W-Geometry and isomonodromic deformations
M. Olshanetsky
Abstract:
We consider new type of times in the isomonodromy preserving
equations coming from deformations of $W$-structures
of the basic spectral curve. $W_k$-structure is the
generalized complex structure defined by $(-k+1,1)$-differentials,
while the $W_2$-structure is the standard complex structure
since $(-1,1)$-differentials are the Beltrami differentials.
We derive these equations via symplectic reduction from the Chern-Simons theory.
Solutions of the moment constraints equations are the Lax operators
of the corresponding linear problem. These equations for $k>2$ are
nonlinear and the Lax operators and the Hamiltonians can be found
only perturbatively. On the quantum level this theory corresponds
to the generalized KZB equations with higher order Casimirs.
As an example, we consider $W_3$ theory on $P^1$
with $n$ marked points (the analog of the SL(3) Schlesinger system).