  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
$\tau$-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.
