Deformations of the heterotic structure
1 Notations: 
, 
, 
Let
denote those forms which vanish on
, and
those forms which vanish on
. Let us denote:
For the flat worldsheet,
is generated by
.
Let
denote a rank 2 vector bundle generated by commutators
of sections of
. This is actually a sub-Lie-algebroid of
. Let
us denote:
There is a nondegenerate pairing between
and
.
Therefore, as a line bundle:
It follows from Eq. (
1) that:
To prove Eq. (
6), consider any section
of
(in flat case we can take
).
It follows from Eq. (
1) that
and
.
Therefore
is proportional to
,
i.e. Eq. (
6).
The structure of
and
can be summarized in the following exact sequences:
Given
—
a section of
:
2 Cotangent bundle to the moduli space of heterotic structures
2.1 General description
An infinitesimal deformation of a point
corresponds to an infinitesimal rotation of the
subspaces
and
inside
:
where
is the subspace consisting of the elements of the form Eq. (
13) below; we have
to factor it out because of the constraint (
1).
The general element of
can be presented as follows:
However, because of the constraint (
1), this description of
is not one-to-one;
some
and
actually correspond to the zero element of
, and
therefore we factor out
in Eq. (
10). We will now describe such zero elements.
2.2 Study of equivalence relation 
Suppose that we are given, for each point
,
a section
. Then we can construct a section of
corresonding to a zero element of
, in the following way. Since
,
we can
view as an element of 
(this space does not depend on a point in
!).
Then it makes sense to consider the variation of
when we vary the point of
:
Because of the constraint (
1):
For any
and
we
define the Lie derivative 
so that:
From them point of view of Eq. (
11), the first two terms on the right hand side
of (
12) correspond to
and
.
Notice that the map
is
-linear. For any
:
Now, returning to Eq. (
10), elements of the “zero subspace”
can be described as follows.
Consider
with sufficiently small compact support, and choose
and
so that:
| |  on support of ; this is possible by Theorem 1
|
| | |
| | | | |
Such elements generate the subspace
which we factor out in Eq. (
10).
2.3 Gauge fixing
As we explained in Eq. (
7) and (
9),
is generated by sections of
and expressions of the form
.
By the same argument,
is also generated by
sections of
and expressions of the form
with
.
Comparing this to Eq. (
13), we conclude that we can gauge fix
the
of Eq. (
11) as follows:
Now we have:
