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2006.311: Dual forms on supermanifolds and Cartan calculus

2006.311: Theodore Voronov (2002) Dual forms on supermanifolds and Cartan calculus. Communications in Mathematical Physics, 228 (1). pp. 1-16. ISSN 1432-0916

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DOI: 10.1007/s002200200655

Abstract

We introduce and study the complex of "stable forms" on supermanifolds. Stable forms on a supermanifold M are represented by Lagrangians of "copaths" (formal systems of equations, which may or may not specify actual surfaces) on M 2 ÂD. Changes of D give rise to stability isomorphisms. The resulting (direct limit) {Cartan-de Rham} complex made of stable forms extends both in positive and negative degree. Its positive half is isomorphic to the complex of forms defined as Lagrangians of paths, studied earlier. Including the negative half is crucial, in particular, for homotopy invariance. For stable forms we introduce (non-obvious) analogs of exterior multiplication by covectors and contraction with vectors and find the anticommutation relations that they obey. Remarkably, the version of the Clifford algebra so obtained is based on the super anticommutators rather than the commutators and (before stabilization) it includes some central element †. An analog of Cartan's homotopy identity is proved, which also contains this "stability operator" †.

Item Type:Article
Subjects:MSC 2000 > 57 Manifolds and cell complexes
MIMS number:2006.311
Deposited By:Miss Louise Stait
Deposited On:16 August 2006

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