Conformal Invariance and Liouville Field Theory

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(1)Universidade de São Paulo Instituto de Física. Invariância Conforme e Teoria de Campo de Liouville Laura Raquel Rado Díaz. Orientador: Prof. Dr. Elcio Abdalla Dissertação de mestrado apresentada ao Instituto de Física para a obtenção do título de Mestre em Ciências. Banca Examinadora: Prof. Dr. Elcio Abdalla (IF-USP) Prof. Dr. Christian Dieter Jäkel (IME-USP) Prof. Dr. Antonio Lima-Santos (CCET-UFSCAR). São Paulo 2015.

(2) FICHA CATALOGRÁFICA Preparada pelo Serviço de Biblioteca e Informação do Instituto de Física da Universidade de São Paulo Rado Diaz, Laura Raquel Conformal invariance and Liouville field theory. São Paulo, 2015. Dissertação (Mestrado) – Universidade de São Paulo. Instituto de Física. Depto. Física Matemática Orientador: Prof. Dr. Élcio Abdalla Área de Concentração: Teoria Geral de Partículas e Campos. Unitermos: 1. Invariantes conformes; 2. Teoria de campos; 3. Funções complexas.. USP/IF/SBI-060/2015.

(3) University of São Paulo Physics Institute. Conformal Invariance and Liouville Field Theory Laura Raquel Rado Díaz. Advisor: Prof. Dr. Elcio Abdalla. Master Thesis submitted to the IFUSP Graduate Program in Physics for the Degree of Master in Physics. Approved by: Prof. Dr. Elcio Abdalla (IF-USP) Prof. Dr. Christian Dieter Jäkel (IME-USP) Prof. Dr. Antonio Lima-Santos (CCET-UFSCAR). São Paulo 2015.

(4) i. A mis padres..

(5) Acknowledments I would like to thank Prof. Elcio Abdalla, my advisor in this work, for his motivation and confidence. Thanks to this work proposal, I could study CFT and, in particular, Liouville field theory, which has awoken my interest deeply. Special acknowledgements to Jonathan Maltz for his detailed correspondence and patience to answer my questions. I would like to thank especially Prof. Gastón Giribet for his kindness, enthusiasm and motivation for helping me. Thanks for his suggestions and sharing his ideas about CFT. To IFUSP for giving me a comfortable place to study and to develope much of my interests in physics. To COSEAS-USP for giving a place to live in USP, and especially to Karla. And to José Renato Sánchez Romero, an acknowledgment is not enough.. ii.

(6) iii.

(7) Abstract In this work, we make a brief review of the Conformal Field Theory in two dimensions, in order to understand some basic definitions in the study of the Liouville Field Theory, which has many application in theoretical physics like string theory, general relativity and supersymmetric gauge field theories. In particular, we focus on the analytic continuation of the Liouville Field Theory, context in which an interesting relation with the Chern-Simons Theory arises as an extension of its well-known relation with the Wess-Zumino-Witten model. Thus, calculating correlation functions by using the complex solutions of the Liouville Theory will be crucial aim in this work in order to test the consistency of this analytic continuation. We will consider as an application the time-like version of the Liouville Theory, which has several applications in holographic quantum cosmology and in studying tachyon condensates. Finally, we calculate the three-point function for the Wess-Zumino-Witten model for the standard Kac-Moody level k > 2 and the particular case 0 < k < 2, the latter has an interpretation in time-dependent scenarios for string theory. Here we will find an analogue relation we find by comparing the correlation function of the time-like and space-like Liouville Field Theory. Keywords Conformal Field Theory, Liouville Field Theory, Analytic Continuation, Time-like Liouville Theory, Wess-Zumino-Witten model.. iv.

(8) Resumo Neste trabalho, nós fazemos uma breve revisão da Teoria de Campo Conforme em duas dimensões, a fim de entender algumas defini¸co˜es básicas do estudo da Teoria de Campo de Liouville, que tem muitas aplica¸co˜es em f´ısica teórica como a teoria das cordas, a relatividade geral e teorias de campo de calibre supersimétricas. Em particular, vamos nos concentrar sobre a continua¸cão anal´ıtica da Teoria de Campo de Liouville, contexto no qual uma interessante rela¸cão com a Teoria de Chern-Simons surge como uma extensão de sua rela¸caõ conhecida com o modelo de Wess-Zumino-Witten. Assim, o cálculo das fun¸co˜es de correla¸caõ usando as solu¸cões complexas da Teoria Liouville será o objectivo fundamental neste trabalho, a fim de testar a consistência da continua¸cão anal´ıtica. Vamos considerar como uma aplica¸cão a versão time-like da Teoria de Liouville, que tem várias aplica¸cões em cosmologia quântica holográfica e no estudo de condensados de tachyon. Finalmente, calculamos a fun¸cão de três pontos para o modelo de Wess-Zumino-Witten no n´ıvel de Kac-Moody k > 2 e o caso particular 0 < k < 2, este u ´ltimo tem uma interpreta¸caõ em cenários dependentes do tempo para a teoria das cordas. Aqui nós vamos encontrar uma rela¸caõ análoga ao que temos para a fun¸caõ de correla¸caõ do space-like e time-like na Teoria de Campo de Liouville. Keywords Teoria de Campo Conforme, Teoria de Campo de Liouville, Continua¸caõ Anal´ıtica , Teoria de Liouville Time-like, Modelo de Wess-Zumino-Witten.. v.

(9) Contents Acknowledments. ii. Abstract. iv. Abstract. v. Índex. vi. List of Figures. ix. List of Tables. x. 1 Introduction. 1. 2 Conformal Field Theories. 3. 2.1. Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.2. Conformal symmetry in 2-dimensions . . . . . . . . . . . . . . . . . . . . .. 9. 2.3. Conformal Invariance in Quantum Field Theory . . . . . . . . . . . . . . .. 13. 2.4. Conformal theories in 2-dimensions . . . . . . . . . . . . . . . . . . . . . .. 15. 2.5. The stress-tensor and the Virasoro algebra . . . . . . . . . . . . . . . . . .. 17. 3 Liouville Theory and Correlation Functions. 20. 3.1. Free Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 3.2. Coupling the Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3.3. Computing the Central Charge . . . . . . . . . . . . . . . . . . . . . . . .. 24. 3.4. Primary Fields and their Conformal Dimension . . . . . . . . . . . . . . .. 25. vi.

(10) vii 3.5. Adding the Liouville Exponential . . . . . . . . . . . . . . . . . . . . . . .. 26. 3.6. The semi-classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. 3.7. Semiclassical Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 3.8. DOZZ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. 3.9. Four-Point Functions and Degenerate Operators . . . . . . . . . . . . . . .. 34. 4 Complex solutions of the Liouville Theory 4.1. 4.2. General Form of Complex Solutions . . . . . . . . . . . . . . . . . . . . . .. 43. 4.1.1. Two-Point Complex Solutions . . . . . . . . . . . . . . . . . . . . .. 48. 4.1.2. Three-Point Complex Solutions . . . . . . . . . . . . . . . . . . . .. 50. Analytic Continuation and Stokes Phenomena . . . . . . . . . . . . . . . .. 55. 4.2.1. Analytic Continuation of the Two-Point Fuction . . . . . . . . . . .. 55. 4.2.2. Analytic Continuation of the Three-Point Function . . . . . . . . .. 63. 4.2.3. Three-Point Function with Light Operators . . . . . . . . . . . . .. 73. 5 Singular solutions of the Liouville Field Theory 5.1. 5.2. 78. Four-Point Functions and the Interpretation of Singular Saddle Points . . .. 78. 5.1.1. Finiteness of the ”Action” . . . . . . . . . . . . . . . . . . . . . . .. 79. 5.1.2. Multivaluedness of the Action . . . . . . . . . . . . . . . . . . . . .. 81. 5.1.3. Degenerate Four-Point Function as a Probe . . . . . . . . . . . . .. 82. Interpretation in Chern-Simons Theory . . . . . . . . . . . . . . . . . . . .. 85. 5.2.1. Liouville Solutions and Flat Connections . . . . . . . . . . . . . . .. 86. 5.2.2. Interpretation in Chern-Simons Theory . . . . . . . . . . . . . . . .. 88. 5.2.3. Liouville Primary Fields and Monodromy Defects . . . . . . . . . .. 89. 5.2.4. Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91. 6 Timelike Liouville 6.1. 43. 92. Proposal of Harlow, Maltz and Witten . . . . . . . . . . . . . . . . . . . .. 92. 6.1.1. The Timelike DOZZ formula . . . . . . . . . . . . . . . . . . . . . .. 94. 6.1.2. Semiclassical Tests of the Timelike DOZZ formula . . . . . . . . . .. 96. 6.1.3. Two-point Function . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96.

(11) viii. 6.2. 6.1.4. Three-point Function with Heavy Operators . . . . . . . . . . . . .. 97. 6.1.5. Three-point Function with Light Operators . . . . . . . . . . . . . .. 98. 6.1.6. An Exact Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. Free-field method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.1. Three-point correlation function in the free-field representation . . . 103. 6.2.2. The three-point function on the spacelike Liouville. 6.2.3. The three-point function on the timelike Liouville . . . . . . . . . . 106. . . . . . . . . . 105. 7 WZNW and the Liouville Theory. 107. 7.1. The WZNW model and the primary operators . . . . . . . . . . . . . . . . 107. 7.2. Three-point correlation function in the free-field representation. 7.3. The three-point function on the nivel k > 2 . . . . . . . . . . . . . . . . . . 111. 7.4. The three-point function on the level 0 < k < 2 . . . . . . . . . . . . . . . 112. . . . . . . 110. Conclusions. 114. A Properties of the Υb Function. 116. B Theory of Hypergeometric Functions. 119. B.1 Hypergeometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 B.2 Hypergeometric Differential Equation . . . . . . . . . . . . . . . . . . . . . 120 B.3 Riemann’s Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . 120 B.4 Particular Solutions of Riemann’s Equation Bibliography. . . . . . . . . . . . . . . . . . 121 126.

(12) List of Figures 4.1. Standard recipe to make an American football ball out of a European one.. 50. 4.2. Diangle that consisting of the two spherical triangular regions A and B. . .. 65. 5.1. Monodromy defects in Σ × I . . . . . . . . . . . . . . . . . . . . . . . . . .. 89. ix.

(13) List of Tables 3.1. Important elements from Liouville CFT . . . . . . . . . . . . . . . . . . . .. 27. 5.1. The multivaluedness of various quantities . . . . . . . . . . . . . . . . . . .. 81. x.

(14) Chapter 1 Introduction Liouville field theory has many applications on several areas of theoretical physics, including string theory [2] [3] [4], general relativity and supersymetric gauge theories [6] [5] [36] [48] [56]. In this work, we are interested in studying the analytic continuation of Liouville theory, or in other words to make complex its parameters. Thus, a principal ingredient for us will be the correlation function for three points of a product of three primary fields on the Riemann sphere, that has an exact formula knows as the DOZZ formula [55] [50]. Liouville theory is suitably parametrized in terms of a quantity b and its central charge c = 1 + 6Q2 , where Q = b + b−1 . In the semiclassical limit, when b → 0, we can consider two cases for the momentum αi , heavy operators whose αi = ηi /b with ηi fixed as b → 0. Insertions of these operators change the saddle points that dominate the functional integral; and light operators αi = σi b, with σi fixed as b → 0. These operators does not affect the saddle points in the semiclassical limit. A saddle point is simply a real solution of the classical field equations [55]. We define the physical region as that where ηi < 1/2 and P i ηi > 1. Here the semiclassical limit at the path integral is determined by a real saddle point [22]. In Chapter 2, we give a brief review of Conformal field theory (CFT), focusing on the case of two-dimensions. In order to introduce some basics concepts that we need to understand the rest of this work. [1] [9] In Chapter 3, we introduce the Liouville field theory and calculate the three-point 1.

(15) CHAPTER 1. INTRODUCTION. 2. function following the seminal work by Zamolodchikov and Zamolodchikov [55], we also compute the four-point function in a modern approach as was proposed by Teschner to obtain the DOZZ formula [50]. The relation proposed by Teschner for the structure constants will be crucial in the Chapter 6. In the Chapter 4, after making a convenient parametrization, we show the form of the complex solution of the Liouville field theory (LFT) and analyze the two-point and threepoint functions in this context. Also we show that the analytic continuation of the DOZZ formula in a restricted region of η, can be interpreted in terms of complex classical solutions [22]. In Chapter 5, we consider singular solutions coming from our last parametrization, which will be interpreted. It is interesting to see that those solutions are related to the Chern-Simons theory [54]. In Chapter 6, we consider the so-called time-like version of Liouville theory, that has other interesting applications. Time-like LFT has been considered in holographic quantum cosmology [15] [43], studying tachyon condensates [19] [41] [44], and time-dependent scenarios for string theory [24]. Here review the proposal given by Harlow, Maltz and Witten [22] as well as the well-known free field method for the DOZZ formula. Finally, in the Chapter 7, we study briefly the Wess-Zumino-Novikov-Witten (WZNW) model focusing on the computation of the three-point function for k > 2 and 0 < k < 2, by means of the free-field method..

(16) Chapter 2 Conformal Field Theories Conformal field theories (CFT’s) are a particular class of field theories characterized by a type of symmetry transformation whose net effect on the metric is to multiply it by a positive function, thus preserving angles. The approach to study conformal field theories is somewhat different from the usual approach for quantum field theories. Indeed, instead of starting with a classical action for the fields and quantising them via the canonical quantisation or the path integral method, one employs the symmetries of the theory [9]. For example, in statistical mechanics as a system approaches a second order phase transition its correlation length diverges. At the critical point the theory possesses no dimensional parameter and is scale or dilatation invariant; in two dimensions the field theory describing the critical point turns out to be, not just dilatation invariant, but conformally invariant. The operator product φ(x)φ(0) of some quantum field theories becomes independent of mass in the limit of small x. This has led to the suggestion that the elementary constituents of matter, which are the relevant degrees of freedom at very small distance, may be described by theories with conformally invariant small distance limits. The short distance behavior of field theories is intimately related to their renormalisation properties. The renormalisation properties of correlation functions are constrained by the need to obey the Callan-Symanzik renormalisation group equations. A necessary, though insufficient, condition for a theory to possess conformal invariance is that the renormalisation group flow has a fixed point; this means that the Callan-Symanzik function β(g) has a zero [37]. 3.

(17) CHAPTER 2. CONFORMAL FIELD THEORIES. 4. The main object of a field theory is the calculation of correlation functions, which are the physically measured quantities. In general, non-trivial theories with conformal invariant correlation functions are very difficult find. However, in two dimensions, where the conformal group is infinite dimensional, the situation is somewhat better [1].. 2.1. Conformal Invariance. Let us consider a d dimensional space-time with flat metric gµν = ηµν . The conformal group is defined as a subgroup of the coordinate transformations that leaves the metric tensor invariant up to a scale, i.e. 0 gµν (x) → gµν (x0 ) = Λ(x)gµν (x),. (2.1). ∂x0ρ ∂x0σ gρσ (x ) µ = Λ(x)gµν (x). ∂x ∂xν 0. 0. The conformal transformation is locally equivalent to a rotation and a dilatation, that also leave the metric invariant up to a scale. The conformal transformation preserves the angle between two crossing curves. The set of conformal transformations forms a group which has the Poincaré group as a subgroup corresponding to the scale factor Λ(x) = 1.. Under an infinitesimal coordinate transformation x0ρ = xρ + ρ (x) + O(2 ). Such a transformation induces a variation of the metric tensor of the form 0ρ 0σ ∂ρ ∂σ 0 0 ∂x ∂x ρ 2 σ 2 ηρσ (x ) µ = ηρσ δµ + µ + O( ) δν + ν + O( ) , ∂x ∂xν ∂x ∂x σ ρ ∂ ∂ = ηµν + ηµσ ν + ηρν µ + O(2 ), ∂x ∂x ∂µ ∂ν = ηµν + + µ + O(2 ). ν ∂x ∂x. (2.2). To get a conformal transformation, the change of variables must satisfy the requirement (2.1), implying ∂µ ν + ∂ν µ = f (x)ηµν ,. (2.3).

(18) CHAPTER 2. CONFORMAL FIELD THEORIES. 5. where f (x) is determined by taking the trace on both sides of last equation η µν (∂µ ν + ∂ν µ ) = f (x)d, 2 f (x) = ∂ · . d. (2.4). So we find the following restriction on the infinitesimal transformation to be conformal 2 ∂µ ν + ∂ν µ = (∂ · )ηµν , d. (2.5). These are the Killing-Cartan equations and the functions µ satisfying these equations are called the conformal Killing vectors. Other useful relations are (d − 1)(∂ · ) = 0, 2∂µ ∂ν ρ =. 2 (−ηµν ∂ρ + ηρµ ∂ν + ηνρ ∂µ )(∂ · ). d. (2.6) (2.7). After having obtained the condition for the infinitesimal transformations that satisfy (2.1), let us determine the conformal group in the case of dimension d > 2. We see that (2.6) implies that (∂ · ) is at most linear in xµ , then it follows that µ is at most quadratic in xν and we can introduce the ansatz µ = aµ + bµν xν + cµνρ xν xρ ,. (2.8). where aµ , bµν , cµνρ 1 are constants.. • The term aµ describes infinitesimal translations x0µ = xµ + aµ with generator is the momentum operator Pµ = −i∂µ . • If we insert the last expression in (2.5), we get ∂µ (aν + bνµ xµ + cνµρ xµ xρ ) + ∂ν (aµ + bµν xν + cµνρ xν xρ ) 2 µ ∂ (aµ + bµν xν + cµνρ xν xρ ) ηµν , d 2 ρ (∂ bρσ xσ )ηµν , = d 2 ρσ = (η bρσ )ηµν , d. = bµν + bνµ bµν + bνµ.

