Integration in two steps

In our approach we integrate some closed form over a cycle in the moduli space of Lagrangian submanifolds modulo gauge symmetries. The form itself is defined in terms of the integration of over the Lagrangian submanifold (path integral). We therefore have a double integral: first the path integral, and then a finite-dimensional integral over a cycle in the moduli space.

However, we suspect that there is actually no fundamental distinction between these two steps. In principle, one can combine them into one integration.

It is often convenient to pull one or more integration out from the integration of into the path integral or vice versa. This is “picture changing”.

In the case of bosonic string we usually choose a Lagrangian submanifold corresponding to a fixed complex strucuture (or metric, depending on the flavour of the formalism). In this case the path integration contains the integration over the antifield to metric/complex structure, which is identified with the -ghost. However, in principle we could also include into the path integral some partial integration over the moduli space of complex structures, and then integrate over the rest later.

It turns out that in the BV formalism, this (at least in some cases) corresponds to changing each Lagrangian submanifolds in the family into a different Lagrangian submanifold. This is a topologically nontrivial change, essentially a change in polarization.