Hitchin integrable systems, deformations of spectral curves, and KP-type equations

An effective family of spectral curves appearing in Hitchin fibrations is determined. Using this family the moduli spaces of stable Higgs bundles on an algebraic curve are embedded into the Sato Grassmannian. We show that the Hitchin integrable system, the natural algebraically completely integrable Hamiltonian system defined on the Higgs moduli space, coincides with the KP equations. It is shown that the Serre duality on these moduli spaces corresponds to the formal adjoint of pseudo-differential operators acting on the Grassmannian. From this fact we then identify the Hitchin integrable system on the moduli space of Sp(2m)-Higgs bundles in terms of a reduction of the KP equations. We also show that the dual Abelian fibration (the SYZ mirror dual) to the Sp(2m)-Higgs moduli space is constructed by taking the symplectic quotient of a Lie algebra action on the moduli space of GL-Higgs bundles.

💡 Research Summary

The paper establishes a deep and explicit bridge between the Hitchin integrable system on the moduli space of stable Higgs bundles over a smooth projective curve and the infinite‑dimensional KP hierarchy. The authors begin by constructing an “effective family” of spectral curves associated with the Hitchin fibration. For a Higgs bundle ((E,\Phi)) on a curve (C) the characteristic equation (\det(\eta-\Phi)=0) defines a curve (\Sigma) inside the total space of the canonical bundle (K_C). By treating the coefficients of this polynomial as global parameters, the authors produce a universal family ({\Sigma_{\mathbf a}}) that varies holomorphically with the Hitchin base (\mathbf a). They then normalize each (\Sigma_{\mathbf a}) and attach a line bundle (L_{\mathbf a}) whose sections encode the Higgs field.

The second major step is to embed the Hitchin moduli space (M_{\mathrm{Higgs}}) into the Sato Grassmannian (\mathrm{Gr}(H)). The Grassmannian parametrises half‑infinite dimensional subspaces (W\subset H=\mathbb C((z))) and provides the geometric arena for the KP hierarchy via the Krichever map. The authors define a Krichever‑type map \