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An Algorithmic Approach to Operator Product Expansions, W-Algebras

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An Algorithmic Approach to Operator Product Expansions, W-Algebras Katholieke Universiteit Leuven Faculteit Wetenschappen Instituut voor Theoretische Fysica An Algorithmic Approach to Operator Product Expansions, -Algebras and -Strings W W Promotor: Dr. W. Troost Proefschrift ingediend tot het behalen van de graad van Doctor in de Wetenschappen door Kris Thielemans Leuven, juni 1994. arXiv:hep-th/9506159v1 23 Jun 1995 Jun 23 arXiv:hep-th/9506159v1 Notations ¯ d d Abstract ∂, ∂ : dz , dz¯. δ : infinitesimal transformation with ε(x) an infinitesimal parameter. String theory is currently the most promising theory to explain ε the spectrum of the elementary particles and their interactions. : T1T2 : : normal ordening. One of its most important features is its large symmetry group, which contains the conformal transformations in two dimensions OPA : Operator Product Algebra, subsection 2.3.3. as a subgroup. At quantum level, the symmetry group of a theory OPE : Operator Product Expansions, section 2.3. gives rise to differential equations between correlation functions of observables. We show that these Ward-identities are equiva- [T1(z)T2(z0)] : the complete OPE. lent to Operator Product Expansions (OPEs), which encode the T1(z)T2(z0) : singular part of an OPE. short-distance singularities of correlation functions with symme- try generators. The OPEs allow us to determine algebraically −n [T1T2]n : coefficient of (z z0) in the OPE [T1(z)T2(z0)], eq. (2.3.3). many properties of the theory under study. We analyse the calcu- − << c 0 2T ∂T >> : list of the operators at the singular poles in an OPE. High- lational rules for OPEs, give an algorithm to compute OPEs, and 2 | | | discuss an implementation in Mathematica. est order pole is given first. First order pole occurs last in the list. There exist different string theories, based on extensions of the n−1 T T (x ) : coefficientof (−1) ∂n−1δ(2)(x x ) in the Poisson bracket T (x)T (x ) conformal algebra to so-called W -algebras. These algebras are { 1 2}n 0 (n−1)! − 0 { 1 2 0 } generically nonlinear. We study their OPEs, with as main results subsection 2.3.5. an efficient algorithm to compute the β-coefficients in the OPEs, Am : mode of the operator A, section 2.4. the first explicit construction of the WB2-algebra, and criteria for L : is L L , L reversed order, chapter 4. the factorisation of free fields in a W -algebra. b{n} np · · · n1 {−n} An important technique to construct realisations of W -algebras g¯ : Lie algebra, see appendix B for additional conventions. is Drinfel’d-Sokolov reduction. The method consists of imposing certain constraints on the elements of an affine Lie algebra. We gˆ : Kaˇc-Moody algebra. quantise this reduction via gauged WZNW-models. This enables e0,e+,e− : generators of sl(2). us in a theory with a gauged W -symmetry, to compute exactly the correlation functions of the effective theory. ± : kernel of the adjoint of e±. K Finally, we investigate the (critical) W -string theories based on Π : projection operator ing ¯, appendix B. an extension of the conformal algebra with one extra symmetry generator of dimension N. We clarify how the spectrum of this Πhw : projection on +. K theory forms a minimal model of the W -algebra. N c, c : index limitation to generators of strictly negative, resp. positive grading, sec- tion 6.4. (a) : Pochhammer symbol, (a) = Γ(a+n) = n−1(a + j) ; a R and n N, n n Γ(a) j=0 ∈ ∈ appendix 2.A. Q Contents 1 Introduction and outline 2 3.7 Otherpackages ............................... 40 2 Conformal Field Theory and OPEs 7 4 -algebras 43 2.1 Conformaltransformations . 7 4.1W Introduction................................. 43 2.2 Correlationfunctionsandsymmetry . .. 8 4.2 Highest weight representations and minimal models . ...... 44 2.2.1 Globalconformalinvariance. 8 4.3 Consequencesofthe globalconformalgroup . ... 45 2.2.2 Wardidentities ........................... 9 4.3.1 ConsequencesforOPEs . 46 2.3 OperatorProductExpansions. 11 4.3.2 Finding the quasiprimaries in an OPE . 46 2.3.1 OPEsandWardidentities. 11 4.4 Consequencesofthefullconformalgroup . ... 48 2.3.2 ConsistencyconditionsforOPEs . 12 4.4.1 Restrictions of conformal covariance on the OPEs . ... 48 2.3.3 OperatorProductAlgebras . 15 4.4.2 Virasoro descendants in a quasiprimary basis . ... 49 2.3.4 OPA–terminology . .. .. .. .. .. .. .. .. .. .. 16 4.4.3 Virasorodescendants of the unit operator . .. 51 2.3.5 Poissonbrackets .. .. .. .. .. .. .. .. .. .. .. 17 4.4.4 FindingtheprimariesinanOPE . 52 2.4 Modealgebra ................................ 18 4.5 An overview of –algebras ........................ 52 2.5 Generatingfunctionals .. .. .. .. .. .. .. .. .. .. .. 19 4.5.1 Directconstruction.W . 52 2.6 Afewexamples............................... 