CONSTRAINED KP HIERARCHIES - ADDITIONAL SYMMETRIES, DAARBOUX-BACKLUNDSOLUTIONS AND RELATIONS TO MULTIMATRIX MODELS

This paper provides a systematic description of the interplay between a specific class of reductions denoted as cKP(r,m) (r, m greater than or equal to 1) of the primary continuum integrable system - the Kadomt sev-Petviashvili (KP) hierarchy and discrete multi-matrix models. The relevant integrable cKP(r,m) structure is a generalization of the fami liar r-reduction of the full KP hierarchy to the SL(r) generalized KdV hierarchy cKP(r,0). The important feature of cKP(r,m) hierarchies is the presence of a discrete symmetry structure generated by successive Darboux-Backlund (DB) transformations. This symmetry allows for expres sing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a cKP(r,l) defines a g eneralized two-dimensional Toda lattice structure. Furthermore, we con sider the class of truncated KP hierarchies (i.e. those defined via Wi lson-Sate dressing operator with a finite truncated pseudo-differentia l series) and establish explicitly their close relationship with DB or bits of cKP(r,m) hierarchies. This construction is relevant for findin g partition functions of the discrete multi-matrix models.The next imp ortant step involves the reformulation of the familiar nonisospectral additional symmetries of the full KP hierarchy so that their action on cKP(r,m) hierarchies becomes consistent with the constraints of the r eduction. Moreover, we show that the correct modified additional symme tries are compatible with the discrete DB symmetry on the cKP(r,m) DB orbits. The above technical arsenal is subsequently applied to obtain complete solutions of the discrete multi-matrix models. The key ingred ient is our identification of q-matrix models as DB orbits of cKP(r,1) integrable hierarchies where r = (p(q) - 1)...(p(2) - 1) with p(1),.. .,p(q) indicating the orders of the corresponding random matrix potent ials. Applying the notions of additional symmetry structure and the te chnique of equivalent hierarchies turns out to be instrumental in impl ementing the string equation and finding closed expressions for the pa rtition functions of the discrete multi-matrix models. As a byproduct, we obtain a representation of the tau-function of the most general DB orbit of cKP(1,1) hierarchy in terms of a new generalized matrix mode l. The present formalism is of direct relevance to the study of variou s random matrix problems in condensed matter physics and other related areas. In particular, we obtain a new type of joint distribution func tion with an additional attractive two-body and a genuine many-body po tentials.