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Topological field theory and Morse theory. (English. Japanese original) Zbl 0897.57021
Sugaku Expo. 10, No. 1, 19-39 (1997); translation from Sugaku 46, No. 4, 289-307 (1994).
This article is a mixture of a survey, the author’s new ideas and a research program on topological field theories. The mathematics classification and the keywords show the wide range of concepts covered. We describe in few words the contents of the paper: §1 gives a simplified definition of a topological field theory. In §2 the Witten complex is constructed; this needs one Morse function. Using several Morse functions and maps of trees into the manifold \(M\) in §3 topological field theories are constructed that correspond to the cup, Massey and higher Massey products. Hence they determine – according to Sullivan – the rational homotopy type of \(M\). In §4 the Floer homology of Lagrangian intersections is introduced and furthermore a product construction analogous to §3. The associative structure of the products from §§3,4 is described in §5 by the notion of a \(A^\infty\)-category. That Morse theory is a special case of the Lagrangian intersection theory is shown in §6. In §7 the author explains what he believes to be a mathematical description of Witten’s open string theory on the cotangent bundle of a 3-manifold; namely counting the number of holomorphic maps from a Riemannian surface \(\Sigma\) with boundary to \(M\). This can be seen as a generalization of §4 from genus 0 to arbitrary genus of \(\Sigma\). According to Witten this open string theory is equivalent to the Chern-Simons gauge theory. The author defines an invariant using the holonomy of a flat vector bundle on \(M\), and conjectures that this coincides with the Chern-Simons gauge theory. In §§8,9 the author goes the way back from Lagrangian intersections to Morse theory and finds a higher genus Morse theory. He conjectures that the found invariant coincides with Chern-Simons perturbation theory. Some more details may be found in [K. Fukaya, Commun. Math. Phys. 181, No. 1, 37–90 (1996; Zbl 0893.58017)].

57R56 Topological quantum field theories (aspects of differential topology)
57R58 Floer homology
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
81T13 Yang-Mills and other gauge theories in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T45 Topological field theories in quantum mechanics
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces