Hodge Theory of Novikov Type Landau--Ginzburg Models

Abstract

We establish a strong Hodge decomposition theorem for Novikov type Landau--Ginzburg models, that is, triples $(M,g,θ)$ consisting of a non-compact complete Kähler manifold $(M,g)$ and a non-trivial closed holomorphic one-form $θ$ on $M$.Under the well-tameness condition and under the assumption that $M$ is Stein, we prove that the space of $L^{2}$-forms on $M$ admits an orthogonal decomposition into the space of harmonic forms of the twisted Laplacian $Δ_θ$ and the images of the twisted Dolbeault operators $\bar\partial+θ\wedge$ and $\bar\partial^{\dagger}+ι_{\barθ^{\sharp}}$; moreover, the harmonic space is concentrated in degree $n=\dim_{\C}M$ and is isomorphic to the Milnor rings of $θ$. As applications, we endow the complement of a hyperplane arrangement with a complete cylindrical Kähler metric, and we show that the differential of a master function with nonresonant weights is well-tamed with respect to this metric. After passing to a log resolution in the non-normal-crossing case; we then apply the results to the Yang--Yang functional of the Gaudin model, whose critical points are the Bethe roots: the top-degree harmonic space is described by the corresponding Milnor rings, and the number of Bethe roots is given by the Varchenko--Orlik--Terao lattice formula.