Extended Conformal Symmetry in Non-Critical String Theory Jan de Boer Extended Conformal Symmetry in Non-Critical String Theory Extended Conformal Symmetry in Non-Critical String Theory Uitgebreide Conforme Symmetrie in Niet-Kritische String Theorie (met een samenvatting in het Nederlands) PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Rijksuniversiteit te Utrecht, op gezag van de Rector Magnificus prof. dr. J.A. van Ginkel, ingevolge het besluit van het College van Decanen in het openbaar te verdedigen op woensdag 9 juni 1993 des namiddags te 12.45 uur door Jan de Boer geboren op 29 juni 1967 te Sint Nicolaasga Promotor: Prof. dr. B. Q. P. J. de Wit Dit proefschrift kwam tot stand met de financi¨elesteun van de Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.). Contents 1 Introduction and Outline 1 2 W Algebras 7 2.1 Conformal Field Theory . 7 2.1.1 Why Conformal Field Theory? . 7 2.1.2 Conformal Field Theory . 9 2.1.3 Operator Product Algebras and Field Algebras . 13 2.1.4 The Free Scalar Field . 16 2.1.5 First-Order Systems . 19 2.1.6 The Wess-Zumino-Novikov-Witten Model . 20 2.2 W Algebras . 25 2.2.1 The c = 2 W3 algebra . 25 2.2.2 General W Algebras . 26 2.2.3 Why W Algebras? . 26 2.2.4 W Algebras From sl2 Embeddings . 27 2.2.5 Why sl2 Embeddings? . 29 2.2.6 Gauge Invariant Polynomials and the Miura Transformation . 31 2.2.7 The Structure of the W Algebra . 32 2.2.8 Quantization of the W Algebra . 35 2.2.9 The Stress-Energy Tensor . 40 2.2.10 The Virasoro Algebra . 41 2.2.11 The W3 Algebra . 41 (2) 2.2.12 The W3 Algebra . 42 2.2.13 Free Field Expressions for W Algebras . 43 2.2.14 Some Explicit Results for sln .................... 45 2.2.15 Extensions of W Algebras . 46 3 Chiral W gravity 48 3.1 Gauging the Chiral Algebra . 48 3.1.1 The Free Scalar Field . 50 3.1.2 The Identity Gauge Field . 51 3.1.3 The Chiral Induced Action . 51 3.2 Solution of the Ward Identity . 55 3.2.1 The Virasoro Algebra . 55 3.2.2 The General Case . 56 8 3.3 Quantization . 58 3.3.1 The Induced Action of W3 Gravity . 60 3.3.2 The Effective Action of W3 Gravity . 67 3.3.3 The Effective Action of the WZNW Model . 68 3.3.4 The Effective Action of W3 Gravity, Continued . 71 3.3.5 Remarks . 73 4 Covariant W Gravity 75 4.1 Gauging Non-Chiral Algebras . 75 4.1.1 Example: Free Scalar Field . 75 4.1.2 General Case . 76 4.1.3 Gauging the WZNW Model . 79 4.2 W Gravity . 81 4.2.1 What is W Gravity? . 81 4.2.2 The Classical Covariant Action for W Gravity . 86 4.2.3 The General Structure of the Covariant Action . 89 4.3 Quantum Aspects of the Covariant Action . 90 4.3.1 All-Order Results for the Covariant Action . 90 4.3.2 KPZ Approach . 94 5 The Moduli Space of W Gravity 95 5.1 W Algebras on Riemann Surfaces . 95 5.1.1 General Case . 96 5.2 The Moduli Space Associated to a W Algebra . 98 5.2.1 The Moduli Space Associated to a Chiral Algebra . 98 5.2.2 A Finite Dimensional Model of the Moduli Space for W Algebras 99 5.2.3 The Connection with Higgs Bundles . 103 5.2.4 Reconstruction of the Metric . 108 5.2.5 Relation to Self-Duality Equations . 110 5.3 Generalized Actions . 112 5.3.1 Classical Actions for Chiral and Covariant Gravity on Higher Genus116 5.3.2 Example: Gravity . 117 5.4 Remarks . 118 5.4.1 The Geometrical Description of W Algebras . 118 5.4.2 The Moduli Space for a Surface with Punctures . 120 5.4.3 From Teichm¨ullerspace to the Moduli Space . 121 6 Towards an Exact Solution? 122 6.1 Continuum Approach . 122 6.1.1 Gravity . 122 9 6.1.2 Generalization to W Gravity: BRST Operator . 125 6.1.3 The Spectrum . 127 6.1.4 Correlators . 129 6.2 Topological Approach . 130 6.2.1 Topological W Conformal Field Theory . 130 6.2.2 N = 2 Structure in W Gravity . 132 6.2.3 Relation with G=G models . 134 6.2.4 Topological Gravity . 138 6.2.5 Topological W Gravity . 138 6.2.6 Intersection Theory . 139 6.3 The Discrete Approach . 141 6.3.1 Review of the p-Matrix Model . 143 6.3.2 Genus-Zero Formulation . 146 6.3.3 The Continuum Limit in Genus Zero . 148 6.3.4 The New Continuum Limit . 