BV awesomeness

Remembering the real good way to think about manifolds and more generally schemes is as a topological space equipped with its sheaf of smooth functions. What if this sheaf locally gave the smooth functions on n dimensional ordinary space. That would give a plain old manifold.

What if it was the tensor product of smooth functions on R^p with the exterior algebra on q variables? Then we would be in superspace with p even directions and q odd directions.

Let's construct lots of examples of supermanifolds. Start with a vector bundle E over M. Now consider the supermanifold whose sheaf of smooth functions is given by sections of the exterior algebra of the dual vector bundle. If E is the tangent bundle to M, this would be the sheaf of differential forms.

Batchelor's theorem says that every supermanifold is isomorphic to the victim of the above construction for some vector bundle. But the above functor from vector bundles to supermanifolds is not fully faithful, so there actually is benefit to be gained by looking at the category of supermanifolds rather than living in vector bundle-land.

For a (n,n) supermanifold construct an odd symplectic structure by 

This is clearly odd, it has one odd variable and one even variable.

You can go ahead and repeat lots of symplectic geometry. You have from this two-form a Poisson bracket, again, you can also get a Laplacian if you provide me with a density function too.

This Laplacian is odd and squares to 0, so we can take its cohomology. This is the beginning for BV-BRST. We can properly define the integrals we need to.

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