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Morse inequalities after E. Witten. (Inégalités de Morse d’après E. Witten.) (French) Zbl 0744.58008
Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 1983-1984, No.I, 32 p. (1984).
This paper is devoted to the study of a part of the article by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)].
The question is a physical interpretation of the Morse inequalities as a quasi-classical limit of the tunnel effect under presence of supersymmetries.
The plan of the article follows the plan of Witten’s one. It is as follows:
The Riemannian calculus on a differentiable manifold \(M\).
A perturbation of the Laplacian with help of a Morse function \(h\in C^ \infty(M,\mathbb{R})\).
Asymptotical analysis of the perturbed Laplacian and asymptotical growth of eigen-functions.
The weak Morse inequality \(B_ p(M)<M_ p(h)\) where \(B_ p(M)\) is the \(p\)-th Betti number of \(M\), and \(M_ p(h)\) is the number of critical points of \(h\) with the Morse indices being equal to \(p\).
Introduction to the calculus of instantons in \(\mathbb{R}^ n\).
The distinguished cohomology model associated with the strong Morse inequalities.
The calculation of matrices of this distinguished model (instantons).
MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
81T60 Supersymmetric field theories in quantum mechanics
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
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