J. W. van de Leur

The n-Component KP Hierarchy and Representation Theory

It is the aim of the present article to give all formulations of the n-component KP hierarchy and clarify connections between them. The generalization to the n-component KP hierarchy is important because it contains many of the most popular systems of soliton equations, like the Davey–Stewartson system (for n=2), the two-dimensional Toda lattice (for n=2), the n-wave system (for n⩾3) and the Darboux–Egoroff system. It also allows us to construct natural generalizations to the Davey–Stewartson and Toda lattice systems. Of course, the inclusion of all these systems in the n-component KP hierarchy allows us to construct their solutions by making use of vertex operators.

Integrable Structure Behind the WDVV Equations

AbstractAn integrable structure behind the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations is identified with the reduction of the Riemann–Hilbert problem for the homogeneous loop group

Multiple sums and integrals as neutral BKP tau functions

\widehat{GL}(N,\mathbb{C})

Super boson-fermion correspondence

. The reduction requires the dressing matrices to be fixed points of an order-two loop group automorphism resulting in a subhierarchy of the

The n-Compo ne nt KP Hierarchy a nd Represe ntatio n Theory

\widehat{gL}(N,\mathbb{C})

Symmetry algebras of Lagrangian Liouville-type systems

hierarchy containing only odd-symmetry flows. The model has Virasoro symmetry; imposing Virasoro constraints ensures the homogeneity property of the Darboux–Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations.