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.


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