Complexification of the worldsheet
We want to define some class of BRST-exact deformations of the theory, which may be
considered “purely left”. We will show that this class is preserved by a subalgebra
in the algebra of vector fields which we call
“left-deforming”.
We will also consider some naturally defined subalgebras:
1 Complexification
In order to consider the left sector from separately from the right sector, we have to
relax the requirement that 
be complex conjugate to 
.
Once we do that, we would actually prefer to consider and both real,
and call them 
and 
.
This does not change formulas, but at the same time allows us to use some geometrical intuition.
In other words, we will consider the worldsheet to be Lorentzial, with the intent to do
the Wick rotation at the end.
2 Geometry of Lorentzian worldsheet
Null-curves on the Lorentzian worldsheet are called characteristics.
Suppose that the metric on the worldsheet is 
.
3 Deforming the Lorentzian metric
Let us consider some deformed metric.
Definition 2:
a deformation of the metric is called
“left deformation” if it has the same left-moving characteristics
as the original metric. In coordinates, left deformations of
are of the form:
The left-moving characteristics remain undeformed; they are tangent to
.
The corresponding deformed action for the bosonic string would be:
4 Left-deforming vector fields
4.1 Definition
Definition 3: A vector field on the worldsheet is called
“left-deforming vector field”
if its flux preserves the class of left deformations of the metric. In other words, the difference
between the metric and its pullback by the flux is its left deformation plus maybe its Weyl rescaling.
In other words:
In coordinates they are of the form:
Left-deforming vector fields form a subalgebra in the algebra of all vector fields.
The Lie derivative of the metric w.r.to such vector field is:
Modulo the Weyl rescaling:
4.2 When left-deforming is conformal?
Eq. (
3) implies the following condition for a left-deforming vector field
to be conformal:
which can be also understood as follows:
| | | | |
| | | ⇒ is conformal |
| |
5 Left vector fields
There is a smaller subalgebra, consisting of those vector fields which are parallel to 
(i.e. to the left-moving characteristic).
We will call them “left vector fields”.
Definition 4: the vector field is called
“left vector field” if it is
parallel to the left-moving characteristic. In coordinates they are of the form:
Left vector fields are automatically left-deforming. They are conformal when
only depends on
(and not on
).
are called left-moving characteristics
are called right-moving characteristics
to be left-deforming does not depend on
defines the variation of
under
