Non-critical string theory

The non-critical string theory describes the relativistic string without enforcing the critical dimension. Because in these cases, the non-critical string theory is not suitable to describe the fundamental nature of interactions, as it does not have the property of consistently quantized Lorenz symmetry and massless excitations. However, there are several applications for such a theory. The most prominent application is the phenomological description of the confinement of strong interaction by QCD strings. There also exists a duality to the 3-dimensional Ising model.

The Critical Dimension and Central Charge

In order for a string theory to be consistent, the worldsheet theory must be conformally invariant. The obstruction to conformal symmetry is known as the Weyl anomaly and is proportional to the central charge of the worldsheet theory. In order to preserve conformal symmetry the Weyl anomaly, and thus the central charge, must vanish. For the bosonic string this can be accomplished by a worldsheet theory consisting of 26 free bosons. Since each boson is interpreted as a flat spacetime dimension, the critical dimension of the bosonic string is 26. A similar logic for the superstring results in 10 free bosons(and 10 free fermions as required by worldsheet supersymmetry). The bosons are again intepreted as spacetime dimensions and so the critical dimension for the superstring is 10.

The non-critical string is not formulated with the critical dimension, but nonetheless has vanishing Weyl anomaly. A worldsheet theory with the correct central charge can be constructed by introducing a non-trivial target space, commonly by giving a spatially varying expectation value to the dilaton. Since the dilaton is related to the string coupling constant, this leads to a region where the coupling is weak(and so perturbation theory is valid) and another region where the theory is strongly coupled.

Two-dimensional String Theory

Perhaps the most studied example of non-critical string theory is that with two-dimensional target space. While clearly not of phenomenological interest, string theories in two-dimensions serve as important toy models. They allow one to probe interesting concepts which would be computationally intractable in a more realistic scenario. Additionally, these models often have fully non-perturbative descriptions in the form of the quantum mechanics of large matrices.