150 Bibliography 153 Samenvatting 170 Dankwoord 173 Curriculum Vitae 174 1 Introduction and Outline The ingenious step that led Einstein from the special to the general theory of rela- tivity, was to extend the invariance under Poincar´etransformations by invariance under general co-ordinate transformations. Later, similar ideas have been successfully applied to other theories. In Maxwell theory, we have the freedom to add the divergence of an arbitrary space-time dependent phase factor to the vector potential. If we replace the phase factor by an arbitrary group, this leads to Yang-Mills theory with a local non-abelian gauge invariance. Any matter theory with a rigid invariance can be made into a gauge theory, by replacing the rigid invariance by a local one. A somewhat more involved example is supergravity, which results from the passage from global to local supersymmetry. As an example, consider the action for a free complex scalar field, Z 4 µ ∗ S ∼ d x @µφ∂ φ (1.1) The action is invariant under the rigid U(1) transformation φ ! eiθφ. To make it invariant under local U(1) transformations, i.e. with θ an arbitrary function of xµ, one can use the principle of `minimal coupling'. Introduce a real gauge field Aµ(x), and modify the action to Z 4 µ ∗ S ∼ d x (Dµφ)(D φ) ; (1.2) where Dµ = @µ + iAµ is the covariant derivative. If the gauge field transforms under a local U(1) transformation as Aµ ! Aµ − @µθ, the modified action has a local U(1) in- variance. Associated to every rigid invariance is a conserved current or Noether current. The integral of this conserved current is a conserved quantity, that can be identified with the generator of the rigid symmetry. An important feature of the extended action (1.2) is that the term linear in Aµ couples to the Noether current of the rigid symmetry. Indeed, (1.2) contains the term Z Z 4 µ ∗ ∗ µ 4 µ i d x Aµ(φ∂ φ − φ @ φ) ≡ d x AµJ ; (1.3) where J µ is the conserved current associated to the rigid U(1) invariance. 2 Introduction and Outline The action (1.1) has another invariance, namely Poincar´einvariance. To make this invariance local, we also have to introduce a gauge field, but now minimal coupling does not work. The role of the gauge field is played by a metric on space-time, and the gauged action reads Z 4 p µν ∗ S ∼ d x −g g @µφ∂νφ (1.4) If we write the metric as gµν = ηµν +hµν, where ηµν is the usual flat space metric, we can expand the action in powers of h. The term linear in h is proportional to the Noether current associated to the global translational invariance, which is the energy-momentum tensor of (1.1). Thus, there is a close analogy between the coupling to a metric and the coupling to a gauge field. In both (1.2) and (1.4) the gauge fields are non-dynamical. To provide them with some dynamics, a special Lagrangian has to be chosen, that does not break the gauge invariance, and gives a kinetic term for them. For the U(1) gauge field this is the usual Maxwell action Z 4 µ ν ν µ SMW ∼ d x (@µAν − @νAµ)(@ A − @ A ); (1.5) whereas for gravity it is the Einstein-Hilbert action Z 4 p SEH ∼ d x −gR: (1.6) The Maxwell action is renormalizable, and the quantization of the U(1) gauge field can proceed without problems. In contrast, the Einstein-Hilbert action is non-renormalizable, and it does not define a consistent quantum theory of gravity. In this thesis we investigate the transition from rigid to local invariance in two- dimensional field theory, treating all symmetries, including gauge symmetries and gen- eral co-ordinate transformations, on an equal footing. The class of two-dimensional field theories we will discuss have the special property that they are invariant under local rescalings or Weyl transformations of the metric. These theories are called conformal field theories. An example is the two-dimensional version of (1.4), Z 2 p µν ∗ S ∼ d x −g g @µφ∂νφ : (1.7) Usually conformal field theories are not presented in a covariant form, but in a form where the symmetries (general co-ordinate invariance and Weyl invariance) have been used to bring the metric in some fixed form. On a flat two-dimensional space, the metric can always be brought in the form ds2 = dt2 − dx2. On a non-trivial curved two-dimensional manifold, the space of metrics modulo general co-ordinate and Weyl transformations is a finite dimensional space, called the moduli space of the manifold.