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Johan van de Leur

Publications and source records attributed to Johan van de Leur.

28 records · Page 2Linked to original sources
arXiv2001

Integrable Structure behind WDVV Equations

An integrable structure behind Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equations is identified with reduction of a Riemann-Hilbert problem for a homogeneous GL(N, C) loop group. Reduction requires the dressing matrices to be fixed points of a loop group automorphism of order two resulting in a sub-hierarchy of gl(N,C) hierarchy containing only odd symmetry flows. The model possesses Virasoro symmetry and imposing Virasoro constraints ensures homogeneity property of the Darboux-Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations.

hep-th↗
arXiv2001

Twisted Toroidal Lie Algebras

Using $n$ finite order automorphisms on a simple complex Lie algebra we construct twisted $n$-toroidal Lie algebras. Thus obtaining Lie algebras wich have a rootspace decomposition. For the case $n=2$ we list certain simple Lie algebras and their automorphisms, which produce twisted 2-toroidal algebras. In this way we obtain Lie algebras that are related to all Extended Affine Root Systems of K. Saito.

math.RT↗
arXiv2001

Matrix integrals and the geometry of spinors

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.

solv-int↗
arXiv2000

Twisted GL_n Loop Group Orbit and Solutions of the WDVV Equations

We show that all (n-component) KP tau-functions, which are related to the twisted loop group of $GL_n$, 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.

nlin.SI↗
arXiv1996

The vector k-constrained KP hierarchy and Sato's Grassmannian

We use the representation theory of the infinite matrix group to show that (in the polynomial case) the $n$--vector $k$--constrained KP hierarchy has a natural geometrical interpretation on Sato's infinite Grassmannian. This description generalizes the the $k$--reduced KP or Gelfand--Dickey hierarchies.

q-alg↗
arXiv1994

The Adler-Shiota-van Moerbeke formula for the BKP hierarchy

We study the BKP hierarchy and prove the existence of an Adler--Shiota--van Moerbeke formula. This formula relates the action of the $BW_{1+\infty}$--algebra on tau--functions to the action of the ``additional symmetries'' on wave functions.

hep-th↗
arXiv1994

The $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and $W_{1+\infty}$ constraints

To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we obtain reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. We show that the following two constraints on a KP $τ$--function are equivalent (1) $τ$ is a $τ$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy which satisfies string equation, $L_{-1}τ=0$, (2) $τ$ satisfies the vacuum constraints of the $W_{1+\infty}$ algebra. Talk given at the V International Conference on Mathematical Physics, String Theory and Quantum Gravity at Alushta, June 10-20 1994

hep-th↗
arXiv1994

The $W_{1+\infty}(gl_s)$--symmetries of the $S$--component KP hierarchy

Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau--functions to an action of a vertex operator of pseudo-differential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchyto the $W_{1+\infty}$--, respectively $W_{1+\infty}\times W_{1+\infty}$--algebra symmeties. In this paper we generalize the results to the $s$--component KP hierarchy. The vertex operators generate the algebra $W_{1+\infty}(gl_s)$, the matrix version of $W_{1+\infty}$. Since the Toda lattice hierarchy is equivalent to the $2$--component KP hierarchy, the results of this paper uncover in that particular case a much richer structure than the one obtained by Adler, Shiota and van Moerbeke.

hep-th↗
arXiv1994

KdV type hierarchies, the string equation and $W_{1+\infty}$ constraints

To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we make reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. Now assuming that (1) $τ$ is a $τ$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and (2) $τ$ satisfies a `natural' string equation, we prove that $τ$ also satisfies the vacuum constraints of the $W_{1+\infty}$ algebra.

hep-th↗