Non-Critical String Theory

Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-point function on S 2 . In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville's equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include "solutions" of Liouville's equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a Chern-Simons theory in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of "timelike" Liouville theory, where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on S 2 can be computed by the original Liouville path integral evaluated on a new integration cycle. arXiv:1108.4417v2 [hep-th]

We compute all string tree level correlation functions of vertex operators in c < 1 string theory. This is done by using the ring structure of the theory. In order to study the multicritical behaviour, we calculate the correlation functions after perturbation by physical vertex operators. We show that the (2k − 1, 2) models can be obtained from the (1, 2) model and the minimal models can be obtained from the (1, p) model by perturbing the action by appropriate physical operators. Our results are consistent with known results from matrix models.

We study c < 1 matter coupled to gravity in the Coulomb gas formalism using the double cohomology of the string BRST and Felder BRST charges. We find that states outside the primary conformal grid are related to the states of non-zero ghost number by means of descent equations given by the double cohomology. Some aspects of the Virasoro structure of the Liouville Fock space are studied. As a consequence, states of non-zero ghost number are easily constructed by "solving" these descent equations. This enables us to map ghost number conserving correlation functions involving non-zero ghost number states into those involving states outside the primary conformal grid.