Action of diffeomorphisms
Formulation of the problem
Let
be the BV Hamiltonian generating the left shift by elements of
; if
is any
function of
, then:
The
,
and
are defined similarly. In particular:
With these notations, when
and
are two even elements of
,
(Even elements are generators of
and
, and also the generators of
and
multiplied
by a Grassmann odd parameter.)
The infinitesimal action of diffeomorphisms is generated by the following BV Hamiltonian
:
In this section we will construct
such that:
It is very easy to construct such
if we don’t care about the global symmetries
of
. (Something like
.) But we will construct a
invariant
under the supersymmetries of
,
i.e. invariant under the right shifts of
.
We believe that such an invariant construction has better chance of satisfying the equivariance
conditions of
Mikhailov:2016myt,
Mikhailov:2016rkp at the quantum level, because supersymmetries restrict quantum corrections.
In particular, the equivariance condition must require that the
correspond, in some sense,
to a primary operator.
Comment on gauge transformations
In this Section we discuss vector fields on the factorspace 
defined by Eq. (24).
They are the same as 
-invariant vector fields on
modulo -invariant vertical vector fields.
All the formulas here are modulo vertical -invariant vector fields.
Subspaces associated to a pair of pure spinors
(Notations of Sec
The projector;
this is a direct sum of two completely independent linear maps.)
For a pair
we decompose
:
where we must use a special representative of the cokernel:
Similarly, any
(assumed to be both TL and STL) can be decomposed:
Explicitly:
Construction of
The generating function
of the infinitesimal worldsheet diffeomorphisms (= vector fields)
, given by Eq. (
44), is BV-exact:
The coefficients
and
satisfy:
Some useful identities
Notice that we have
in denominators. At the same time:
associated to pure spinors. For 
and 
,
let 
denote the map:
denotes the component of
which belongs to
; the label “split” is because we could not invent any better notation.
