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Semidensities on odd symplectic supermanifolds. (English) Zbl 1057.58002

Semidensities on a supermanifold are sections of a square root of the Berezinian sheaf. The paper considers semidensities on a supermanifold endowed with an odd symplectic structure and establishes a relationship between semidensities and differential forms on Lagrangian surfaces (assuming that the underlying manifold is oriented). A new \(\Delta\)-operator acting on semidensities is introduced (\(\Delta\)-operators were first introduced by Batalin and Vilkovisky). These results are applied to give an explicit description of the Batalin-Vilkovisky formalism. Finally \((1,1)\)-codimensional surfaces in the supermanifold are studied and certain integral invariants for those surfaces are constructed.

MSC:

58A50 Supermanifolds and graded manifolds
58C50 Analysis on supermanifolds or graded manifolds
53D05 Symplectic manifolds (general theory)
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