stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1
$a$ | 8 | $(-1,1)\times(-1,0)\times(1,1)$ | 0 | 0 | -5 | 9 | -4 | 0 | 0
$b$ | 4 | $(-2,1)\times(-2,1)\times(-4,1)$ | 13 | 51 | - | - | 51 | -45 | -1
$c$ | 4 | $(1,0)\times(-13,-2)\times(-1,1)$ | 11 | -11 | - | - | - | - | 0
1 | 10 | $(1,0)\times(1,0)\times(1,0)$ | $x_{A}=64x_{B}=\frac{128}{13}x_{C}=64x_{D}$
|  |  | $\beta^{g}_{1}=-5$
|  |  |  $\chi_{1}=8\sqrt{\frac{2}{13}}$, $\chi_{2}=4\sqrt{26}$, $\chi_{3}=16\sqrt{\frac{2}{13}}$
Table 34: D6-brane configurations and intersection numbers of Model 34, and its gauge coupling relation is $g^{2}_{a}=\frac{2176}{53}g^{2}_{b}=15g^{2}_{c}=\frac{25}{11}(\frac{5}{3}g^{2}_{Y})=\frac{192}{53}\sqrt[4]{3}5^{3/4}\pi e^{\phi^{4}}$. model 34 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(6)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1
$a$ | 8 | $(-1,1)\times(-1,0)\times(1,1)$ | 0 | 0 | -5 | 9 | -4 | 0 | 0
$b$ | 4 | $(-2,1)\times(-2,1)\times(-4,1)$ | 13 | 51 | - | - | 57 | -55 | -1
$c$ | 4 | $(1,0)\times(-15,-2)\times(-1,1)$ | 13 | -13 | - | - | - | - | 0
1 | 6 | $(1,0)\times(1,0)\times(1,0)$ | $x_{A}=72x_{B}=\frac{48}{5}x_{C}=72x_{D}$
|  |  | $\beta^{g}_{1}=-5$
|  |  |  $\chi_{1}=4\sqrt{\frac{3}{5}}$, $\chi_{2}=6\sqrt{15}$, $\chi_{3}=8\sqrt{\frac{3}{5}}$
Table 35: D6-brane configurations and intersection numbers of Model 35, and its gauge coupling relation is $g^{2}_{a}=\frac{2736}{59}g^{2}_{b}=17g^{2}_{c}=\frac{85}{37}(\frac{5}{3}g^{2}_{Y})=\frac{64}{177}170^{3/4}\pi e^{\phi^{4}}$. model 35 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1
$a$ | 8 | $(-1,1)\times(-1,0)\times(1,1)$ | 0 | 0 | -5 | 9 | -4 | 0 | 0
$b$ | 4 | $(-2,1)\times(-2,1)\times(-4,1)$ | 13 | 51 | - | - | 63 | -65 | -1
$c$ | 4 | $(1,0)\times(-17,-2)\times(-1,1)$ | 15 | -15 | - | - | - | - | 0
1 | 2 | $(1,0)\times(1,0)\times(1,0)$ | $x_{A}=80x_{B}=\frac{160}{17}x_{C}=80x_{D}$
|  |  | $\beta^{g}_{1}=-5$
|  |  |  $\chi_{1}=4\sqrt{\frac{10}{17}}$, $\chi_{2}=2\sqrt{170}$, $\chi_{3}=8\sqrt{\frac{10}{17}}$

##  Appendix B Four-Family Standard Models from Intersecting D6-Branes with
One Tilted Torus

Table 36: D6-brane configurations and intersection numbers of Model 36, and its gauge coupling relation is $g^{2}_{a}=g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=2\sqrt{2}\pi e^{\phi^{4}}$. model 36 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)^{2}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2 | 4
$a$ | 8 | $(1,1)\times(0,-1)\times(1,1)$ | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0
$b$ | 4 | $(1,0)\times(-4,1)\times(-1,-1)$ | -3 | 3 | - | - | 0 | 0 | 0 | 4
$c$ | 4 | $(0,-1)\times(4,1)\times(1,1)$ | 3 | -3 | - | - | - | - | 4 | 0
2 | 2 | $(1,0)\times(0,1)\times(0,-2)$ | $x_{A}=4x_{B}=x_{C}=4x_{D}$
4 | 2 | $(0,1)\times(0,1)\times(-2,0)$ |  $\beta^{g}_{2}=-2$, $\beta^{g}_{4}=-2$
|  |  |  $\chi_{1}=1$, $\chi_{2}=4$, $\chi_{3}=2$
