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 L2-forms on M admits an orthogonal decomposition into the space of harmonic forms of the twisted Laplacian Δθ and the images of the twisted Dolbeault operators ∂̅+θ∧ and ∂̅†+ιθ̅♯; moreover, the harmonic space is concentrated in degree n=dimCM 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.
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BibTeX
@misc{fan2026hodge,
title = {{Hodge Theory of Novikov Type Landau--Ginzburg Models}},
author = {Huijun Fan and Jieran Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27517},
eprint = {2610.27517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27517},
}
RIS
TY - UNPB
TI - Hodge Theory of Novikov Type Landau--Ginzburg Models
AU - Huijun Fan
AU - Jieran Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27517
UR - https://arxiv.org/abs/2610.27517
AB - 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 ∂̅+θ∧ and ∂̅^†+ι_θ̅^♯; moreover, the harmonic space is concentrated in degree n=dim_CM 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.
ER -