1}} \oplus \Gamma_{\text{2,2}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([4 \text{D}_4 \mathbb{Z}_2^2]\) | \(\text{F on }\frac{T^4 \times S^1}{\mathbb{Z}_2}\) | Table |
| \(\text{(7,3)}\) | \(2 \text{A}_2 \oplus \Gamma_{\text{3,3}}\) | \(\mathbb{Z}_3\) | \([2 \text{E}_6]\) | \(\text{F on }\frac{\text{K3} \times S^1}{\mathbb{Z}_3}\text{ ($\mathbb{Z}_3$-triple)}\) | Table |
| \(\text{(1,3)}\) | \(-\text{A}_2 \oplus \Gamma_{\text{1,1}}^{\text{(3)}}\) | \(\mathbb{Z}_3\) | \([3 \text{E}_6]\) | \(\text{F on }\frac{(T^4 \times S^1)'}{\mathbb{Z}_3}\) | Table |
| \(\text{(1,3)}\) | \(-\text{A}_2 \oplus \Gamma_{\text{1,1}}\) | \(\mathbb{Z}_3\) | \([3 \text{E}_6 \mathbb{Z}_3]\) | \(\text{F on }\frac{T^4 \times S^1}{\mathbb{Z}_3}\) | Table |
| \(\text{(5,3)}\) | \(2 \text{A}_1 \oplus \Gamma_{\text{3,3}}\) | \(\mathbb{Z}_4\) | \([2 \text{E}_7]\) | \(\text{F on }\frac{\text{K3} \times S^1}{\mathbb{Z}_4}\text{ ($\mathbb{Z}_4$-triple)}\) | Table |
| \(\text{(1,3)}\) | \(-2 \text{A}_1 \oplus \Gamma_{\text{1,1}}^{\text{(2)}}\) | \(\mathbb{Z}_4\) | \([\text{D}_4 \oplus 2 \text{E}_7]\) | \(\text{F on }\frac{(T^4 \times S^1)'}{\mathbb{Z}_4}\) | Table |
| \(\text{(1,3)}\) | \(-2 \text{A}_1 \oplus \Gamma_{\text{1,1}}\) | \(\mathbb{Z}_4\) | \([\text{D}_4 \oplus 2 \text{E}_7 \, | \mathbb{Z}_2]\) | \(\text{F on }\frac{T^4 \times S^1}{\mathbb{Z}_4}\) | Table |
| \(\text{(3,3)}\) | \(\Gamma_{\text{3,3}}\) | \(\mathbb{Z}_5\) | \([2 \text{E}_8]\) | \(\text{F on }\frac{\text{K3} \times S^1}{\mathbb{Z}_5}\text{ ($\mathbb{Z}_5$-triple)}\) | Table |
| \(\text{(1,3)}\) | \(\Gamma_{\text{1,1}}-\text{A}_2^{\text{(2)}}\) | \(\mathbb{Z}_6\) | \([\text{D}_4 \oplus \text{E}_6 \oplus \text{E}_8]\) | \(\text{F on }\frac{T^4 \times S^1}{\mathbb{Z}_6}\) | Table |
| \(\text{(3,3)}\) | \(\Gamma_{\text{3,3}}\) | \(\mathbb{Z}_6\) | \([2 \text{E}_8]\) | \(\text{F on }\frac{\text{K3} \times S^1}{\mathbb{Z}_6}\text{ ($\mathbb{Z}_6$-triple)}\) | Table |
| \((r_+,r_-)\) | Charge lattice | \( \mathcal{G}\) | Singularity | Theory | Gauge groups |
| (20,4) | \( \Gamma_{4,4}\oplus 2\text{E}_8 \) | Heterotic string on \(T^4\) | Table | ||
| \(\text{(12,4)}\) | \(2 \text{D}_4 \oplus \Gamma_{\text{3,3}} \oplus \Gamma_{\text{1,1}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([8 \text{A}_1 \mathbb{Z}_2]\) | \(\text{M on }\frac{\text{K3} \times S^1}{\mathbb{Z}_2}\text{ ($\mathbb{Z}_2$-triple)}\) | Table |
| \(\text{(8,4)}\) | \(\text{D}_4 \oplus \Gamma_{\text{4,4}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([12 \text{A}_1 \mathbb{Z}_2]\) | \(?\) | Table |
| \(\text{(8,4)}\) | \(4 \text{A}_1 \oplus \Gamma_{\text{2,2}} \oplus \Gamma_{\text{2,2}}^{\text{(2)}}\) | \(\mathbb{Z}_2^2\) | \([12 \text{A}_1 \mathbb{Z}_2^2]\) | \(\mathbb{Z}_2 \times \mathbb{Z}_2\text{-quadruple}\) | Table |
| \(\text{(6,4)}\) | \(2 \text{A}_1 \oplus \Gamma_{\text{4,4}}^{\text{(2)}}\) | \(\mathbb{Z}_2^2\) | \([14 \text{A}_1 \mathbb{Z}_2^2]\) | \(?\) | Table |
| \(\text{(6,4)}\) | \(2 \text{A}_1 \oplus \Gamma_{\text{1,1}} \oplus \Gamma_{\text{3,3}}^{\text{(2)}}\) | \(\mathbb{Z}_2^3\) | \([14 \text{A}_1 \mathbb{Z}_2^3]\) | \(?\) | Table |
| \(\text{(5,4)}\) | \(\text{A}_1 \oplus \Gamma_{\text{4,4}}^{\text{(2)}}\) | \(\mathbb{Z}_2^3\) | \([15 \text{A}_1 \mathbb{Z}_2^3]\) | \(?\) | Table |
| \(\text{(5,4)}\) | \(\text{A}_1 \oplus \Gamma_{\text{1,1}} \oplus \Gamma_{\text{3,3}}^{\text{(2)}}\) | \(\mathbb{Z}_2^4\) | \([15 \text{A}_1 \mathbb{Z}_2^4]\) | \(?\) | Table |
| \(\text{(4,4)}\) | \(\Gamma_{\text{3,3}}^{\text{(2)}}\) | \(\mathbb{Z}_2^2\) | \([16 \text{A}_1 \mathbb{Z}_2^4]\) | \(?\) | Table |
| \(\text{(4,4)}\) | \(\Gamma_{\text{4,4}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([16 \text{A}'_1 \mathbb{Z}_2^4]\) | \(\text{F on }S^1\times\frac{(T^4 \times S^1)'}{\mathbb{Z}_2}\) | Table |
| \(\text{(4,4)}\) | \(\Gamma_{\text{1,1}} \oplus \Gamma_{\text{3,3}}^{\text{(2)}}\) | \(\mathbb{Z}_2\) | \([16 \text{A}_1 \mathbb{Z}_2^5]\) | \(\text{M on }\frac{T^4 \times S^1}{\mathbb{Z}_2}\) | Table |
| \(\text{(3,4)}\) | \(-\text{A}_1 \oplus \Gamma_{\text{3,3}}^{\text{(2)}}\) | \(\mathbb{Z}_2^3\) | \([17 \text{A}_1 \mathbb{Z}_2^5]\) | \(?\) | Table |
| \(\text{(2,4)}\) | \(-2 \text{A}_1 \oplus \Gamma_{\text{2,2}}^{\text{(2)}}\) | \(\mathbb{Z}_2^2\) | \([18 \text{A}_1 \mathbb{Z}_2^6]\) | \(?\) | Table |
| \(\text{(2,4)}\) | \(-2 \text{A}_1 \oplus \Gamma_{\text{2,2}}^{\text{(2)}}\) | \(\mathbb{Z}_2^2\) | \([18 \text{A}'_1 \mathbb{Z}_2^6]\) | \(?\) | Table |
| \(\text{(1,4)}\) | \(-3 \text{A}_1 \oplus \Gamma_{\text{1,1}}^{\text{(2)}}\) | \(\mathbb{Z}_2^3\) | \([19 \text{A}_1 \mathbb{Z}_2^7]\) | \(?\) | Table |
| \(\text{(8,4)}\) | \(2 \text{A}_2 \oplus \Gamma_{\text{3,3}} \oplus \Gamma_{\text{1,1}}^{\text{(3)}}\) | \(\mathbb{Z}_3\) | \([6 \text{A}_2 \mathbb{Z}_3]\) | \(\text{M on }\frac{\text{K3} \times S^1}{\mathbb{Z}_3}\text{ ($\mathbb{Z}_3$-triple)}\) | Table |
| \(\text{(4,4)}\) | \(\Gamma_{\text{2,2}} \oplus \Gamma_{\text{2,2}}^{\text{(3)}}\) | \(\mathbb{Z}_3^2\) | \([8 \text{A}_2 \mathbb{Z}_3^2]\) | \(?\) | Table |
| \(\text{(2,4)}\) | \(-\text{A}_2 \oplus \Gamma_{\text{2,2}}^{\text{(3)}}\) | \(\mathbb{Z}_3\) | \([9 \text{A}_2 \mathbb{Z}_3^2]\) | \(\text{F on }S^1\times\frac{(T^4 \times S^1)'}{\mathbb{Z}_3}\) | Table |
| \(\text{(2,4)}\) | \(-\text{A}_2 \oplus \Gamma_{\text{1,1}} \oplus \Gamma_{\text{1,1}}^{\text{(3)}}\) | \(\mathbb{Z}_3\) | \([9 \text{A}_2 \mathbb{Z}_3^3]\) | \(\text{M on }\frac{T^4 \times S^1}{\mathbb{Z}_3}\) | Table |
| \(\text{(6,4)}\) | \(2 \text{A}_1 \oplus \Gamma_{\text{3,3}} \oplus \Gamma_{\text{1,1}}^{\text{(4)}}\) | \(\mathbb{Z}_4\) | \([2 \text{A}_1 \oplus 4 \text{A}_3 \, | \mathbb{Z}_4]\) | \(\text{M on }\frac{\text{K3} \times S^1}{\mathbb{Z}_4}\text{ ($\mathbb{Z}_4$-triple)}\) | Table |
