arXiv subjects

N. M. J. Woodhouse

Publications and source records attributed to N. M. J. Woodhouse.

3 recordsLinked to original sources

arXiv2001

Tau-functions, twistor theory, and quantum field theory

This article is concerned with obtaining the standard tau function descriptions of integrable equations (in particular, here the KdV and Ernst equations are considered) from the geometry of their twistor correspondences. In particular, we will see that the quantum field theoretic formulae for tau functions can be understood as arising from geometric quantization of the twistor data. En route we give a geometric quantization formulation of Chern-Simons and WZW quantum field theories using the Quillen determinant line bundle construction and ingredients from Segal's conformal field theory. The $τ$-functions are then seen to be amplitudes associated with gauge group actions on certain coherent states within these theories that can be obtained from the twistor description.

math-ph↗

arXiv2000

The symplectic and twistor geometry of the general isomonodromic deformation problem

Hitchin's twistor treatment of Schlesinger's equations is extended to the general isomonodromic deformation problem. It is shown that a generic linear system of ordinary differential equations with gauge group SL(n,C) on a Riemann surface X can be obtained by embedding X in a twistor space Z on which sl(n,C) acts. When a certain obstruction vanishes, the isomonodromic deformations are given by deforming X in Z. This is related to a description of the deformations in terms of Hamiltonian flows on a symplectic manifold constructed from affine orbits in the dual Lie algebra of a loop group.

nlin.SI↗

arXiv1998

From 2D Integrable Systems to Self-Dual Gravity

We explain how to construct solutions to the self-dual Einstein vacuum equations from solutions of various two-dimensional integrable systems by exploiting the fact that the Lax formulations of both systems can be embedded in that of the self-dual Yang--Mills equations. We illustrate this by constructing explicit self-dual vacuum metrics on $\R^2\times Σ$, where $Σ$ is a homogeneous space for a real subgroup of $SL(2, \C)$ associated with the two-dimensional system.

solv-int↗