On this page:
1 Solution of Master Equation
2 Changing polarization
3 Lagrangian submanifold
4 Singularities of correlation functions
5 Insertions of unintegrated vertices
5.1 Massless states
5.2 Tachyons


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

    2 Changing polarization

    3 Lagrangian submanifold

    4 Singularities of correlation functions

    5 Insertions of unintegrated vertices

      5.1 Massless states

      5.2 Tachyons

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:

(8)

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:

(9)

(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:

(10)

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

This follows from the variation of the action under the infinitesimal shift :

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:

(11)

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:

(12)

where  

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: