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System of a D-brane in bosonic string theory on a constant $B$ fieldbackground is studied in order to obtain further insight into the bulk-boundaryduality. Boundary states which describe arbitrary numbers of open-stringtachyons and gluons are given. UV behaviors of field theories on thenon-commutative world-volume are investigated by using these states. We takezero-slope limits of generating functions of one-loop amplitudes of gluons (andopen-string tachyons) in which the region of the small open-string proper timeis magnified. Existence of $B$ field allows the limits to be slightly differentfrom the standard field theory limits of closed-string. They enable us tocapture world-volume theories at a trans-string scale. In this limit thegenerating functions are shown to be factorized by two curved open Wilson lines(and their analogues) and become integrals on the space of paths with aGaussian distribution around straight lines. These indicate a possibility thatfield theories on the non-commutative world-volume are topological at such atrans-string scale. We also give a proof of the Dhar-Kitazawa conjecture bymaking an explicit correspondence between the closed-string states and thepaths. Momentum eigenstates of closed-string or momentum loops also play animportant role in these analyses.Comment: 110 pages, 5 figures. References are added. Typos are correcte

Open Wilson Lines as States of Closed String