(19) CHAPTER 2. CONFORMAL FIELD THEORIES. 6. bµν can be split a symmetric and antisymmetric part bµν = αηµν + mµν , where mµν = −mνµ and corresponds to infinitesimal rotations x0µ = (δνµ + mµν )xν , for which the generator is the angular momentum operator Lµν = i(xµ ∂ν − xν ∂mu ). The symmetric term αηµν describes infinitesimal scale transformations x0µ = (1 + α)xµ with generator D = −ixµ ∂µ . • The quadratic term can be studied using (2.7), ∂µ ∂ν (aρ + bρσ xσ + cρσγ xσ xγ ) = ∂µ {bρσ δνσ + cρσγ δνσ xγ + cρσγ xσ δνγ } , = cρσγ δνσ δµγ + cρσγ δµσ δνγ , = 2cρµν .. (2.9). ∂ · = ∂ µ {aµ + bµσ xσ + cµσγ xσ xγ } , = bµσ ∂ µ xσ + cµσγ (∂ µ xσ )xγ + cµσγ xσ (∂ µ xγ ), = bµµ + 2cµµγ xγ .. (2.10). Replacing (2.9) and (2.10) in (2.7) 1 (−ηµν ∂ρ + ηρµ ∂ν + ηνρ ∂µ ) cµµγ xγ , d. 1 λ = cλγ −ηµν δργ + ηρµ δνγ + ηνρ δµγ , d. 1 ηρµ cλλν + ηνρ cλλµ − ηµν cλλρ , = d. cρµν =. if bµ = d1 cλλµ cµνρ = ηµν bρ + ηµρ bν − ηνρ bµ . These transformations are called Special Conformal Transformations (SCT) with the following infinitesimal form x0µ = xµ + 2(x · b)xµ − (x · x)bµ , for which the generator is written as Kµ = −i(2xµ xν ∂ν − (x · x)∂µ ).. (2.11).

(20) CHAPTER 2. CONFORMAL FIELD THEORIES. 7. We identified the infinitesimal conformal transformation but, in order to determine the conformal group, we will need the finite version of the last transformation which is x0µ =. x µ − bµ x 2 . 1 − 2(b · x) + b2 x2. (2.12). Summarizing, x → x0µ = xµ + aµ , x → x0µ = Λµν xµ. Λνµ ∈ SO(d + 1, 1) ,. x → x0µ = αxµ , x µ − bµ x 2 x → x0µ = . 1 − 2b · x + b2 x2. (2.13). The first and second lines correspond to the Poincaré transformations. The third is a dilatation and the fourth are named special conformal transformations. Comutation Rules: • [D, Pµ ] = (−ixρ ∂ρ )(−i∂µ ) − (−i∂µ )(−ixρ ∂ρ ), = −xρ ∂ρ ∂µ + δµρ ∂ρ + xρ ∂ρ ∂µ = i(−i∂µ ), [D, Pµ ] = iPµ .. (2.14). • [D, Lµν ] = (−ixρ ∂ρ ) {i(xµ ∂ν − xν ∂µ )} − {i(xµ ∂ν − xν ∂µ )} (−ixρ ∂ρ ), = xρ δµρ ∂ν + xρ xµ ∂ ρ ∂ν − xρ δνρ ∂µ − xρ xν ∂ ρ ∂µ , −xµ δνρ ∂ρ − xµ xρ ∂ν ∂ρ + xν δµρ ∂ρ + xν xρ ∂µ ∂ρ , [D, Lµν ] = 0.. (2.15). . • [D, Kµ ] = (−ixρ ∂ρ ) 2xµ xν ∂ν − x2 ∂µ + 2xµ xν ∂ν − x2 ∂µ ) (xρ ∂ρ ), = −2xρ δµρ xν ∂ν − 2xρ xµ δρν ∂ν − 2xρ xµ xν ∂ρ ∂ν +xρ ∂ρ (x2 )∂µ + xρ x2 ∂ρ ∂µ + 2xµ xν δνρ ∂ρ +2xµ xν xρ ∂ν ∂ρ − x2 δµρ ∂ρ − x2 xρ ∂µ ∂ρ , = −2xµ xν ∂ν + x2 ∂µ , [D, Kµ ] = −iKµ .. (2.16).

