Chiral non - critical strings ∗

It is shown that conformal matter with cL 6= cR can be consistently coupled to two-dimensional ‘frame’ gravity. The theory is quantized in conformal gauge, following David, and Distler and Kawai. There is no analogue of the c = 1 barrier found in nonchiral non-critical strings. A non-critical heterotic string is constructed—it has 744 states in its spectrum, transforming in the adjoint representation of (E8) 3. Correlation functions are calculated in this example, revealing the existence of extra discrete states. Talk presented by RCM at the International Workshop on String Theory, Quantum Gravity, and the Unification of the Fundamental Interactions, held in Rome, Italy, 21-26 September 1992; and at the CAP/NSERC Summer Institute on Quantum Groups, Integrable Models and Statistical Systems, held at Queen’s University, Kingston, Ontario, July 13-18, 1992. [email protected] [email protected]/[email protected] Conformal matter in two dimensions couples to quantum gravity via the conformal anomaly. The coupling is characterized by the central charge of the conformal matter. The central charges for holomorphic and anti-holomorphic fields in a conformal field theory may differ. Hence, it is natural to ask how such theories, with cL 6= cR, interact with quantum gravity. Since these theories have a Lorentz anomaly, in addition to the conformal anomaly, there must be degrees of freedom other than the conformal factor that become dynamical. Under combined scale (ρ) and Lorentz (χ) transformations, the zweibein transforms e±μ → exp (ρ± iχ) e±μ . Just as the conformal anomaly provides dynamics for the scale factor ρ, the Lorentz anomaly provides dynamics for local Lorentz field χ. It is the action that governs these fields that we study here. This paper is organized as follows: Sect. 1 briefly reviews results from the Liouville field treatment of nonchiral non-critical strings. In sect. 2, we determine an analogous conformal field theory representation of chiral gravity. We consider the additional Lorentz moduli, and the gravitational dressings in this new theory. Finally, we derive some critical exponents. In sect. 3 we give an example, the –string, and compute the partition function and correlation functions in this example. Sect. 4 contains some concluding remarks. The present paper is primarily a review of the work appearing in Ref. [1], and we refer the reader there for a more detailed account of this work. Sect. 3 presents some new results about scattering amplitudes for the –string. We note that the coupling of chiral matter to two-dimensional theories of quantum gravity has been previously considered by several authors[2, 3, 4]. Of particular relevance to the present work are Ref. [3], which studied chiral non-critical strings in light-cone gauge, and Ref. [4], which also derived the conformal field theory presented in sect. 2. 1 Review of non-critical strings The initial success in producing an analytic understanding of non-critical strings made use of light-cone gauge[5]. The present discussion follows the David–Distler– Kawai (DDK) approach[6] in conformal gauge. Scattering amplitudes in a (nonsupersymmetric) string theory are calculated with a Polyakov path integral over metrics gμν and matter fields ψ on the two-dimensional string world-sheet ∫ Dg Dψ vol.(symmetries) exp(−So[g, ψ]) O1 . . .On . (1.1) The classical symmetries include both diffeomorphisms and local Weyl rescalings (i.e., g → eg). In the quantum theory, the path integral measure introduces an anomaly for the latter transformations. Polyakov[7] showed that this anomaly is proportional to cm − 26, where cm is the central charge for the matter fields.