# nLab Sandbox2

$\gamma_\Omega \;=\; \left( \array{ 0_{16} & 1_{16} \\ -1_{16} & 0_{16} } \right)$
$(\gamma_\Omega)^{-1} \;=\; \gamma_\Omega$
$\gamma_{1} \;=\; diag \big( \array{ 1_8, & i_{8}, & -1_{8}, & -i_{8} } \big)$
$(\gamma_{\Omega})^{-1} \gamma_1 (\gamma_{\Omega}) \;=\; \gamma_1$
$(\gamma_{\Omega})^{-1} (A)^{T} (\gamma_{\Omega}) \;=\; -A$
\begin{aligned} (\gamma_\Omega A)^T & = (A)^T (\gamma_\Omega)^T \\ & = - (A)^T (\gamma_\Omega) \\ & = (\gamma_\Omega) (A) \end{aligned}

$e_{b_1 b_2} \;=\; F_{a [b_1} \delta_{b_2]}^5 \, d x^a + \tfrac{1}{3!} \epsilon_{b_1 b_2 b_3 a_1 a_2} F^{a_1 a_2} d x^{b_3}$
$\,$
\begin{aligned} & \Big( \underset{ \mathclap{ 0 \leq a_i \leq 5 } }{\sum} e_{a_1}{}^{a_2} \wedge e_{a_2}{}^{a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \Big( 3 \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 5} \wedge e_{a_3 5} \wedge e^{a_1 a_3} + \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 a_2} \wedge e^{a_2 a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \phantom{+}\; \tfrac{1}{2} F_{a_1 b_1} d x^{a_1} \wedge F_{a_2 b_2} d x^{a_2} \wedge g_{a_3 b_3} d x^{a_3} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{c_1 c_2} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \\ & \phantom{=}\; + \epsilon^{b_1}{}_{b_2 c_1 c_2 c_3 } \delta^{b_1}_{[b_2} \delta^{a_1}_{a'_1} \delta^{a_2}_{a'_2} \delta^{a_3}_{a'_3]} F^{a'_1 a'_2} d x^{a'_3} F^{c_1 c_2} d x^{c_3} F_{a_1 a_2} g_{a_4 a_3} d x^{a_4} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \\ & = \phantom{+}\; \tfrac{1}{2} F_{a_1 b_1} d x^{a_1} \wedge F_{a_2 c_2} d x^{a_2} \wedge g_{a_3 b_3} d x^{a_3} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{c_1 b_2} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \;+\; \cdots \end{aligned}

\begin{aligned} & \Big( \underset{ \mathclap{ 0 \leq a_i \leq 5 } }{\sum} e_{a_1}{}^{a_2} \wedge e_{a_2}{}^{a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \Big( 3 \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 5} \wedge e_{a_3 5} \wedge e^{a_1 a_3} + \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 a_2} \wedge e^{a_2 a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \tfrac{1}{2} F_{a_1 b_1} d x^{a_1} \wedge F_{a_2 b_2} d x^{a_2} \wedge g_{a_3 b_3} d x^{a_3} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{c_1 c_2} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \\ & = \tfrac{1}{2} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{a_1 b_1} F_{a_2 b_2} g_{a_3 b_3} F_{c_1 c_2} F_{d_1 d_2} \epsilon^{a_1 a_2 a_3 d_1 d_2} \cdot \mathrm{dvol} \;+\; \cdots \end{aligned}

\begin{aligned} & \phantom{= +}\, \epsilon^{a_1 a_2 a_3 d_1 d_2} \big( F_{a_1 b_1} F_{a_2 b_2} g_{a_3 b_3} F_{c_1 c_2} F_{d_1 d_2} \big) \epsilon^{b_1 b_2 b_3 c_1 c_2} \\ & = - \epsilon^{a_1 a_2 a_3 d_1 d_2} \big( F_{a_1 b_1} F_{a_2 c_2} g_{a_3 b_3} F_{c_1 b_2} F_{d_1 d_2} \big) \epsilon^{b_1 b_2 b_3 c_1 c_2} \\ & = + \epsilon^{a_1 a_2 a_3 d_1 d_2} \big( F_{a_1 b_1} F_{d_1 c_2} g_{a_3 b_3} F_{c_1 b_2} F_{a_2 d_2} \big) \epsilon^{b_1 b_2 b_3 c_1 c_2} \end{aligned}

Last revised on September 2, 2019 at 15:04:46. See the history of this page for a list of all contributions to it.