(X, ψ) duality and enhanced dKdV on a Riemann surface

Given a dKdV potential V, arising from a finite zone KdV situation on aRiemann surface Σ, one can create an enhanced dispersionless context dKdVϵ with an expanded V (retaining powers of ϵ) in which various formulas in the (X, ψ) duality theory of Faraggi and Matone [Phys. Rev. Lett. 78 (1997) 163–166] have representations, and a natural symplectic form (dX/ϵ) ∧ dQ(P = iQ) in the Hamilton-Jacobi theory for dKdV has a representation in terms of the prepotential F of Faraggi and Matone. The theory establishes relations between an expanded Fϵ = F and the free energy Fϵ of dKdVϵ wwhich lead to formulas relating the duality variables ai,aiD of the Seiberg-Witten type on Σ to Q = Im P = −(12 Re F).