A Realization of Matrix KP Hierarchy by Coincident D-brane States

D-branes constitute an important part of the superstring theory in its nonperturbative description. Especially coincident multi D-brane states enable us to reproduce nonabelian guage theory in the low energy. From the mathematical point of view, on the other hand, a non-perturbative field theory must be formulated within the framework of a completely integrable system, namely a system whose whole possible solutions can be determined analytically. If a system is nonintegrable there remain some domains which refuse our nonperturbative analysis in principle. Therefore it is natural to ask if there exists a way to describe a D-brane dynamics in terms of integrable systems. The correspondence between open string correlation functions and the KP hierarchy of soliton theory has been known for a long time[1]. Recently we have extended[2] the relation to the case of closed strings and shown that a boundary state of conformal strings satisfies Hirota-Miwa equation. Moreover we generalized Hirota-Miwa equation in order to characterize dynamics of D-branes on which tachyon fields are condensed[3]. We would like to extend our argument further, in this paper, to establish a correspondence between multi D-brane states and integrable systems. In particular we will show that a multi D-brane state satisfies the matrix generalization of the KP hierarchy, thus leading to a natural interpretation of coincident D-branes as a sourse of non-abelian gauge theory.