Topology of the Real Part of the Hyperelliptic Jacobian Associated with the Periodic Toda Lattice

We study the topology of the isospectral real manifold of the ${\mathfrak sl}(N)$ periodic Toda lattice consisting of $2^{N-1}$ different systems. The solutions of these systems contain blow-ups, and the set of these singular points defines a divisor of the manifold. With the divisor added, the manifold is compactified as the real part of the $(N-1)$-dimensional Jacobi variety associated with a hyperelliptic Riemann surface of genus $g=N-1$. We also study the real structure of the divisor and provide conjectures on the topology of the affine part of the real Jacobian and on the gluing rule over the divisor to compactify the manifold based on the sign representation of the Weyl group of ${\mathfrak sl}(N)$.