  We study the single spin asymmetry (SSA) induced by purely gluonic
correlation inside a nucleon, in particular, by the three-gluon correlation
functions in the transversely polarized nucleon, $p^\uparrow$. This
contribution is embodied as a twist-3 mechanism in the collinear factorization
framework and controls the SSA to be observed in the $D$-meson production with
large transverse-momentum in semi-inclusive DIS (SIDIS), $ep^\uparrow
\rightarrow eDX$. We define the relevant three-gluon correlation functions in
the nucleon, and determine their complete set at the twsit-3 level taking into
account symmetry constraints in QCD. We derive the single-spin-dependent cross
section for the $D$-meson production in SIDIS, taking into account all the
relevant contributions at the twist-3 level. The result is obtained in a
manifestly gauge-invariant form as the factorization formula in terms of the
three-gluon correlation functions and reveals the five independent structures
with respect to the dependence on the azimuthal angle for the produced $D$
meson. We also demonstrate the remarkable relation between the twist-3
single-spin-dependent cross section and twist-2 cross sections for the
$D$-meson production, as a manifestation of universal structure behind the SSA
in a variety of hard processes.