| \(\text{(4,4)}\) | \(\Gamma_{\text{2,2}} \oplus \Gamma_{\text{1,1}}^{\text{(2)}} \oplus \Gamma_{\text{1,1}}^{\text{(4)}}\) | \(\mathbb{Z}_2 \mathbb{Z}_4\) | \([4 \text{A}_1 \oplus 4 \text{A}_3 \, | \mathbb{Z}_2 \mathbb{Z}_4]\) | \(?\) | Table |
| \(\text{(2,4)}\) | \(-2 \text{A}_1 \oplus \Gamma_{\text{1,1}}^{\text{(2)}} \oplus \Gamma_{\text{1,1}}^{\text{(4)}}\) | \(\mathbb{Z}_4\) | \([6 \text{A}_1 \oplus 4 \text{A}_3 \, | \mathbb{Z}_2 \mathbb{Z}_4]\) | \(\text{F on }S^1\times\frac{(T^4 \times S^1)'}{\mathbb{Z}_4}\) | Table |
| \(\text{(2,4)}\) | \(-2 \text{A}_1 \oplus \Gamma_{\text{1,1}} \oplus \Gamma_{\text{1,1}}^{\text{(4)}}\) | \(\mathbb{Z}_4\) | \([6 \text{A}_1 \oplus 4 \text{A}_3 \, | \mathbb{Z}_2^2 \mathbb{Z}_4]\) | \(\text{M on }\frac{T^4 \times S^1}{\mathbb{Z}_4}\) | Table |
| \(\text{(2,4)}\) | \(\Gamma_{\text{2,2}}-2 \text{A}_1^{\text{(2)}}\) | \(\mathbb{Z}_4^2\) | \([6 \text{A}_3 \mathbb{Z}_4^2]\) | \(?\) | Table |
| \(\text{(4,4)}\) | \(\Gamma_{\text{3,3}} \oplus \Gamma_{\text{1,1}}^{\text{(5)}}\) | \(\mathbb{Z}_5\) | \([4 \text{A}_4 \mathbb{Z}_5]\) | \(\text{M on }\frac{\text{K3} \times S^1}{\mathbb{Z}_5}\text{ ($\mathbb{Z}_5$-triple)}\) | Table |
| \(\text{(2,4)}\) | \(\Gamma_{\text{1,1}}-\text{A}_2^{\text{(2)}} \oplus \Gamma_{\text{1,1}}^{\text{(6)}}\) | \(\mathbb{Z}_6\) | \([5 \text{A}_1 \oplus 4 \text{A}_2 \oplus \text{A}_5 \, | \mathbb{Z}_6]\) | \(\text{M on }\frac{T^4 \times S^1}{\mathbb{Z}_6}\) | Table |
| \(\text{(4,4)}\) | \(\Gamma_{\text{3,3}} \oplus \Gamma_{\text{1,1}}^{\text{(6)}}\) | \(\mathbb{Z}_6\) | \([2 \text{A}_1 \oplus 2 \text{A}_2 \oplus 2 \text{A}_5 \, | \mathbb{Z}_6]\) | \(\text{M on }\frac{\text{K3} \times S^1}{\mathbb{Z}_6}\text{ ($\mathbb{Z}_6$-triple)}\) | Table |
| \(\text{(2,4)}\) | \(-\text{A}_2 \oplus \Gamma_{\text{2,2}}^{\text{(2)}}\) | \(\mathbb{Z}_6\) | \([3 \text{A}_1 \oplus 3 \text{A}_5 \, | \mathbb{Z}_6]\) | \(?\) | Table |
| \(\text{(2,4)}\) | \(-\text{A}_1 \oplus \Gamma_{\text{2,2}}-\text{A}_1^{\text{(3)}}\) | \(\mathbb{Z}_2 \mathbb{Z}_6\) | \([3 \text{A}_1 \oplus 3 \text{A}_5 \, | \mathbb{Z}_2 \mathbb{Z}_6]\) | \(?\) | Table |
| \(\text{(2,4)}\) | \(\left(\begin{smallmatrix}-2 & 1 \\ 1 & -4 \end{smallmatrix}\right) \oplus \Gamma_{\text{2,2}}\) | \(\mathbb{Z}_7\) | \([3 \text{A}_6 \mathbb{Z}_7]\) | \(\text{M on }\frac{\text{K3} \times S^1}{\mathbb{Z}_7}\) | Table |
| \(\text{(2,4)}\) | \(-\text{A}_1 \oplus \Gamma_{\text{2,2}}-\text{A}_1^{\text{(2)}}\) | \(\mathbb{Z}_8\) | \([\text{A}_1 \oplus \text{A}_3 \oplus 2 \text{A}_7 \, | \mathbb{Z}_8]\) | \(\text{M on }\frac{\text{K3} \times S^1}{\mathbb{Z}_8}\) | Table |
We only show the groups with maximal rank.
Simple algebras always have short roots of length 2 except when we write a superscript indicating half of their length.
Created by Bernardo Fraiman and Hector Parra de Freitas.
For any queries please contact us at bernardo.fraiman@cern.ch and hparradefreitas@fas.harvard.edu