Superfields and their components
We think of the superfields as functions on
. In particular, those which lift from
are called
chiral and those which lift from
are called
antichiral.
1 Matter fields
The matter fields are five chiral and five antichiral:
Here the index
runs in
; we assume that the target space is a complex five-dimensional manifold.
2 Action for matter fields
Let us concentrate on the heterotic case. In this case, we have to remember about the right
(“antiholomorphic”) coordinate
.
Theorem 1 tells us that the holomorphic
part of the Berezinian is trivial; therefore we can just integrate
:
3 
superconformal transformations of superfields
By definition, the action (
5) is invariant finite
transformations:
These transformations can be understood as the result of integrating the vector fields (
2), (
3).
Explicitly:
In components:
and for antichirals:
