arXiv subjects

Gayana Jayasinghe

Publications and source records attributed to Gayana Jayasinghe.

6 recordsLinked to original sources

arXiv2026

Spectral asymmetry, supersymmetry and the equivariant Riemann-Roch defect

We investigate the relationship between two interpretations of equivariant Riemann-Roch defects of complex spaces with conic singularities; as (i) equivariant $\eta_{T}$ and $\xi_{T}$ invariants, and as (ii) supertraces over local cohomology groups. This leads to a novel threefold partitioning of the $L^{2}$-spinor space on the link and a corresponding splitting of $\xi_{T}$. Two partitions correspond to cohomological contributions coming from $\bar\partial$-Neumann and $\bar\partial$-Dirichlet operators on the cone, while the third partition makes no contribution to the equivariant index defect, which we show is due to supersymmetric cancellations on the cone that we call lifted supersymmetry. We use this to define complex equivariant $\xi_T$ and $\eta_T$ invariants, which are equivalent to the usual invariants but are easier to compute. We highlight connections to related algebraic and analytic descriptions of Riemann-Roch defects in the literature, both at the level of numbers and their categorifications, and explore connections to existing notions of supersymmetric cancellations in physics and mathematics.

math.DG↗

arXiv2025

Logarithmic wave decay for short range wavespeed perturbations with radial regularity

We establish logarithmic local energy decay for wave equations with a varying wavespeed in dimensions two and higher, where the wavespeed is assumed to be a short range perturbation of unity with mild radial regularity. The key ingredient is H\"older continuity of the weighted resolvent for real frequencies $\lambda$, modulo a logarithmic remainder in dimension two as $\lambda \to 0$. Our approach relies on a study of the resolvent in two distinct frequency regimes. In the low frequency regime, we derive an expansion for the resolvent using a Neumann series and properties of the free resolvent. For frequencies away from zero, we establish a uniform resolvent estimate by way of a Carleman estimate.

math.AP↗

arXiv2025

Supersymmetric harmonic oscillators on singular geometries

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 (K\"ahler 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/Schr\"odinger type operators, reducing the problem to the study of Sturm-Liouville operators and one dimensional Schr\"odinger 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 K\"ahler 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.

math.DG↗

arXiv2024

Witten instanton complex and Morse-Bott inequalities on stratified pseudomanifolds

In this paper we construct Witten instanton complexes on stratified pseudomanifolds with wedge metrics, for all choices of mezzo-perversities which classify the self-adjoint extensions of the Hodge Dirac operator. In this singular setting we introduce a generalization of the Morse-Bott condition and in so doing can consider a class of functions with certain non-isolated critical point sets which arise naturally in many examples. This construction of the instanton complex extends the Morse polynomial to this setting from which we prove the corresponding Morse inequalities. This work proceeds by constructing Hilbert complexes and normal cohomology complexes, including those corresponding to the Witten deformed complexes for such critical point sets and all mezzo-perversities; these in turn are used to express local Morse polynomials as polynomial trace formulas over their cohomology groups. Under a technical assumption of `flatness' on our Morse-Bott functions we then construct the instanton complex by extending the local harmonic forms to global quasimodes. We also study the Poincar\'e dual complexes and in the case of self-dual complexes extract refined Morse inequalities generalizing those in the smooth setting. We end with a guide for computing local cohomology groups and Morse polynomials.

math.DG↗

arXiv2024

Holomorphic Witten instanton complexes on stratified pseudomanifolds with K\"ahler wedge metrics

We construct Witten instanton complexes for K\"ahler Hamiltonian Morse functions on stratified pseudomanifolds with wedge K\"ahler metrics satisfying a local conformally totally geodesic condition. We use this to extend Witten's holomorphic Morse inequalities for the $L^2$ cohomology of Dolbeault complexes, deriving versions for Poincar\'e Hodge polynomials, the spin Dirac and signature complexes for which we prove rigidity results, in particular establishing the rigidity of $L^2$ de Rham cohomology for these circle actions. We study formulas for Rarita Schwinger operators, generalize formulas studied by Witten and Gibbons-Hawking for the equivariant signature and extend formulas used to compute NUT charges of gravitational instantons. We discuss conjectural inequalities extending known Lefschetz-Riemann-Roch formulas for other cohomology theories including those of Baum-Fulton-Quart. This article contains the first extension of Witten's holomorphic Morse inequalities and instanton complexes to singular spaces.

math.DG↗

arXiv2023

An analytic approach to Lefschetz and Morse theory on stratified pseudomanifolds

We develop an analytic framework for Lefschetz fixed point theory and Morse theory for Hilbert complexes on stratified pseudomanifolds. We develop formulas for both global and local Lefschetz numbers and Morse, Poincar\'e polynomials as (polynomial) supertraces over cohomology groups of Hilbert complexes, developing techniques for relating local and global quantities using heat kernel and Witten deformation based methods. We focus on the case where the metric is wedge and the Hilbert complex is associated to a Dirac-type operator and satisfies the Witt condition, constructing Lefschetz versions of Bismut-Cheeger $\mathcal{J}$ forms for local Lefschetz numbers of Dirac operators, with specialized formulas for twisted de Rham, Dolbeault and spin$^{\mathbb{C}}$ Dirac complexes as supertraces of geometric endomorphisms on cohomology groups of local Hilbert complexes. We construct geometric endomorphisms to define de Rham Lefschetz numbers for some self-maps for which the pullback does not induce a bounded operator on $L^2$ forms. A de Rham Witten instanton complex is constructed for Witt spaces with stratified Morse functions, proving Morse inequalities related to other results in the literature including Goresky and MacPherson's in intersection cohomology. We also prove a Lefschetz-Morse inequality for geometric endomorphisms on the instanton complex that is new even on smooth manifolds. We derive $L^2$ Lefschetz-Riemann-Roch formulas, which we compare and contrast with algebraic versions of Baum-Fulton-Quart. In the complex setting, we derive Lefschetz formulas for spin Dirac complexes and Hirzebruch $\chi_y$ genera which we relate to signature, self-dual and anti-self-dual Lefschetz numbers, studying their properties and applications including instanton counting. We compute these invariants in various examples with different features, comparing with versions in other cohomology theories.

math.DG↗