  System of a D-brane in bosonic string theory on a constant $B$ field
background is studied in order to obtain further insight into the bulk-boundary
duality. Boundary states which describe arbitrary numbers of open-string
tachyons and gluons are given. UV behaviors of field theories on the
non-commutative world-volume are investigated by using these states. We take
zero-slope limits of generating functions of one-loop amplitudes of gluons (and
open-string tachyons) in which the region of the small open-string proper time
is magnified. Existence of $B$ field allows the limits to be slightly different
from the standard field theory limits of closed-string. They enable us to
capture world-volume theories at a trans-string scale. In this limit the
generating functions are shown to be factorized by two curved open Wilson lines
(and their analogues) and become integrals on the space of paths with a
Gaussian distribution around straight lines. These indicate a possibility that
field theories on the non-commutative world-volume are topological at such a
trans-string scale. We also give a proof of the Dhar-Kitazawa conjecture by
making an explicit correspondence between the closed-string states and the
paths. Momentum eigenstates of closed-string or momentum loops also play an
important role in these analyses.
