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

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.