Open Wilson lines as states of closed string

A 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. The UV behavior of field theories on the non-commutative world-volume is investigated by using these states. We take the zero-slope limits of the 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. The existence of a B field allows the limits to be slightly different from the standard field theory limits of a closed-string. These limits enable us to obtain world-volume theories at a trans-string scale. In this limit the generating functions are shown to be factorized into two curved open Wilson lilies (and their analogues) and become integrals on the space of paths with a Gaussian distribution around straight lilies. These facts indicate the 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 determining 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.