Using metric instead of complex structure
It is also possible to use the worldsheet metric, instead of complex structure, as a dynamical variable.
This requires Weyl invariance. We will just list the formulas here.
1 Solution of Master Equation
(the subindex
stands for “matter”, although this action also
involves the dynamical metric
). Let
denote the space parametrized
by
and
.
It follows from the
general theory, that the following functional on
solves the Master Equation:
where
is the ghost for the Weyl transformations.
2 Changing polarization
Let us choose some reference metric
and change the coordinates in the following way:
This is just the definition of the new pair of canonically conjugate fields
and
which replace
and
.
With these new notations, the action becomes:
(Notice that in the computation of
we have to use the metric
.)
3 Lagrangian submanifold
We now choose the Lagrangian submanifold in the following way:
On this Lagrangian submanifold the action is quadratic. Indeed, first notice that
the integral over
makes
traceless with respect to
.
Then in the last line of Eq. (
9)
only the first term survives and becomes
4 Singularities of correlation functions
5 Insertions of unintegrated vertices
5.1 Massless states
The case of massless states is special, because the corresponding vertex operators are free of divergencies
and do not require regularization. Consider the scattering of
gravitons, which are parametrized
by their momenta
and polarizations
. The corresponding insertion is:
Eq. (
11) implies that there are no divergencies. Straightforward calculations gives:
in ether classical or quantum theory. The matter part of the path integral:
can be computed exactly using Wick theorem (as the theory is free), and the result is a finite expression. The computation is
explicitly Weyl-invariant.
Now we will proceed to tachyons, which are the simples example of massive state.
5.2 Tachyons
Let us regularize
by a point-splitting procedure,
splitting by a small
as measured with the worldsheet metric.
In other words, let us define the insertion of
tachyon vertices as the
limit of the following expression:
This regularization breaks diffeomorphisms to the stabilizer of
.
On the other hand, the breaking of the Weyl symmetry is much worse; but assuming that
, we
observe the following facts:
In the classical theory, the expression (12) does not have a good limit when
But in quantum theory (i.e. inside the path integral) the limit of the expression (12) is well-defined
Moreover:
where
is the Weyl ghost.
To prove this facts, we observe that we can explicitly evaluate the matter path integral:
| | | | |
| | | | |
| | where  is the scalar propagator |
| | |
Consider the following expression:
It depens on the metric
through explicit factors
and also because the distances
and
are measured using
. Notice that:
:
