3 Summary

We can summarize the results we have obtained in the ``flow chart'' below.

$A_\mu{}^i_j\left( x \right)$	$\Rightarrow$	${\mbox{\boldmath {$A$}}}_\mu{}^i_j$	$\mapsto$	${\mathcal{A}}_\mu\left( \sigma \right)$	$\hookrightarrow$	$X_\mu\left( \sigma \right)$
gauge field	$\Rightarrow$	matrix	$\mapsto$	Weyl symbol	$\hookrightarrow$	brane coordinate
$S^{\mathrm{GYM}}$	$\Rightarrow$	$S^{\mathrm{q, GYM}}_{\mathrm{red.}}$	$\mapsto$	$W^{\mathrm{GYM}}$	$\hookrightarrow$	$S^{p=4k-1}_{\mathrm{DT}}=$volume term
$S^\theta$	$\Rightarrow$	$S^{q, \theta}_{\mathrm{red.}}$	$\mapsto$	$W^\theta$	$\hookrightarrow$	$S^{p=4k-2}_{\mathrm{CS}}=$boundary term

The various arrows respectively represent:

$\Rightarrow$

quenching and zero volume limit;

$\mapsto$

Weyl-Wigner-Moyal mapping;

$\hookrightarrow$

large-$N$ limit.

Through all these operations we transformed the generalized Yang-Mills theory, described by equation 1 and equation 2 into an effective theory for higher dimensional, $(4k-1)$-dimensional, vacuuh domain of large-$N$, generalized Yang-Mills theory, bounded bd a $(4k-2)$-dimensional Chern-Simons brane.