Lectures on Liouville Theory and Matrix Models

Lectures on Liouville Theory and Matrix Models Alexei Zamolodchikov and Alexander Zamolodchikov Lecture1. Introduction 1. The Liouville gravity 1. Theory of gravity. Since Einstein the term gravity means the dynamic theory of the space-time metric structure. This dynamics may be either classical (classical gravity) or quantum, in which case we talk about quantum gravity. The main dynamical variable is the components of the metric tensor gab(x): In general this theory of gravity is very complicated structure, both from mathematical point of view and conceptually. Even in classical gravity the equations of motion imposed on the metric are highly non-linear and lead to soluthions which typically develop singularities where the space-time becomes highly curved and the classical Einstein theory itself fails to describe the physics near such singularities. In quantum gravity the situation is much worse, especially from the point of view of interpretations. Having lost the calssical \rigid" space-time frame to settle his experimental equipment, the virtual observer feels himself somewhat \dissolved" and is forced to look for new interpretational possibilities. The simplest (and quite common) solutionis to forget about coordinates and consider only coordinate-independent observables. Such approach, which can be called the topological gravity in some extended sense, is reasonably consistent and su®ers from the only problem: how to make contact with the semiclassical limit, where, as each of us know, the everyday life has apparently nothing to do with the topological gravity. Anyhow, the problem of interpretations, the problem of correct choice of observables, is still of primary importance in quantum gravity. In other words we still don't know what are correct questions to be asked. To this order any simpli¯ed model, which softens the severe mathematical problems of gravity but shares the same questions of interpretation, can be considered useful and worth studying. Below I'll concentrate on gravity in two dimensions (2D) where many technical simpli¯cations are immediately come to play. Even there only few very particticular and most simple tasks are taken, mainly to illustrate the general pattern of problems coming even in this very simpli¯ed model. 2. Two-dimensional gravity. From now on we imply a two-dimensional manifold equipped with a metric gab: Morover, I restrict myself to the so called euclidean gravity, where the metric is positive de¯nite g > 0. I remind here the peculiarities and simpli¯cations of two-dimensional metric geometry. 1. In 2D the Riemann curvature is completely described by the scalar curvature R. 2. Metric gab contains only three independent components. Therefore by an appropriate choice of the coordinate system (a two-parameter freedom) it can be described by only one ¯eld-like dynamical variable. As an example, we can take the so called isothermic (or conformal) coordinate system, which can always be chosen locally in two dimensions and 2 g ab g ab where the metric tensor has the form σ(x) gab(x) = ±abe (1.1) and σ(x) characterises completely the metric structure of the manifold. E.g. the scalar curvature reads ¡σ(x) 2 R(x) = ¡e @aσ(x) 3. The action functional. To construct a classical covariant theory of gravity we have ¯rst of all to choose the action, which must a covariant (coordinate independent) functional of the metric A [gab]. At ¯rst sight it seems natural to to take a local action, i.e., with the density a local function of metric and its derivatives. General covariance prescribes this density to be constructed from coordinate tensors such as metric and Riemann curvature, like Z Z µ ¶ p p terms of higer degree in R A [g ] = ¹ gd2x + k R gd2x + (1.2) ab and it's derivatives The ¯rst term here is simply the 2-volume of the surface. Coupling ¹ is called therefore the cosmological coupling constant. Second quoted term is nothing but the famous Eistein action. It is another peculiarity of the two-dimensional gravity, that the Einsten action in 2D does not lead to any essential local dynamics: the Gauss-Bonnet theorem permits to reduce the Einstein action to a number which depens on the topology only. In particular, it doesn't inﬂuence the local equations of motion. In principle we can consider next terms in (1.2) to create a non-trivial dynamics. I will not follow this line here. First, these terms play small role in the most interesting case of big surfaces (such terms are called irrelevant). The second and more important reason is that it seems more natural to construct the gravitational action as the e®ective one induced by certatin matter ¯elds living on the surface. Such induced action is not nessesserily local, the series of local terms (1.2) being nothing but its long-wave expansion. 