2025/12/22 by Prasoon Saurabh, Saurabh, Prasoon
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Quantum chaos and dynamical systems #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2512.19487
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 L(k) over a complex parameter manifold X canonically defines a \textbfregular holonomic DX-module M. By identifying the physical coherence order with the Hodge filtration and the decay rate hierarchy with the Kashiwara filtration, we show that the open quantum system underlies a Mixed Hodge Module (MHM) structure in the sense of Saito. This identification allows us to apply the Grothendieck six-functor formalism rigorously to dissipative dynamics. We prove that the divergence corresponds to a non-trivial cohomology class in Ext1DX, 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 Brieskorn lattice associated with the vanishing cycles functor.