Abstract
In his famous paper (Witten E. Super-symmetry and Morse Theory. J Diff Geom 1982; 17:661–692), Witten used a twisted Laplacian, twisted by a Morse function, to develop his Morse theory as a super-symmetric quantum mechanics. Afterwards, in their studies on puits multiples en mecanique semi-classique, Helffer and Sjöstrand (Helffer B, Sjöstrand J. Puits Multiples en Mecanique Semi-classique IV, Etude du Complexe de Witten. Comm. Partial Differential Equations 1985; 10:245–340) made a detailed investigation of Witten's complexes. In this paper, we use a twisted Laplacian, twisted by a versal deformation of an isolated singularity (invariant under a finite unitary reflection group), to construct a duality between a pair of polynomial twisted de Rham cohomology groups associated with the isolated singularity. The construction is based on a twisted version of Hodge-Kodaira decomposition derived through a pseudo-differential calculus of Witten's twisted Laplacian. Discussion is also made on the real structure and super-symmetry, compatible with the duality constructed, of the twisted de Rham cohomology.
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