Table 37: D6-brane configurations and intersection numbers of Model 37, and its gauge coupling relation is $g^{2}_{a}=2g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=4\pi e^{\phi^{4}}$. model 37 | $U(4)\times U(4)_{L}\times U(4)_{R}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$
$a$ | 8 | $(1,-1)\times(-1,0)\times(-1,-1)$ | 0 | 0 | 0 | 4 | 0 | -4
$b$ | 8 | $(0,1)\times(-1,-2)\times(1,1)$ | -2 | 2 | - | - | 0 | 0
$c$ | 8 | $(1,0)\times(1,-2)\times(1,1)$ | 2 | -2 | - | - | - | -
|  |  | $x_{A}=\frac{1}{2}x_{B}=x_{C}=\frac{1}{2}x_{D}$
|  |  |  $\chi_{1}=1$, $\chi_{2}=\frac{1}{2}$, $\chi_{3}=2$
Table 38: D6-brane configurations and intersection numbers of Model 38, and its gauge coupling relation is $g^{2}_{a}=2g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=4\pi e^{\phi^{4}}$. model 38 | $U(4)\times U(4)_{L}\times U(4)_{R}$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$
$a$ | 8 | $(1,0)\times(-1,1)\times(-1,-1)$ | 0 | 0 | 4 | 0 | 0 | -4
$b$ | 8 | $(1,-2)\times(0,-1)\times(-1,1)$ | 2 | -2 | - | - | 0 | 0
$c$ | 8 | $(1,-2)\times(-1,0)\times(-1,-1)$ | 2 | -2 | - | - | - | -
|  |  | $x_{A}=x_{B}=\frac{1}{2}x_{C}=\frac{1}{2}x_{D}$
|  |  |  $\chi_{1}=\frac{1}{2}$, $\chi_{2}=1$, $\chi_{3}=2$
Table 39: D6-brane configurations and intersection numbers of Model 39, and its gauge coupling relation is $g^{2}_{a}=2g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=\frac{8}{3}\sqrt[4]{2}\pi e^{\phi^{4}}$. model 39 | $U(4)\times U(4)_{L}\times U(2)_{R}\times USp(2)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1
$a$ | 8 | $(1,-1)\times(1,0)\times(1,1)$ | 0 | 0 | 4 | 0 | -4 | 0 | 0
$b$ | 8 | $(0,1)\times(-1,2)\times(-1,1)$ | 2 | -2 | - | - | 0 | 4 | 4
$c$ | 4 | $(-1,0)\times(-1,-4)\times(1,-1)$ | -3 | 3 | - | - | - | - | 0
1 | 2 | $(1,0)\times(1,0)\times(-2,0)$ | $x_{A}=\frac{1}{2}x_{B}=2x_{C}=\frac{1}{2}x_{D}$
|  |  | $\beta^{g}_{1}=-2$
|  |  |  $\chi_{1}=\sqrt{2}$, $\chi_{2}=\frac{1}{2\sqrt{2}}$, $\chi_{3}=2\sqrt{2}$
Table 40: D6-brane configurations and intersection numbers of Model 40, and its gauge coupling relation is $g^{2}_{a}=g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{8}{3}\sqrt[4]{2}\pi e^{\phi^{4}}$. model 40 | $U(4)\times U(2)_{L}\times U(4)_{R}\times USp(2)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2
$a$ | 8 | $(-1,-1)\times(0,-1)\times(-1,-1)$ | 0 | 0 | 4 | 0 | -4 | 0 | 0
$b$ | 4 | $(-1,0)\times(4,1)\times(-1,1)$ | 3 | -3 | - | - | 0 | -4 | 0
$c$ | 8 | $(0,-1)\times(2,-1)\times(-1,1)$ | -2 | 2 | - | - | - | - | 4
2 | 2 | $(1,0)\times(0,1)\times(0,-2)$ | $x_{A}=2x_{B}=x_{C}=4x_{D}$
|  |  | $\beta^{g}_{2}=-2$
|  |  |  $\chi_{1}=\sqrt{2}$, $\chi_{2}=2\sqrt{2}$, $\chi_{3}=\sqrt{2}$
Table 41: D6-brane configurations and intersection numbers of Model 41, and its gauge coupling relation is $g^{2}_{a}=2g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=\frac{8}{3}\sqrt[4]{2}\pi e^{\phi^{4}}$. model 41 | $U(4)\times U(4)_{L}\times U(2)_{R}\times USp(2)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 2
