coordinates. (An odd symplectic form should not be confused with a Grassmann-even symplectic form on a supermanifold. In contrast, the Darboux version of an even symplectic form is where are even coordinates, odd coordinates and are either +1 or −1.) Antibracket Given an odd symplectic 2-form ω one may define a Poisson bracket known as the antibracket of any two functions F and G on a supermanifold by Here and are the right and left derivatives respectively and z are the coordinates of the supermanifold. Equipped with this bracket, the algebra of functions on a supermanifold becomes an antibracket algebra. A coordinate transformation that preserves the antibracket is called a P-transformation. If the Berezinian of a P-transformation is equal to one then it is called an SP-transformation. P and SP-manifolds Using the Darboux theorem for odd symplectic forms one can show that P-manifolds are constructed from open sets of superspaces glued together by P-transformations. A manifold is said to be an SP-manifold if these transition functions can be chosen to be SP-transformations. Equivalently one may define an SP-manifold as a supermanifold with a nondegenerate odd 2-form ω and a density function ρ such that on each coordinate patch there exist Darboux coordinates in which ρ is identically equal to one. Laplacian One may define a Laplacian operator Δ on an SP-manifold as the operator which takes a function H to one half of the divergence of the corresponding Hamiltonian vector field. Explicitly one defines . In Darboux coordinates this definition reduces to where xa and θa are even and odd coordinates such that . The Laplacian is odd and nilpotent . One may define the cohomology of functions H with respect to the Laplacian. In Geometry of Batalin-Vilkovisky quantization, Albert Schwarz has proven that the integral of a function H over a Lagrangian submanifold L depends only on the cohomology class of H and on the homology class of the body of L in the body of the ambient supermanifold. SUSY A pre-SUSY-structure on a supermanifold of dimension (n,m) is an odd m-dimensional distribution . With such a distribution one associates its Frobenius tensor (since P is odd, the skew-symmetric Frobenius tensor is a symmetric operation). If this tensor is non-degenerate, e.g. lies in an open orbit of , M is called a SUSY-manifold. SUSY-structure in dimension (1, k) is the same as odd contact structure. See also Superspace Supersymmetry