4.Induced action. If certain generally covariant matter lives over the surface, it gener- ates an e®ective gravitational action, which although in general non-local, is authomatically covariant. Since long people argue (A.D.Sakharov) that the standard 4d Einstein action is simply a ¯rst term in the short wave expansion of the e®ective action generated by massive 3 τ τ σ σ material degrees of freedom. Another, more relevant in 2D example, is how 2D gravity ap- pears in the string theory context. The trajectory of a string in the target space-time is a two-dimensional surface (the world sheet), either with a boundary in the case of open string, or compact for closed strings. Embedding coordinates X~ (σ; ¿) can be considered as ¯elds on the world seet. In the simplest example of purely bosonic string the dynamics is prescribed by the standard string action Z h i 1 p A g ; X~ = gab@ X@~ X~ gd2x (1.3) string ab 2 a b Of course, once the \matter ¯elds" X~ are integrated out, this results in certain e®ective action dependent on the metric only. This e®ective action is highly non-local, because, as one can see immediately from (1.3) the ¯elds X~ are massless and thus have in¯nite correlation length. More generally, one can plug-in any relativistic ¯eld theory, massless or massive. Let me write down explicilty an example of the generator of the e®ective gravity the familiar two-dimensional sin-Gordon model immersed to a general relativity background Z µ ¶ 1 m2 p A [g ;'] = gab@ '@ ' ¡ cos(¯') gd2x ab 2 a b ¯2 From the point of view of e®ective gravity we have to disinguish the \heavy" matter theories, which have correlation length (or inverse mass scale) much less then the characteristic scale L of the surface itself Rc ¿ L, and the \light" matter theories, either massless or having the inverse mass scale compatible with the scale L. With \havy" theories the situation is simple. From the scale L the induced action is almost local and we're back to the long 4 wave expansion. As a result, the \heavy" matter contributes only to the cosmological term and topological Einstein action. Further local terms are less relevant. For \light" matter the situation is much more complicated: the action is no more local and much less universal. Very important simpli¯cations taking place in two dimensions with a special matter content are considered in the next subsections. To this order I remind few facts about conformal ¯eld theory (CFT). 5. Conformal matter. Among two-dimensional relativistic ¯eld theories there is a class of massless theories which are scale covariant, i.e. they don't have any distinguished mass scale and behave self-similarly as the scale changes. Typically such theories posess, in addition to ordinary relativistic and scale covariance, much higher conformal symmetry, which in 2D can be enlarged to in¯nite dimensional Virasoro symmetry. Such theories are called the conformal theories. Well familiar examples of conformal theories are the two dimensional free bosonic and fermionic ¯elds 1 L = (@ ')2 c = 1 2 a ¹ a L = iÃγ @aÃ c = 1=2 These ¯elds are free and it is not a big deal to treat them explicitly. There are however interacting non-trivial conformal theories. Due to their enlarged symmetry, conformal theories are studied much better then general relativistic ¯eld theories. Many conformal ¯eld theories are constructed explicitly. All conformal theories are characterized by certain number c called the central charge and a set of local observables which are called the primary ¯elds f©i; ¢ig with their characteristic \dimensions" ¢i. This dimensions describe variations of the ¯elds ©i with respect to the scale transformations ½ c charge centrale CFT = f©i; ¢ig champs primaires One of the most important properties of conformal theories is the very explicit and simple way they are coupled to curved spacetime background and simpli¯ed reaction on the variation of the metric background. This in order is due to the following simple statements [1] Stress tensor anomaly. Consider the stress tensor as a succeptibility of the system with respect to the variations of the background metric Z 1 ±A[g ] = ¡ T (x)±gab(x)g1=2(x)d2x ab 4¼ ab In 2D CFT the trace of the stress tensor Tab(x) is in fact a c-number (i.e., proportional to the identity operator) and reads explicitly c θ(x) = gabT = ¡ R + ¹ (cosm. constant) (1.4) ab 12 where R is the scalar curvature of the background metric and c is precisely the central charge mentioned above. 5 Primary ¯elds. The primary Ái ¯elds mentioned above vary very simply (here the primary ¯elds are supposed to be scalars) ±©i(x) = ¡¢i©i(x)±σ(x) (1.5) under the Weil variations of the metric ±gab(x) = gab(x)±σ(x) (1.6) All other local ¯elds behave less simple but all of them can be constructed as the operator product expansions of primaries and the nontrivial stress tensor components.

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