The tangent vector to at a point corresponds to an infinitesimal deformation of . As in Classical Mechanics, such deformations are in one-to-one correspondence with functions on . Since is odd, even deformations correspond to odd functions and vice versa. In BV literature is called “infinitesimal gauge fermion”.
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The right way to deal with this problem is to consider Lagrangian submanifolds with marked points, and require the vanishing of at the marked point.
Another possibility is to use ghost number symmetry, which is a symmetry typically present in string theory. If we require Lagrangian submanifolds to be -invariant, then must have ghost number and cannot be constant.