The Polyakov action for a string worldsheet has Weyl invariance. In the conformal gauge augmented with Weyl gauge-fixing, we can always impose a flat worldsheet metric in Minkowski coordinates. The residual gauge symmetries take on the form of conformal Virasoro and anti-Virasoro algebras. This is equivalent to a two dimensional conformal field theory. Later, we impose the Virasoro constraints.
However, can't we come up with more general worldsheet actions which are invariant under diffeomorphisms but not Weyl transformations? The conformal gauge is still possible, but the worldsheet volume factor is dynamical. Residual diffeomorphisms taking on the same form as conformal transformations, minus the compensating Weyl transformation, still exist. They still take on the form of Virasoro algebras and Virasoro constraints. This is no longer a conformal field theory because we have a characteristic length scale on the worldsheet.
Some of these theories are also modular invariant. Do they describe valid string theories?
Best Answer
Dear Jenkins, the theories you want to construct are "noncritical string theories" and they're less interesting and less consistent than the "critical string theories".
First, the Nambu-Goto action - the proper area of the world sheet - is nonlinear. It includes square roots etc. It's much better to introduce an auxiliary metric tensor on the world sheet and the action for the coordinates $X$ becomes nice and bilinear - a free theory.
However, we don't want new degrees of freedom to be added. The 2D metric tensor has three independent components. Two of them may be set to a standard form by the 2 degrees of freedom in the 2D coordinate reparameterization symmetry; and the third by the Weyl symmetry if it exists.
If it doesn't exist, it's too bad. The auxiliary world sheet metric may only be brought to the form of $e^\phi \eta_{ab}$. That means that $\phi$, determining the overall scaling, becomes another function of the world sheet coordinates $(\sigma,\tau)$, very analogously to the spacetime coordinates $X(\sigma,\tau)$. In fact, it is really valid to say that the parameter determining the overall scaling of the metric is another spacetime coordinate.
If this coordinate were totally identical to the other coordinates, then there would also be a translation symmetry in the $\phi$ direction - but that's equivalent to the Weyl symmetry (multiplicative scaling of $e^\phi$ is the same thing as additive shifts to $\phi$). Because by assumption, the Weyl symmetry doesn't hold in your theory, the new spacetime coordinate $\phi$ can't have quite the same properties as the other spacetime coordinates.
However, in normal circumstances, you obtain the violations of the Weyl invariance as a disease. In particular, if you try to study string theory in a non-critical dimension, i.e. $D\neq 26$ or $D\neq 10$, you will find out that the field $\phi$ doesn't decouple and the path integral, when calculated including the one-loop accuracy, still depends on $\phi$. So the Weyl symmetry, equivalent to an additive shift of $\phi$ by a function of the world sheet, is not a symmetry.
As I said, this can be interpreted as $\phi$'s becoming a new spacetime coordinate. But if you try to calculate the effective action in the new spacetime that has an additional dimension $\phi$, you will find out that the laws of physics are not invariant under translations in $\phi$ - that's nothing else than the failure of the theory to be Weyl-invariant.
In particular, you will find out that the dilaton linearly depends on $\phi$: search for papers about "linear dilaton". The squared gradient of the dilaton is related to the surplus or excess (if it is time-like or space-like) of the spacetime coordinates, relatively to the critical dimension.
If the spacetime has two dimensions, one may choose the dilaton to depend on the (only) spacelike coordinate $\phi=X^1$ in such a way that the theory including $\phi$ is Weyl-invariant again. In this case, it's useful to consider not only the right linear dilaton - solving the equations of motion - but also a non-trivial background for the tachyon. One ends up with the so-called "Liouville theory" - a "linear dilaton" theory with some extra tachyonic profile in a non-critical stringy $D=2$ spacetime - which is slightly more consistent than other noncritical string theories. The Liouville theory may also be described by a quantum mechanical model with a large matrix - the old matrix theory.