Familty of Lagrangian submanifolds
1 Families with the structure of vector bundle
1.1 A subset of 
with additional structure
Consider a subset
invariant under the group of diffeomorphisms
.
Suppose that:
Every Lagrangian submanifold
comes with a subspace of functions on it:
(in our application this subspace will be finite-dimensional)
There is an action of
on this subspace, more precisely for each
we are given
a linear map
:
such that
1.2 Continuation problem
Question: we want to continue every function from
from the Lagrangian submanifold
to the full BV phase space:
so that the continuation satisfies the following two properties:
For every fixed
, the extended functions are all in involution w.r.to the odd Poisson bracket on
:
(but for two different
and
in
, the functions extended from
are
not
required to be in involution from the functions extended from
)
The extension commutes with the action of diffeomorphisms:
— is this possible?
Suppose that the answer is “yes”. Then we can enlarge to
a larger subset of which has a structure of a vector bundle over , as
we will now describe.
1.3 A vector bundle over
We define 
as the total space of the following vector bundle over : the fiber
over 
is the orbit of under the Hamiltonian vector fields
generated by all the Hamiltonians from 
.
Eq. (
26) guarantees that:
is diffeomorphism-invariant
2 Application to RNS formalism
2.1 Base family
Let
be a set of
all split worldsheets,
i.e. worldsheets
with
split
superconformal structure:
2.2 Subspaces
In this case, we will choose
as follows:
generated by the integrals:
with
running over the representatives of the Dolbeault realization of
.
2.3 Are there solutions to continuation problem?
We do not have an explicit solution of the continuation problem. Actually, we do not even have a proof
that the solution exists.
2.4 Split family
But if the continuation problem has solution, then we can construct a
split 
-dimensional
cycle in the following way:
pick any
- dimensional cycle in the moduli space of bosonic Riemann surfaces
and restrict
to this cycle
(in our application this subspace will be finite-dimensional)
such that
(but for two different
being in involution with each other, we could have required that the space
would not be a vector bundle, just some nonlinear bundle.
have 
zero modes (see our notations).
To follow the standard notations, we denote 
the leading component of 
