Hodge Theory of Novikov Type Landau--Ginzburg Models

We establish a strong Hodge decomposition theorem for Novikov type Landau–Ginzburg models, that is, triples $(M,g,\theta )$ consisting of a non-compact complete Kähler manifold $(M,g)$ and a non-trivial closed holomorphic one-form $\theta$ 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 ${\mathrm{\Delta }}_{\theta }$ and the images of the twisted Dolbeault operators $\bar{\mathrm{\partial }}+\theta \wedge$ and ${\bar{\mathrm{\partial }}}^{†}+{\iota }_{{\bar{\theta }}^{\mathrm{♯}}}$; moreover, the harmonic space is concentrated in degree $n={\dim }_{\text{C}}M$ and is isomorphic to the Milnor rings of $\theta$. 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.