## 2 The Setup

With the embedding coordinates and the Polyakov action of the
string in conformal gauge () takes the form ()
where we have suppressed the fermionic terms in the action, as they will not be relevant
in our discussion of classical solutions (the full fermionic action is stated in [89, 100]).
A natural choice of coordinates for the space (“global coordinates”)
is
Moreover, we have directly written the string action with the help of the effective string tension
of (2). It is helpful to picture the space-time as a bulk cylinder with a
four-dimensional boundary of the form (see Figure 2).
A consequence of the conformal gauge choice are the Virasoro constraints, which take the form
where the dot refers to and the prime to derivatives. Of course the contractions in the above are
to be performed with the metrics of (5).
As mentioned in the introduction, it is presently unknown how to perform an exact quantization of this
model. It is, however, possible to perform a quantum fluctuation expansion in . For this one expands
around a classical solution of (4) and integrates out the fluctuations in the path-integral loop
order by order. This is the route we will follow. Of course, in doing so, we will only have a
patch-wise access to the full spectrum of the theory, with each patch given by the solution expanded
around.