$a$ | 8 | $(-1,0)\times(1,1)\times(-1,1)$ | 0 | 0 | 0 | 4 | 0 | -4 | 0
$b$ | 8 | $(1,2)\times(1,0)\times(1,-1)$ | -2 | 2 | - | - | 0 | -4 | 4
$c$ | 4 | $(-1,4)\times(0,-1)\times(1,-1)$ | 3 | -3 | - | - | - | - | 0
2 | 2 | $(1,0)\times(0,1)\times(0,-2)$ | $x_{A}=\frac{1}{2}x_{B}=\frac{1}{4}x_{C}=\frac{1}{4}x_{D}$
|  |  | $\beta^{g}_{2}=-2$
|  |  |  $\chi_{1}=\frac{1}{2\sqrt{2}}$, $\chi_{2}=\frac{1}{\sqrt{2}}$, $\chi_{3}=\sqrt{2}$
Table 42: D6-brane configurations and intersection numbers of Model 42, and its gauge coupling relation is $g^{2}_{a}=g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{8}{3}\sqrt[4]{2}\pi e^{\phi^{4}}$. model 42 | $U(4)\times U(2)_{L}\times U(4)_{R}\times USp(2)$
---|---
stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 1
$a$ | 8 | $(-1,0)\times(-1,-1)\times(1,-1)$ | 0 | 0 | 0 | 4 | -4 | 0 | 0
$b$ | 4 | $(-1,-4)\times(-1,0)\times(1,-1)$ | -3 | 3 | - | - | 4 | 0 | 0
$c$ | 8 | $(-1,-2)\times(0,1)\times(1,1)$ | -2 | 2 | - | - | - | - | 4
1 | 2 | $(1,0)\times(1,0)\times(-2,0)$ | $x_{A}=2x_{B}=\frac{1}{2}x_{C}=\frac{1}{2}x_{D}$
|  |  | $\beta^{g}_{1}=-2$
|  |  |  $\chi_{1}=\frac{1}{2\sqrt{2}}$, $\chi_{2}=\sqrt{2}$, $\chi_{3}=2\sqrt{2}$
Table 43: D6-brane configurations and intersection numbers of Model 43, and its gauge coupling relation is $g^{2}_{a}=\frac{8}{3}g^{2}_{b}=g^{2}_{c}=(\frac{5}{3}g^{2}_{Y})=\frac{16}{9}2^{3/4}\pi e^{\phi^{4}}$. model 43 | $U(4)\times U(2)_{L}\times U(2)_{R}\times USp(2)$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$ | 3
$a$ | 8 | $(0,1)\times(1,-1)\times(1,-1)$ | 0 | 0 | 1 | 3 | -4 | 0 | 0
$b$ | 4 | $(-1,1)\times(0,1)\times(-2,4)$ | 2 | -2 | - | - | -9 | -5 | 2
$c$ | 4 | $(4,-1)\times(1,0)\times(1,1)$ | -3 | 3 | - | - | - | - | 0
3 | 2 | $(0,1)\times(1,0)\times(0,2)$ | $x_{A}=x_{B}=\frac{1}{2}x_{C}=4x_{D}$
|  |  | $\beta^{g}_{3}=-4$
|  |  |  $\chi_{1}=\sqrt{2}$, $\chi_{2}=2\sqrt{2}$, $\chi_{3}=\frac{1}{\sqrt{2}}$
Table 44: D6-brane configurations and intersection numbers of Model 44, and its gauge coupling relation is $g^{2}_{a}=\frac{16}{5}g^{2}_{b}=2g^{2}_{c}=\frac{10}{7}(\frac{5}{3}g^{2}_{Y})=\frac{16}{5}\sqrt{2}\pi e^{\phi^{4}}$. model 44 | $U(4)\times U(2)_{L}\times U(2)_{R}\times$
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stack | $N$ | $(n^{1},l^{1})\times(n^{2},l^{2})\times(n^{3},l^{3})$ | $n_{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $n_{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $b$ | $b^{\prime}$ | $c$ | $c^{\prime}$
$a$ | 8 | $(0,1)\times(1,-1)\times(1,-1)$ | 0 | 0 | 1 | 3 | -4 | 0
$b$ | 4 | $(-1,1)\times(0,1)\times(-2,4)$ | 2 | -2 | - | - | -6 | -6
$c$ | 4 | $(2,-1)\times(1,0)\times(2,2)$ | -2 | 2 | - | - | - | -
|  |  | $x_{A}=x_{B}=\frac{1}{2}x_{C}=2x_{D}$
|  |  |  $\chi_{1}=1$, $\chi_{2}=2$, $\chi_{3}=1$