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Circular symmetry and stationary-phase approximation. (English) Zbl 0578.58039
Colloq. Honneur L. Schwartz, Éc. Polytech. 1983, Vol. 1, Astérisque 131, 43-59 (1985).
[For the entire collection see Zbl 0566.00010.]
The Duistermaat-Heckman exact integration formula \[ \int_{M}e^{-tH} \omega^ n/n!=\int_{X}(e^{-tH(X)} e^{\omega}/\prod^{k}_{j=\quad 1}(tm_ j-i\alpha_ j)) \] for computation of oscillatory integrals is explained in the first part of the lecture. Then, following E. Witten [J. Differ. Geom. 17, 661- 692 (1982; Zbl 0499.53056)], an application of this formula to the infinite-dimensional loop manifold \(M=Map(S',X)\) is heuristically discussed. It turns out that the Duistermaat-Heckmann formula for the infinite-dimensional loop space is reduced to the index theorem for the Dirac operator.
Reviewer: V.Yumaguzhin

58J20 Index theory and related fixed-point theorems on manifolds