Witten Complex Research Articles

Witten Complex of Transitive Digraph and Its Convergence

Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.

Cellular homology of real flag manifolds

Let FΘ=G∕PΘ be a generalized flag manifold, where G is a real non-compact semi-simple Lie group and PΘ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow FΘ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in Rn and use them to compute the boundary operator ∂ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of ∂ are 0 or ±2 (so that Z2-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.

Morse–Conley–Floer homology

The gradient flow of a Morse function on a smooth closed manifold generates, under suitable transversality assumptions, the Morse–Smale–Witten complex. The associated Morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical Morse relations. In this paper we define Morse–Conley–Floer homology, which is an analogous homology theory for isolated invariant sets of smooth, not necessarily gradient-like, flows. We prove invariance properties of the Morse–Conley–Floer homology, and show how it gives rise to the Morse–Conley relations.

Orbifold Morse–Smale–Witten complexes

Given a Morse–Smale function on an effective orientable orbifold, we construct its Morse–Smale–Witten complex. We show that critical points of a certain type have to be discarded to build a complex properly, and that gradient flows should be counted with suitable weights. Its homology is proven to be isomorphic to the singular homology of the quotient space under the self-indexing assumption. For a global quotient orbifold [M/G], such a complex can be understood as the G-invariant part of the Morse complex of M, where the G-action on generators of the Morse complex has to be defined including orientation spaces of unstable manifolds at the critical points. Alternatively in the case of global quotients, we introduce the notion of weak group actions on Morse–Smale–Witten complexes for non-invariant Morse–Smale functions on M, which give rise to genuine group actions on the level of homology.

The Witten complex for singular spaces of dimension two with cone‐like singularities

AbstractThe Witten deformation is a method proposed by Witten which, given a function \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$f:M \longrightarrow \mathbb {R}$\end{document} on a smooth compact manifold M, allows to prove the Morse inequalities. Witten’s proof of the Morse inequalities is analytical and can thus be applied to situations where the Thom‐Smale method is not accessible. In these notes we generalise the Witten deformation to certain singular Riemannian manifolds X which are metric models for singular algebraic curves, and functions on X which we call admissible Morse functions. They are particular examples of stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

This article follows the previous works \cite{HKN} by Helffer-Klein-Nier and \cite{HelNi1} by Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of $\Delta_{f,h}^{(0)}=-h^{2}\Delta +\left|\nabla f(x)\right|^{2}-h\Delta f(x)\;,$ are considered as the small parameter $h>0$ goes to $0$. The function $f$ is assumed to be a Morse function on some bounded domain $\Omega$ with boundary $\partial\Omega$. Neumann type boundary conditions are considered. With these boundary conditions, some simplifications possible in the Dirichlet problem studied in \cite{HelNi1} are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is carried out.

The Witten complex for algebraic curves with cone-like singularities

The Witten deformation is an analytical method proposed by Witten which, given a function f : M → R on a smooth compact Riemannian manifold M, leads to a proof of the Morse inequalities. In this Note we generalise the Witten deformation to singular complex algebraic curves X with cone-like singularities, and functions on X which we call admissible Morse functions. They are particular examples of stratified Morse functions in the sense of the theory developed by Goresky/MacPherson. To cite this article: U. Ludwig, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary

This article is a continuation of previous works by Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein and Helffer-Klein-Nier. It is concerned with the analysis of the exponentially small eigenvalues of a semiclassical Witten Laplacian. We consider here the case of riemanian manifolds with boundary with a Dirichlet realization of the Witten Laplacian. A modified version of this preprint has been published in Memoires de la SMF vol. 105, (2006)

The Morse–Witten complex via dynamical systems

Given a smooth closed manifold M, the Morse–Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in Weber [Der Morse–Witten Komplex, Diploma Thesis, TU Berlin, 1993] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman–Hartman theorem and the λ -lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.

Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach.

Nous calculons de facon precise les premieres valeurs propres, exponentiellement petites, du Laplacien de Witten sur les 0-formes. Ces quantites sont liees au taux de retour a l\'equilibre pour des processus stochastiques reversibles.Lapproche proposee ici repose sur la structure du complexe de Witten et donne des resultats plus precis et generaux que les methodes probabilistes.

Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach

Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach

Supersymmetry, Witten Complex and Asymptotics for Directional Lyapunov Exponents in Zd

We study the long distance asymptotic for random walks in random potentials. We use the analytical formulation. We relate the problem to Witten Laplacians via using supersymmetry. We obtain the asymptotic for the directional Lyapunov exponents.

Commuting Hamiltonians from Seiberg–Witten Θ-functions

Elementary MAPLE calculations are used to support the claim of hep-th/9906240 that the ratios of theta-functions, associated with the Seiberg–Witten complex curves, provide Poisson-commuting Hamiltonians which describe the dual of the original Seiberg–Witten integrable system.

COVER OF THE BROWNIAN BRIDGE AND STOCHASTIC SYMPLECTIC ACTION

We define the universal cover of the Brownian bridge of a symplectic manifold. This allows us to define a non-trivial functional over it called the stochastic symplectic action, and to define local Sobolev spaces over the universal cover, such that the symplectic action belong to them. This is done in the purpose of an analytical Morse theory in the sense of Witten over the loop space associated to this symplectic action. In this purpose, a stochastic Witten complex is constructed over the universal cover of the loop space, and modulo some weights, it is shown that its cohomology is equal to the stochastic cohomology of the universal cover.

Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in 𝐙 d

Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>𝐙</mml:mi> <mml:mi>d</mml:mi> </mml:msup></mml:math>

Stochastic Wess-Zumino-Witten model over a symplectic manifold

Over the path space of a symplectic manifold with end points in two Lagrangian submanifolds, we define a measure and a stochastic symplectic action in the simply connected case. We define a regularized Wess-Zumino-Witten Laplacian over the forms of finite degree over the path space. We perform a short time asymptotic near the critical points and find a limit Brownian harmonic oscillator: we can diagonalize it explicitly, and find the limit ground state of the Laplacian. We define a stochastic Witten complex, and its algebraic counterpart at the level of Chen forms.

Instanton Effects and Witten Complex in Supersymmetric Quantum Mechanics on SO(4)

We examine supersymmetric quantum mechanics on $SO(4)$ to realize Witten's idea. We find instanton solutions connecting approximate vacuums. We calculate Hessian matrices for these solutions to determine true vacuums. Our result is in agreement with de Rham cohomology of $SO(4)$. We also give a criterion for cancellation of instanton effects for a pair of instanton paths.

Witten complex and supersymmetric quantum mechanics on SO(3)

A supersymmetric quantum mechanics on SO(3) is examined. It was found that instanton solutions and the normal coordinate expansions around them allow the explicit path integration. The Witten complex is obtained by calculating the matrix elements of the generalized exterior derivative.

Witten's complex and infinite-dimensional Morse theory

We investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space. The results are applied to an infinite dimensional problem involving the symplectίc action function.

The Witten complex and the degenerate Morse inequalities

On utilise le complexe introduit par Witten pour demontrer des inegalites de Morse dans le cas ou les points critiques de la fonction de Morse sont isoles, et aussi les inegalites de Morse degenerees de Bott