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A tutorial on Quantum Cohomology

Let (M, f,G) be a manifold, a function and a Riemann metric on the manifold. Topologists would use these data in order to analyze the manifold by means of Morse theory, that is by studying the dynamical system ẋ = ±∇f . Many recent applications of physics to topology are based on another point of view suggested in E. Witten’s paper Supersymmetry and Morse theory J. Diff. Geom. (1982). Given the data (M, f,G), physicists introduce some super-lagrangian whose bosonic part reads