It is shown that the action of the Liouville field theory describing random surfaces contains terms not noticed before. Critical exponents are predicted and an analogue of the "c = 1 barrier" of two-dimensional gravity is derived. These terms ensure, ordered in the interaction coupling constant, that the theory is scale invariant.
Critical exponents are predicted and the analogue of the c = 1 barrier of two-dimensional gravity is derived.
The Trace Anomaly and DDK
The subscripts ge Examples in two dimensions are the normally ordered operator cos px, or the operator eaf/J of the last subsection. This measures the average extent From the analysis of the c = 1 matrix model, the correlation functions of (4.10) are known to all orders in the string loop expansion, that is, summed over all genera, and beyond. Further confirmation comes from the agreement between the results of the continuum approach and the matrix model approach. As will be shown in this part of the thesis, this implies that new terms must be added to the action (3.16). They have not been discussed before, but are important for understanding the flow of the renormalization group and can be seen in recent matrix model results for the gravity-coupled Sine-Gordon model phase diagram. The action is the corresponding conformal invariant free action and interaction terms that are usually assumed to have form. The operators m (1.1) are not exactly marginal.t They should be, because the Liouville theory must be independent of the background as a consequence of the general covariance.14•51 Therefore, the beta functions of the theory must be zero for all coupling orders . For the case of the Sine-Gordon model coupled to gravity, it will be seen that this flow qualitatively agrees with the recent results of Moore's matrix model. Equation (1.2) should be viewed as a second-order correction to the gravity dressing of the i(x). That the correction (1.2) is essentially unique is argued in Appendix A by considering the marginality conditions as equations of motion. of string theory. In subsections 3.2 and 3.3 this is applied to the Sine-Gordon model and the resulting phase boundaries are compared with those found with the non-perturbative matrix model tecb.niques.l31l It is seen that the presence of the terms (1.2) is decisive is, even for qualitative agreement of the matrix model and the Liouville theory approximations. In particular, it is argued that the relationship between the correlation functions in the matrix model and in the Liouville approach is more complex than is often assumed. Using the above method, we can check whether the second term really appears as a correction to the first term.~. As already mentioned, we need to look for almost quadratic singularities such that cjk are universal constants. Note also that from the point of view of string theory (2.8) describes the back-reaction of the tachyon to itself and to the graviton. Here the integrations are along contours in the z-plane surrounding the operators on which IJ±,H3 acts. If the interadion tim<.l>jm[x]~jm[x] is added to the Lagrangian of matter, the dressed interaction is first order in the coupling constants. The jm can be rescaled so that the operator algebra of the 1/jm has the w00 structurel24•25l. Next, we must ask whether the modifications (1.2) of the operators (1.1) are the unique modifications that achieve marginality to order (T)2. The situation is largely clarified by thinking of the marginality conditions as equations of motion of string theory, as in ref. In the Sine-Gordon model coupled to gravity, no unwanted terms with cos 2px are induced because the OPEs are "softer" than in free-field theory (see (A.4-5)). These conclusions will be confirmed in section 3 by observing the agreement with matrix model results. As mentioned above, the action (3.2) corresponds to a classical solution of string theory with two-dimensional target space (x,. The equations of motion of classical string theory thus play the role of the Gell-1\Iann-Low equations in The presence of gravity .135l They contain second (and higher) order derivatives of ¢, which we have just interpreted as 'group time renormalization'. about metrics.l37l We now apply the previous procedure to the examples worked out in section 2, starting with the Sine-Gordon model. When deriving E(A), the term >.m2(ox)2 is included in a redefinition of x and then t. This is more complicated with Til, but to find phase boundaries T'(>.) is good enough. The coupling constant current is qualitatively the same as in flat space and is given by the Kosterlitz-Thouless diagram (Figure 1). We see that the m2 By normalizing m and 10 as in (3.3), we obtain the slope V'i/1r for the phase boundary. It will also be interesting to see whether the logarithm in (3.5) derives from the higher-order modifications in m needed to keep the interaction close to p = V2. As mentioned in subsection 2.4, the cosmological constant cannot be neglected there and further investigation is needed. The latter becomes comparable to the background cosmological constant at Let us tentatively* write the leading-order action as:. Thus, switching the jmCI>j'm' operators with j' > 1 will in general induce an infinite set of higher spin operators jm CI>jm al 0(T2), the unions of which had originally turned into oil. This is what is expected of these non-renormalizable operators, but it would not occur without the modification 0(T2) 8£. In the Liouville theory approach to 2D quantum gravity coupled to an interacting scalar field, new terms appear in the Lagrangian at higher orders in the coupling constants. The new terms are crucial for obtaining the correct phase diagram, as found with the nonperturbative matrix model techniques in the case of the Sine-Gordon model. The cosmological constant must be treated more strictly, and the cubic terms in the beta function (2.2), which is also universal, must be derived. It is useful to think of 2D quantum gravity as classical string theory.l6l Let us first discuss the example of the Sine-Gordon model. 6], we will adopt boundary conditions given (i) in the ultraviolet by bare action and (ii) in the infrared by the regularity requirement. Given a solution of (A.3) for y(l) and h1w, the other solutions are obtained by adding linear combinations of O(m2) of the on-shell tachyons and the two discrete gravitons. A.5) Limit condition (ii) essentially means that the operators with the more negative Liouville bound must be dropped. Including the discrete operators from section 2.3 as interactions is equivalent to enabling higher spin backgrounds in the sigma model, and the same arguments seem to apply. Gravitational relations in the presence of a cosmological constant J.l can be found in principle as follows (see [6,20] for some details). One includes the cosmological constant in the tachyon of string theory, replacing, for example, for the Sine-Gordon model, the ansatz (A.2) with . -L has the form of a fold centered at the free parameter ¢, which is related to J.l by J.l = eVZ¢ and with a bare cosmological constant of 6. Like TI-L, this difference decays exponentially in the UV. For the Sine-Gordon model, this modification of the OPEs seems to solve the problem of the new lrl-2 singularities that would naively appear in (2.7) under p = ~.J2. As in two dimensions, the determinants can be separated from JgF is independent of The consistency conditions of invariance when rescaling the background metric (in particular that the theory is at a fixed point of the renormalization group) mean that the integrands of SR2,. The result for the size of the operator eP¢> is consistent with the result of ref. First, in [45] the conformal factor theory was studied as a 'mini-superspace' theory, rather than as gravity with a self-duality constraint. This justifies the use of the minisuperspace approximation in the UV but not in the IR. As in two dimensions, if This is suggested by the fact that the moduli space of the latter theory appears to be precisely the moduli space of conformal self-dual metrics that arose here.Applied Liouville Theory
Exactly Marginal Operators
Running Coupling Constants
Outlook
