Regularization
The “minimalistic action” (41) cannot be regularized in a way that would preserve the
symmetries of 
; it is impossible to choose a 
-invariant
Lagrangian submanifold so that the restriction of the Master Action of Eq. (41) to it be
non-degenerate. Let us therefore return to the original action of Eqs. (25), (40), but in a way
preserving the worldsheet diffeomorphisms. The construction is somewhat similar to the
description of the topological A-model in Alexandrov:1995kv.
Adding more fields
Add a pair of bosonic 1-form fields
and
,
taking values in
and
, respectively, and their antifields
and
, also 1-forms:
(where
is the differential in the field space, not on the worldsheet!). In other words,
for any “test 1-forms”
and
:
We define the BV Master Action as follows:
and the BV Hamiltonian for the action of diffeomorphisms as follows:
where
is the Lie derivative.
The expression 
defines a symplectic structure on the space of 1-forms
with values in 
. The expression 
is the
corresponding Poisson bivector.
The Lie derivative preserves this (even) symplectic structure, and 
is
the corresponding Hamiltonian.
A canonical transformation
Let us do the canonical transformation by a flux of the following odd Hamiltonian:
This is the Hamiltonian of
in the same sense as
is the
Hamiltonian of
; we again use the same procedure of passing from Eq. (
4) to Eq. (
17),
actually in reverse.
The effect of the flux of
on the BV Master Action
of Eq. (
47) is:
Notice that the terms of the form
cancelled. This is automatic, because such terms
would contradict the Master Equation (the bracket
would have nothing to
cancel against).
The purpose of this canonical transformation was, essentially, to introduce the compensator
term 
into the action of 
on 
, cp. Eq. (37). We will discuss this in a more
general context in Section Gluing charts.
We are now ready to construct the Lagrangian submanifold.
Constraint surface and its conormal bundle
The configuration space
of this new theory is parametrized by
,
,
and
. Let us consider a subspace
defined by the constraints:
Consider the
odd conormal bundle
of
in the BV phase space
.
As any conormal bundle, this is a Lagrangian submanifold.
The restriction of
on this Lagrangian submanifold is still degenerate. But let us deform
it by the following generating function:
The restriction of
to this deformed Lagrangian submanifold is equal to:
where
,
,
Notice that the terms:
vanish on
. Indeed, the vector field:
is tangent to the constraint surface (
51); the conormal bundle, by definition, consists of
those one-forms which vanish on such vectors.
The term
computes the contrubution to the action from the fiber
. The coordinates of the fiber enter without derivatives, and decouple.
We therefore return to the original action of Eq. (25).
But now we understand how the worldsheet diffeomorphisms act, at the level of the BV phase space.
Gluing charts
In our construction we used a lift of to (Section Gauge fixing 
).
This is only possible locally. Therefore, we have to explain how to glue together overlapping
patches. This is a particular case of a general construction, which we will now describe.
The idea is to build a theory which is
locally (on every patch of
)
a direct product of two theories
and
:
but transition functions between overlapping patches mix
and
.
Consider the following data, consisting of two parts. The first part is a Lie group 
and a principal -bundle 
with base 
. Suppose that comes with a nilpotent vector
field 
and a
-invariant action 
. Then 
satisfies
the Master Equation on the BV phase space 
.
The second part of the data is a symplectic vector space 
which is a representation of .
This means that is equipped with an even -invariant symplectic form .
Let us cover
with charts
and trivialize
over each chart:
At the intersection
we identify
with
if
All this comes from
. We will now construct a new odd symplectic manifold, which is
locally
, with some transition functions, which we will now describe.
Technical assumption: in this Section we assume that all are bosons, and
that is a “classical” ( i.e. not super) Lie group. This is enough for our
considerations.
Transition functions
Let
be the Lie algebra of
. For each
consider the following BV Hamiltonian:
Here
is the representation of the Lie algebra corresponding to the representation
of the
group, and
is the symplectic form of
. Eq. (
54) defines
as the
Hamiltonian of the infinitesimal action of
on
,
i.e. the “usual” (even) moment map.
(Here we use our assumption that
is
-invariant.) The explicit formula for
is:
Notice that:
The flux of the BV-Hamiltonian vector field
is a canonical transformation,
and Eq. (
53) implies that this canonical transformation is a symmetry of
. This canonical
transformation does not touch
, it only acts on
. We identify
on chart
with
on chart
when
is the flux of
by the time
along the vector field
where
is the log of
,
i.e. 
. Explicitly:
These gluing rules are consistent on triple intersections because of Eq. (
55).
Lagrangian submanifold
Eqs. (
57) and (
58) look somewhat unusual. In particular, the “standard”
Lagrangian submanifold
is not well-defined, because it is incompatible with our transition functions.
One simple example of a well-defined Lagrangian submanifold is
. We will now
give another example, which repairs the ill-defined
.
The construction requires a choice of a
connection in the principal bundle
.
To specify a connection, we choose on every chart
some
-valued 1-form
,
with the following identifications on the intersection
:
and in particular:
On every chart, let us pass to a new set of Darboux coordinates, by doing the canonical
transformation with the following gauge fermion:
Notice that
does not depend on antifields; therefore this canonical transformation
is just a shift:
This canonical transformation does not preserve
, therefore
the
expression for the action will be different in different charts, see Eq. (
50). In
particular, it will contain the term
, which means that the action of the BRST
operator on
involves the connection. On the other hand, the transition functions simplify:
These are the usual transition functions of the odd cotangent bundle
, where
is the vector bundle with the fiber
, associated to the principal vector bundle
.
In particular, the “standard” Lagrangian submanifold
is compatible with gluing.
The corresponding BRST operator is defined by the part of the BV action linear in the antifields:
After this canonical transformation of Eqs. (
60), (
61) and (
62), the new
is such that
this
is nilpotent on-shell.
Gluing together 
Let us consider the relation between the functions
defined by Eq. (
48) on two overlapping
charts. It is enough to consider the case of infinitesimal transition function,
i.e.
, where
is infinitesimally small. With
defined in Eq. (
54),
the difference between
on two coordinate charts is:
The first term on the RHS is zero:
since
is diffeomorphism-invariant. Let us study the second term. We have:
where
is any connection, transforming as in Eq. (
59). Therefore the following expression:
is consistent on intersections of patches.
The correcting term 
is the infinitesimal gauge transformation (see Eqs. (53)
and (54)) with the parameter 
.
Back to
In our case
is the pure spinor bundle over super-
; the coordinates
are functions from the worldsheet to
(defined in Eq. (
24)).
The total space
is the space of maps from the worldsheet to
.
Notice that
is a principal
-bundle over
.
It has a natural
-invariant connection, which for every tangent vector:
declares its vertical component to be
,
i.e. the projection of
on the
denominator of (
22) using the Killing metric. This defines, pointwise, the connection
on the space of maps.
It is natural to use this connection as in Eq. (66).
Notice that we do not need a connection to write the BV Master Action (Eq. (47)).
But the connection is needed to construct 
(and also in our construction of the
Lagrangian submanifold).
