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Local rings of singularities and \(N=2\) supersymmetric quantum mechanics. (English) Zbl 0724.58066

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.

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
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