Open Quantum Systems as Regular Holonomic $\mathcal{D}$-Modules: The Mixed Hodge Structure of Spectral Singularities

TL;DR

This paper introduces a geometric framework for open quantum systems at spectral singularities, using mixed Hodge modules and $ ext{D}$-modules to rigorously analyze dissipative dynamics and spectral degeneracies.

Contribution

It establishes a novel connection between open quantum systems and mixed Hodge modules, providing a rigorous geometric and algebraic structure at spectral singularities.

Findings

Regular holonomic $ ext{D}$-modules describe spectral singularities.

Quantum Geometric Tensor regularized via cohomology classes.

Residue of connection on Brieskorn lattice captures singular components.

Abstract

The geometric description of open quantum systems via the Quantum Geometric Tensor (QGT) traditionally relies on the assumption that the physical states form a differentiable vector bundle over the parameter manifold. This framework becomes ill-posed at spectral singularities, such as Exceptional Points, where the eigen-bundle admits no local trivialization due to dimension reduction. In this work, we resolve this obstruction by demonstrating that the family of Liouvillian superoperators $\mathcal{L}(k)$ over a complex parameter manifold $X$ canonically defines a \textbf{regular holonomic $\mathcal{D}_X$-module} $\mathcal{M}$. By identifying the physical coherence order with the Hodge filtration and the decay rate hierarchy with the \textbf{Kashiwara filtration}, we show that the open quantum system underlies a \textbf{Mixed Hodge Module (MHM)} structure in the sense of Saito. This identification allows us to apply the \textbf{Grothendieck six-functor formalism} rigorously to dissipative dynamics. We prove that the divergence corresponds to a non-trivial cohomology class in $\text{Ext}^1_{\mathcal{D}_X}$, thereby regularizing the Quantum Geometric Tensor without ad-hoc cutoffs. Specifically, the ``singular component'' of the Complete QGT arises as the residue of the connection on the \textbf{Brieskorn lattice} associated with the vanishing cycles functor.