21 4.5.2 Subalgebras of known –algebras ................ 53 2.6.1 Thefreemasslessscalar . 21 4.5.3 Constructing a –algebraviaarealisationW . 53 2.6.2 ThefreeMajoranafermion . 23 4.5.4 Superconformalalgebras.W . 54 2.6.3 Otherfirstordersystems . 24 4.5.5 Attempts towards a classification . 54 2.6.4 Wess-Zumino-Novikov-Witten models . 24 4.6 An example : B ............................ 54 Wc 2 2.A Appendix:Combinatorics . 26 4.6.1 The cB2-algebra......................... 54 4.6.2 CoulombGasRealisationW . 55 3 OPEs in Mathematica 28 4.6.3 Highestweightrepresentations . 57 3.1 Intentionandhistory............................ 28 4.7 Discussion.................................. 57 3.2 Designconsiderations . .. .. .. .. .. .. .. .. .. .. .. 28 4.A Appendix .................................. 57 3.3 Implementation............................... 29 3.3.1 Algorithm.............................. 29 5 Factoring out Free Fields 62 3.3.2 The internals of OPEdefs ..................... 32 5.1 Algorithmsforfactorisation . .. 62 3.3.3 Operatorhandling . 33 5.1.1 Freefermions ............................ 62 3.3.4 Performance ............................ 37 5.1.2 Symplecticbosons . 63 3.4 User’sGuide ................................ 37 5.1.3 U(1)currents............................ 64 3.5 Example : The conformal anomaly in superstring theory . ...... 39 5.2 Generatingfunctionals . 64 3.6 Futuredevelopments . .. .. .. .. .. .. .. .. .. .. .. 40 5.3 Examples .................................. 65 i 5.3.1 N =3superconformalalgebras . 65 C A Mathematica primer 117 5.3.2 N =4superconformalalgebras . 68 5.4 Discussion.................................. 70 Bibliography 120 6 –algebras and gauged WZNW models 72 6.1W Classical Drinfeld-Sokolov reduction . ..... 72 6.2 Quantumreduction............................. 74 6.2.1 GaugedWZNWModel .. .. .. .. .. .. .. .. .. 74 6.2.2 Theinducedaction. 75 6.3 An example : osp(N 2) .......................... 76 6.4 Cohomology ................................| 78 6.4.1 Computing the quantum cohomology . 78 6.4.2 Classicalcohomology. 80 6.4.3 The quantum –algebra ..................... 81 6.5 An example : osp(N 2),continuedW .................... 82 6.6 Quantum correctionsto| the extended action . .... 82 6.7 Discussion.................................. 83 7 Renormalisation factors in –Gravity 85 7.1 Introduction.................................W 85 7.2 All–orderresults .............................. 86 7.3 An example : so(N)–supergravity. 89 7.4 Semiclassical evaluation for the linear superconformalalgebras . 91 7.4.1 N =1................................ 93 7.4.2 N =2................................ 94 7.4.3 N =3, 4............................... 95 7.5 Discussion.................................. 95 8 Critical –strings 98 8.1 ThebosonicstringW ............................. 98 8.2 –strings.................................. 99 W 8.3 2,s–strings.................................101 8.4 MinimalW models and –strings.....................103 W2,s 8.4.1 The 2,4–string ..........................104 8.4.2 The W string...........................105 W2,5 8.4.3 The 2,6 string...........................106 8.5 HierarchiesofstringembeddingsW . .107 8.6 Conclusionanddiscussion . 110 A Green’s function for the Laplacian in 2d 112 B Superconventions 114 0 Chapter 1 Introduction and outline The first chapter of a recent PhD. thesis in contemporary high-energy physics number of space-time dimensions to 26. Somehow, this makes one hope that a more necessarily stresses the importance of symmetry [42, 51, 45]. The reason for this is realistic string theory would “explain” why we are living in a four-dimensional world. that symmetry is the most powerful organising principle available, and a theoretical The third surprise is that, while we started with a free theory, interactions seem also physicist wants to assume as little as possible. This has the peculiar consequence to be fixed by the action (1.1), with only one parameter T . This is in contrast with that he or she ends up making the far-reaching assumption that “nature” has the the grand-unified theories where interactions have to be put in by hand, requiring largest symmetry we are able to find. the introduction of a number of parameters that have to be fixed by comparing with A striking example is provided by string theory. The universe seems to contain experiments. A last surprise which we wish to mention, is that the spectrum of the a large number of “elementary” particles. It is an appealing idea to think of these physical states contains a particle with the correct properties for a graviton. String particles as different states of one single object. This would enable us to treat them theory thus seems to incorporate quantum gravity. This is particularly fortunate in a symmetric way. The simplest objects in every-day experience which have such because no other theory has been found
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