Lagrangian submanifolds
1 A gauge slice in 
Let 
denote the 
-dimensional submanifold of 
.
Suppose it is parametrized by 
and 
.
We then choose the following Lagrangian submanifold:
where
denotes the normal subbundle (the subbundle of the cotangent bundle
to
consisting of those elements which restrict to zero on the tangent
space to
). The integration measure on
is:
where
runs unconstrained, while
and
should be orthogonal
to the tangent directions to
. Alternatively we could say that
and
are also unconstrained but insert the
-functions; then the measure is, in the
PDF notation:
In this form it can be interpreted as the integration over the family of
Lagrangian submanifolds. Namely,
is the integration over
a fixed Lagrangian submanifold, while
is for integrating over the family.
2 Equations of motion and BRST
2.1 Equations of motion
The integration over
gives:
The integration over
and
gives the equations of motion for the ghosts:
Eqs. (
19), (
20) and (
21) imply that
where
denotes the vector fields preserving the heterotic structure.
TODO: DOUBLE-CHECK
More explicitly:
2.2 Algebra of superconformal transformations
The anticommutator of
with itself can be expressed through the structure
constants of the superconformal algebra:
This formula may be slightly confusing because the relative sign between
the first two terms appears to be different from Eq. (
22); a better way
to present these formulas is to pick two constant Grassmann numbers
and
and write:
3 BRST transformations
Let us integrate out
(this means that we substitute into the action the
given by Eq. (
19)). Then the action (on the Lagrangian submanifold, at the
flat point) is:
with the BRST transformations (somewhat schematically):
3.1 In components: ghost fields
On shell:
The BRST transformations of
and
are read from Eq. (
24):
The BRST transformation of
is easy to read from Eq. (
25):
3.2 In components: matter fields
and 
; we agree
with their Equation (3.20).
