Relation to the approach of Sen
In approach to bosonic string, we integrate over
worldsheet complex structures.The approach of Sen in OffShellAmplitudes uses patches and integration over transition functions.
We will now outline the derivation of Sen’s approach from our approach. It turns out that the derivation is not completely straightforward.
Locally all complex structures are the same. This means that, on a small enought patch, we can
undo the variation of the complex structure by applying a diffeomorphism. Generally speaking, this
diffeomorphism does not vanish on the boundary of the patch, but becomes a conformal
transformation (this is under the assumption that the variation of the complex structure is only nonzero
in a compact region inside the patch, i.e. vanishes near the boundary of the patch).
This means that we have a map:
Here we must consider the variations of the transition functions modulo those which can be extended to conformal transformation inside the patch.
In other words:
one can always undo the variation of the metric by applying a diffeomorphism, but at the
price of changing the transition functions between patches.
We, however, face a subtlety: the map (10) is only a surjection.
This raizes a question:
can we lift Sen’s integration cycles to closed integration cycles in the space of metrics?
This requires factoring out large diffeomorphisms.