The construction of Frobenius manifolds from KP tau-functions

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux-Egoroff equations. This system of partial differential equations appears as a specific subset of the $n$-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux-Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function.Comment: 29 pages, latex2e, no figure

Geometric B\"acklund--Darboux transformations for the KP hierarchy

We shown that, if you have two planes in the Segal-Wilson Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by B\"acklund-Darboux transformations (BDT). The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from elementary BDT's and is given here in a closed form. The geometric description of elementary DBT's requires that one has a geometric interpretation of the dual wavefunctions involved. This is done here with the help of a suitable algebraic characterization of the wavefunction. The BDT's also induce transformations of the tau-function associated to a plane in the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric characterization of the BDT'ss that preserves these subsystems of the KP hierarchy. This generalizes the classical Darboux-transformations. we also determine an explicit expression for the squared eigenfunction potentials. Next a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy. It is shown that infinite flags in the Grassmannian yield solutions of the latter hierarchy. these flags can be constructed by means of BDT's, starting from some plane. Other applications of these BDT's are a geometric way to characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy and the construction of a vast collection of orthogonal polynomials, playing a role in matrix models.Comment: 44 pages Latex2

WDVV Equations, Darboux-Egoroff Metric and the Dressing Method

Dressing technique is used to construct commuting Lax operators which provide an integrable (canonical) structure behind Witten--Dijkgraaf--Verlinde--Verlinde equations. The commuting flows are related to the isomonodromic flows. Examples of the canonical integrable structure are given in two- and three-dimensional cases. The three-dimensional example is associated with the rational Landau-Ginzburg potentials.Comment: Contribution to the conference "Workshop on Integrable Theories, Solitons and Duality", Unesp2002, LaTeX file w. JHEP style fil

Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows

We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy the 2n-component KP hierarchy. The hierarchy equations can be formulated in terms of pseudo-differential equations for n × n matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato-Wilson relations. A reduction process leads to the AKNS, two-component Camassa-Holm and Cecotti-Vafa models and the formalism provides simple formulas for their solutions

Enumeration of hypermaps and Hirota equations for extended rationally constrained KP

We consider the Hurwitz Dubrovin--Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin--Frobenius manifold is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy. This statement was conjectured by Liu, Zhang, and Zhou. We also provide a partial enumerative meaning for this partition function associating one particular set of times with enumeration of rooted hypermaps.Comment: 39 page

p̲ -reduced Multicomponent KP Hierarchy and Classical W -algebras W(glN,p̲)

For each partition p̲ of an integer N≥ 2 , consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p̲-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical W-algebra W(glN,p̲), and write down explicit formulas for evolution of these generators along the commuting Hamiltonian flows