Klimek, Slawomir; Lesniewski, Andrzej Local rings of singularities and \(N=2\) supersymmetric quantum mechanics. (English) Zbl 0724.58066 Commun. Math. Phys. 136, No. 2, 327-344 (1991). Summary: We investigate the Kähler structure arising in \(n\)-component, \(N=2\) supersymmetric quantum mechanics. We define \(L^ 2\)-cohomology groups of a modified \({\bar \partial}\)-operator and relate them to the corresponding spaces of harmonic forms. We prove that the cohomology is concentrated in the middle dimension, and is isomorphic to the direct sum of the local rings of the singularities of the superpotential. In the physics language, this means that the number of ground states is equal to the absolute value of the index of the supercharge, and each ground state contains exactly \(n\) fermions. Cited in 1 ReviewCited in 10 Documents MSC: 58J90 Applications of PDEs on manifolds 81Q60 Supersymmetry and quantum mechanics 58A14 Hodge theory in global analysis 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F25 Classical real and complex (co)homology in algebraic geometry 32E10 Stein spaces 58A10 Differential forms in global analysis 81T60 Supersymmetric field theories in quantum mechanics 58J20 Index theory and related fixed-point theorems on manifolds Keywords:smooth cohomology; Stein manifold; Hodge-type theorem; vanishing theorem; index theorem; residues of meromorphic n-forms; supersymmetric quantum mechanics; \(L^ 2\)-cohomology; \({\bar \partial }\)-operator; spaces of harmonic forms PDFBibTeX XMLCite \textit{S. Klimek} and \textit{A. Lesniewski}, Commun. Math. Phys. 136, No. 2, 327--344 (1991; Zbl 0724.58066) Full Text: DOI References: [1] [AGV] Arnold, V. I., Gusein-Zade, S. M., Varchenko, A. 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