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1 Notations:   , ,
2 Cotangent bundle to the moduli space of heterotic structures
2.1 General description
2.2 Study of equivalence relation
2.3 Gauge fixing

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:

(5)

(6)

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:

(7)

Given a section of :

(8)

(9)

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 :

(10)

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:

(11)

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):

(12)

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

(13)

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:

(14)

Now we have:

(15)

instead of Eq. (10).