Heterotic Structure
1 Definition of worldsheet heterotic structure
We will consider super-worldsheet parametrized
by two even coordinates 
and 
, and one odd coordinate 
. It is
natural to denote it 
.
We treat and its complex conjugate as independent complex numbers,
and therefore denote them and 
(instead of and 
).
The matter action is a functional of the following three objects:
Ten scalar superfields 
, in other words
An odd vector field 
(a “superconformal structure”)
An even vector field 
Let
and
denote the sheaves of vector fields which are proportional
to
and
, respectively.
We will also impose the following additional constraint:
Definition 1:
the data 
will be called the
worldsheet heterotic structure.
2 Any heterotic structure is locally flat
Eq. (1) is actually the condition of local flatness. The flat
heterotic structure is defined as follows:
Definition 2:
Let
and
be the distributions generated by the following two
vector fields:
This is called flat heterotic structure.
Theorem 1:
Any heterotic structure is locally flat.
Proof: Let us pick any sections
and
.
Eq. (
1) implies the existence of an even function
and an odd function
such that:
Locally, we can always find some even functions
and
such that:
Now we have:
We can always find a coordinate
such that
.
Let us consider
—
the factorspace of
by the orbits
of
. Eq. (
4) implies that
consistently projects from
to
. We can always
introduce on
the coordinates
such that:
We can then consider
as coordinates on
.
In these coordinates
and
are given by the same
formulas as
and
in Eqs. (
2) and (
3).
This
completes the proof of
Theorem 1.
3 Worldsheet action
We will now explain how the matter action 
is defined.
The pair 
defines a section
in the following way.
Remember that a section of the Berezinian is a function of a pair
where 
is a point of , and three
vectors 
form a basis of 
; moreover we require that this function depends on
in a “controllable way”, namely when we change 
, it
gets multiplied by the super-determinant of 
.
We define 
so that it is
equal 
for the basis 
.
In other words:
Now we are ready to write the action:
This action is invariant under the rescaling of
and
:
It is also invariant under the super-diffeomorphisms; for any vector field
:
leaves the action invariant. (This follows from everything being geometrically defined.)
