Supersymmetric harmonic oscillators on singular geometries
Gayana Jayasinghe
TL;DR
The paper develops a rigorous framework for equivariant localization on singular spaces by extending twisted de Rham and Dolbeault complexes to singular warped product (swp) geometries and formulating $\,\mathcal{N}=2\,$ SUSY quantum mechanics on Hilbert complexes. It introduces radial Kähler Hamiltonian Morse data, adapted geometry, and a generalized Cheeger ansatz that reduces high-dimensional spectral problems to one-dimensional singular Sturm–Liouville problems, enabling the construction of renormalized Lefschetz/Morse trace formulas. A key contribution is proving the existence of orthonormal eigensection bases for Witten-deformed Laplacians in the discrete-type setting and deriving renormalized McKean–Singer identities, which unify local and global invariants even in the presence of singularities. The work covers both de Rham and Dolbeault settings, including boundary conditions, Serre duality, and Kähler Hamiltonian actions, and provides illustrative examples (rotation, spinning sphere, cusp) to demonstrate the tractability and breadth of the framework. Overall, it yields a robust method for extracting global invariants from local geometric data on singular spaces, with potential applications in geometric analysis, index theory, and mathematical physics.
Abstract
Equivariant localization expresses global invariants in terms of local invariants, and many of them appearing in equivariant index theory, (holomorphic) Morse theory, geometric quantization and supersymmetric localization can be characterized as renormalized supertraces over cohomology groups of Hilbert complexes associated to local model geometries. This paper extends such local invariants, introducing and studying twisted de Rham and Dolbeault complexes (including Witten deformed versions) on singular spaces equipped with generalized radial (Kähler Hamiltonian) Morse functions and singular metrics arising naturally in algebraic geometry and moduli problems. We use the $\mathcal{N}=2$ supersymmetry and nilpotency properties of these complexes to extend an ansatz of Cheeger for the eigensections of the associated Laplace/Schrödinger type operators, reducing the problem to the study of Sturm-Liouville operators and one dimensional Schrödinger operators corresponding to different choices of domains, including those with del-bar Neumann boundary conditions for Dolbeault complexes. We define renormalized Lefschetz numbers and Morse polynomials generalizing those established in the smooth and conic settings where they have been used to compute many invariants of interest in physics and mathematics. We study structures on links of topological cones with singular Kähler metrics, which we use to describe associated analytic invariants including local cohomology groups. The techniques and results collected here are broadly applicable in the study of global analysis on singular spaces, including proofs of localization theorems with numerous applications.