Non-critical string theory

The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory does not describe a Lorenz invariant background. Thus it is not a suitable theory of everything.

However, there are several applications of the non-critical string. Through the AdS/CFT correspondence it provides a holographic description of gauge theories which are asymptotically free. It may then have applications to the study of the QCD, the theory of strong interactions between quarks. Another area of much research is two-dimensional string theory which provides simple toy models of string theory. 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 an expectation value to the dilaton which varies linearly along some spacetime direction. For this reason non-critical string theory is sometimes called the linear dilaton theory. Since the dilaton is related to the string coupling constant, this theory contains 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.

References

• Polchinski, Joseph (1998). String Theory, Cambridge University Press. A modern textbook.
  • Vol. 1: An introduction to the bosonic string. ISBN 0-521-63303-6.
  • Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4.

See also

• String Theory, for general information about critical superstrings
• Weyl anomaly
• Central charge