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, Poincaré 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 J forms for local Lefschetz numbers of Dirac operators, with specialized formulas for twisted de Rham, Dolbeault and spin-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 L2 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 L2 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 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.
Articles on some extensions are in preparation. A few related figures with a little exposition can be found here. Some slides from a talk on part of this work can be found here.
Holomorphic Witten instanton complexes on stratified pseudomanifolds with Kahler wedge metrics.
We construct Witten instanton complexes for Kähler Hamiltonian Morse functions on stratified pseudomanifolds with wedge Kähler metrics satisfying a local conformally totally geodesic condition. We use this to extend Witten's holomorphic Morse inequalities for the L2 cohomology of Dolbeault complexes, deriving versions for Poincaré Hodge polynomials, the spin Dirac and signature complexes for which we prove rigidity results, in particular establishing the rigidity of L2 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.
Here's an interesting video where Bott explains the key ideas of the proof (in the smooth, de Rham case), and discusses naming the complexes at 00:08:40. Here's a longer version of the talk, where at 58:30, there's more questions on naming some of the related complexes. These are notes accompanying his talk.
Here are some notes on some lectures by Witten explaining some relations to physics in a simple manner. Here are some more accessible notes by David Tong. Here are some slides from a talk I gave on this work.
Witten instanton complex and Morse-Bott inequalities on stratified pseudomanifolds.
Joint with Hadrian Quan and Xinran Yu.
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 Poincaré 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.
This work is part of a program to extend Witten's instanton complexes, Morse inequalities and applications to stratified spaces with various types of critical point sets and for all admissible self-adjoint extensions of de Rham (and Signature) operators. Very much related to the earlier work, which showcase concrete and potential applications. More work coming up.
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 (Kähler Hamiltonian) Morse functions and singular metrics arising naturally in algebraic geometry and moduli problems.
We use the 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.
This was an exploration of some technical analytical results, relevant to the study of Dolbeault complexes and their equivariant index/Morse theory on singular spaces with various degenerate metrics, and the relevant notions of Hamiltonians for adapted symplectic structures. Local Lefschetz numbers and local (holomorphic) Morse polynomials are defined, extending the definitions from the conic case. This keeps the focus on the analysis of operators which includes topological twists and Witten deformation. The global versions coming out of this will be written elsewhere. Applications similar to those in the conic (wedge) would be interesting for these metrics as well.
Perspectives from semi-classical analysis and supersymmetry, as well as examples showing connections to geometric quantization are included.
Logarithmic wave decay for short range wavespeed perturbations with radial regularity
Joint with Katrina Morgan, Jacob Shapiro and Mengxuan Yang.
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 Holder continuity of the weighted resolvent for real frequencies λ, modulo a logarithmic remainder in dimension two as λ → 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.