Johan van de Leur

Bosonic and fermionic realizations of the affine algebra\(g\hat l_n \)

AbstractWe give an explicit description of all inequivalent Heisenberg subalgebras of the affine Lie algebra

Twisted GLn Loop Group Orbit and Solutions of the WDVV Equations

g\hat l_n (\mathbb{C})

Solutions of the WDVV Equations and Integrable Hierarchies of KP type

and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra. The construction uses multicomponent fermionic fields and yields a correspondence between bosons (elements of the Heisenberg subalgebra) and fermions.

Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

Abstract We obtain the collection of symmetric and symplectic matrix integrals and the collection of Pfaffian tau-functions, recently described by Peng and Adler and van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector of a fermionic Fock space. This fermionic Fock space is the same space as one constructs to obtain the KP and 1-Toda lattice hierarchy.

Irreducible highest weight representations of the simple n -Lie algebra

We show that all (n-component) KP tau-functions, which are related to the twisted loop group of GLn, give solutions of the Darboux-Egoroff system of PDE’s. Using the Geometry of the Grassmannian we construct from the corresponding wave function the deformed flat coordinates of the Egoroff metric and from this the corresponding solution of the Witten–Dijkgraaf–E. Verlinde–H. Verlinde equations

Pfaffian and determinantal tau functions

We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this proves a conjecture of Milanov-Tseng that relates the total descendent potential of the orbifold

A family of second Lie algebra structures for symmetries of a dispersionless Boussinesq system

C_{k,m}

Level one representations of the twisted affine algebrasan(2) anddn(2)

with a tau function of the bigraded Toda hierarchy.