(21) CHAPTER 2. CONFORMAL FIELD THEORIES. 8. . . • [Kµ , Pν ] = − 2xµ xρ ∂ρ − x2 ∂µ {∂ν } + ∂ν 2xµ xρ ∂ρ − x2 ∂µ ) , = −2xµ xρ ∂ρ ∂ν + x2 ∂µ ∂ν + 2ηµν xρ ∂ρ + 2xµ δνρ ∂ρ +2xµ xρ ∂ρ ∂ν − 2xν ∂µ − x2 ∂µ ∂ν , = 2ηµν xρ ∂ρ + 2xµ ∂ν − 2xν ∂µ , [Kµ , Pν ] = 2i {ηµν D − Lµν } . • [Kρ , Lµν ] =. . (2.17). . 2xρ xλ ∂λ − x2 ∂ρ {xµ ∂ν − xν ∂µ } − {xµ ∂ν − xν ∂µ } 2xρ xλ ∂λ − x2 ∂ρ ,. = 2xρ xλ ∂λ (xλ ηµλ )∂ν + 2xρ xµ xλ ∂λ ∂µ − 2xρ xλ ∂λ (xλ ηνλ )∂µ −2xρ xν xλ ∂λ ∂µ − x2 ∂ρ (xρ ηµρ )∂ν − x2 xµ ∂ν ∂ρ +x2 ∂ρ (xρ ηνρ )∂µ + x2 xν ∂µ ∂ρ − 2xµ ∂ν (xν ηνρ )xλ ∂λ −2xµ xρ δνλ xλ − 2xν xρ xλ ∂λ ∂ν + 2xµ xν ∂ρ +x2 xµ ∂ν ∂ρ + 2xν ∂µ (xµ ηµρ )xλ ∂λ + 2xν xρ δµλ xλ +2xν xρ xλ ∂λ ∂µ − 2xν xµ ∂ρ − x2 xν ∂µ ∂ρ + ..., = ηµρ (2xν xλ ∂λ − x2 ∂ν ) − ηνρ (2xµ xλ ∂λ − x2 ∂µ ) + 2xρ (xµ ∂ν − xν ∂µ ), [Kρ , Lµν ] = i {ηµρ Kν − ηνρ Kµ } .. (2.18). We also get some other ones • [Pρ , Lµν ] = i(ηρµ Pν − ηρν Pµ ). • [Lµν , Lρσ ] = i(ηνρ Lµσ + ηνρ Lµσ + ηµσ Lνρ − ηµρ Lνσ − ηνσ Lµρ ).. (2.19) (2.20). In order to put the above commutation rules into a simpler form, we define the following generators Jµν = Lµν , J−1,0 = D, 1 (Pµ − Kµ ) , J−1,µ = 2 1 J0,µ = (Pµ + Kµ ) , 2 where Jab = −Jba , with {a, b ∈ {−1, 0, 1, ...d}}. These new generators obey the SO(d+1, 1) commutation relations: [Jab , Jcd ] = i {ηad Jbc + ηbc Jad − ηac Jbd − ηbd Jac } .. (2.21).

(22) CHAPTER 2. CONFORMAL FIELD THEORIES. 9. The metric used above is ηab = diag(−1, 1, ..., 1) for Euclidean d-dimensional space Rd,0 . For the case of dimension d = p + q > 3 the conformal group or Rp,q is SO(p + 1, q + 1). Notice that the Poincare group plus with dilations form a subgroup of the full conformal group. This means that a theory invariant under translations, rotations and dilations is not necessarily invariant under special conformal transformation (SCT).. 2.2. Conformal symmetry in 2-dimensions. Conformal invariance in d = 2 is more interesting since it implies stronger restrictions on the correlation functions. We begin studying the conformal group of two dimensions. Using the condition (2.5) for invariance, in two dimensions we have ∂0 0 = +∂1 1 ,. ∂0 1 = −∂1 0 .. These equations are similar to the Cauchy-Riemann equations for holomorphic complex functions. We introduce complex variables in the following way: z = x0 + ix1 = 0 + i1 ∂z = 21 (∂0 − i∂1 ), z¯ = x0 + ix1 ¯ = 0 − i1 ∂z¯ = 12 (∂0 + i∂1 ). We express the infinitesimal conformal transformations as z → f (z) and z¯ → f¯(¯ z ),. (2.22). which implies that the metric tensor transforms as ds2 = dzd¯ z 7→. ∂f ∂f dzd¯ z. ∂z ∂ z¯. Every analytic map on the complex plane is known to be conformal and to preserve the angles. The conformal algebra in d = 2 is infinite dimensional as can be seen from an infinitesimal transformation z 0 → z + (z), then we perform a Laurent expansion of (z) around say z = 0, z 0 = z + (z) = z +. X. z¯0 = z¯ + ¯(¯ z ) = z¯ +. X. n (−z n+1 ),. n∈Z. n∈Z. ¯n (−¯ z n+1 )..

(23) CHAPTER 2. CONFORMAL FIELD THEORIES. 10. We have also introduced the generators ln = −z n+1 ∂z ,. ¯ln = −¯ z n+1 ∂z¯. (n ∈ Z).. (2.23). Let us calculate some commutators • [lm , ln ] = z m+1 ∂z (z n+1 ∂z ) − z n+1 ∂z (z m+1 ∂z ) = (n + 1)z m+n+1 ∂z − (m + 1)z m+n+1 ∂z = (m − n)lm+n , • ¯lm , ¯ln = (m − n)¯lm+n , • lm , ¯ln = 0.. (2.24). Since we can show that the holomorphic and antiholomorphic parts commute, the algebra is a direct sum of two isomorphic subalgebras and we can treat z and z¯ as independent. The algebra (2.24) is named de-Witt algebra.. There are only three generators (2.23) that are globally defined on the Riemann sphere S 2 = C ∪ ∞. This set of conformal transformations correspond to the special conformal group with associated subalgebra given l−1 , l0 , l1 . It can be seen that l−1 generate the translations, l1 generate the special conformal transformations and l0 + ¯l0 generate the dilatations while l0 − ¯l0 generate the rotations on the real plane. The eigenvalue of ¯ the operator l0 (¯l0 ) is called the holomorphic (antiholomorphic) conformal dimension h(h). The conformal dimension (or weight) ∆ and the spin s are respectively the eigenvalues ¯ and s = h − h. ¯ ∆=h+h. The finite form of the global conformal transformations is f (z) =. az + b , cz + d. (2.25). with a, b, c, d ∈ C. For these transformations to be invertible, we have to require that ad − bc is non zero. So we can scale that constant such that ad − bc = 1. Note that the expression is invariant under (a, b, c, d) 7→ (−a, −b, −c, −d). These are the three global holomorphic and the three global antiholomorphic conformal transformations and form.

(24) CHAPTER 2. CONFORMAL FIELD THEORIES. 11. the Mobius group SL(2, C)/Z2 ≈ SO(3, 1) in correspondence with the conformal transformation in d > 2 explained in the previous section, where there are only global conformal transformations.. Virasoro Algebra: V irasoro = W itt. L. C The central extension g˜ = g. Lie algebra g by C is characterized by the commutations rules, [˜ x, y˜]g˜ = [x, y]g + c p(x, y) [˜ x, c]g˜ = 0. x, y ∈ g,. [c, c]g˜ = 0. c ∈ C,. x˜, y˜ ∈ g˜,. where p : g × g → C is bilinear (linear to ”g and g”). For the deWitt algebra, h i ˆ m, L ˆ n = (m − n)L ˆ m+n + cm,n , L. m, n ∈ Z.. Using the Jacobi identity, we find i hh i i i i hh i hh ˆ p, L ˆm , L ˆ n = 0, ˆm + L ˆ n, L ˆp , L ˆp + L ˆ m, L ˆn , L L (m − n)cm+n,p + (n − p)cn+p,m + (p − m)cp+m,n = 0. The algebra does not change if one adds a constant operator, ˆm → L ˆ 0m = L ˆ m + b(m). L Proof. [L0m , L0n ] = [Lm + b(m), Ln + b(n)] = [Lm , Ln ] , = (m − n)Lm+n + cm,n , = (m − n)L0m+n − (m − n)b(m + n) + cm,n , = (m − n)L0m+n + c0m,n . Therefore c0m,n = cm,n − (m − n)b(m + n). Choosing  cm,0 0 b(m) = m → cm,0 = 0  without lost b(0) = c−1,1 → c0 = 0 of generality 2. −1,1. L. C of a.

(25) CHAPTER 2. CONFORMAL FIELD THEORIES. 12. thus, from the beginning one can choose cm,0 for all m and c−1,1 = 0. From the Jacobi identity we find (m − n)cm+n,0 + ncn,m + (−m)cm,n = 0. Since the algebra has to be satisfied, we have cm,n = −cn,m , then, (m + n)cm,n = 0 ⇒ cm,n = c(m)δm,−n . In order to find the expression for c(m), we put p = −(m + 1) and n = 1 in the Jacobi identity, (m − 1)cm+1,−(m+1) + (m + 2)c−m,m + (−(2m + 1))c1,1 = 0, (m − 1)c(m + 1) + (m + 2)c(−m) = 0, (m − 1)c(m + 1) − (m + 2)c(m) = 0,   c(1) = 0 m+2 c(m + 1) = c(m) c(0) = 0, m−1 for m > 1 4 c(2) = 1 5 c(3) = •c(4) = 2 6 •c(5) = c(4) = 3 •c(3) =. Solution: c(m) =. (m+1)! c(2), 3!(m−2)!. 4 3·2 4! c(2) = c(2), 1 3·2 3! 5 4 3·2 5! · · c(2) = c(2), 2 1 3·2 3!2! 6 5 4 3·2 6! · · · c(2) = c(2). 3 2 1 3·2 3!3!. defining c(2) = 2c , c(m) =. m(m2 − 1) · c. 12. Finally h. i ˆ ˆ ˆ m+n + c m(m2 − 1)δm+n,0 . Lm , Ln = (m − n)L 12. The central extension of the Witt algebra has the following form c [lm , ln ] = (m − n)lm+n + m(m2 − 1), 12 ¯lm , ¯ln = (m − n)¯lm+n + c m(m2 − 1), 12 ¯lm , ln = 0..

(26) CHAPTER 2. CONFORMAL FIELD THEORIES. 13. This is the quantum version of the Witt algebra. We can note that cm,n = 0 for m, n = −1, 0, 1. It is still true that l−1 generates translation, l0 generates dilations and rotations and l1 generates the special conformal transformations. So {l−1 , l0 , l1 }V IR are generators of SL(2, C)/Z2 transformations.. 2.3. Conformal Invariance in Quantum Field Theory. There are certain functions that are invariant under the transformations (2.13), which are important in the construction of n-point correlation functions. Invariance under translations and rotations enforce these functions to depend on the relative distance between pairs of different points |xi − xj |. Scale invariance allows these functions to be only quotients between these distances,. |xi −xj | |xk −xl |. . Finally, special conformal transformations invariance. imposes that these functions depend only on the expressions |x1 − x2 | |x3 − x4 | , |x1 − x3 | |x2 − x4 |. |x1 − x2 | |x3 − x4 | , |x2 − x3 | |x1 − x4 |. (2.26). called anharmonic ratios or cross-ratios. Conformal invariance at the quantum level not only implies invariance of the action but also invariance of the measure of path integrals. That means that the expectation value of the trace of the energy-momentum tensor must be zero. We show how conformal invariance is imposed on n-point correlation functions. Under 0. a conformal transformation x → x a field φ(x) of spin zero transforms as

(27) 0

(28) −∆/d

(29) ∂x

(30)

(31) φ(x) → φ0 (x0 ) =

(32)

(33) φ(x), (2.27) ∂x

(34)

(35) 0

(36)

(37) is the Jacobian corresponding to the where ∆ is the conformal dimension of φ and

(38) ∂x ∂x coordinate transformation. Fields transforming in this way are named quasi-primaries. Correlation functions of a theory covariant under the transformation (2.27) must satisfy

(39) 0

(40) ∆1 /d

(41) 0

(42) ∆n /d

(43) ∂x

(44)

(45) ∂x

(46)

(47)

(48)

(49) ... hφ1 (x01 )...φn (x0n )i . (2.28) hφ1 (x1 )...φn (xn )i =

(50)

(51)

(52)

(53)

(54) ∂x x=x1 ∂x x=xn The expectation value hOi is defined as h 0 | O | 0 i, where | 0 i is the vacuum. By definiR tion hOi = Dφ e−S O , where S is the Euclidean action..

(55) CHAPTER 2. CONFORMAL FIELD THEORIES. 14. The vacuum | 0 i must be invariant under the conformal group. Due to (2.28) a 2-point correlation function invariant under translations, rotations and dilatations satisfies

(56) 0

(57) ∆1 /d

(58) 0

(59) ∆2 /d

(60) ∂x

(61)

(62) ∂x

(63)

(64)

(65)

(66) hφ1 (x1 )φ2 (x2 )i =

(67)

(68) hφ1 (x01 )φ2 (x02 )i . ∂x

(69) x=x1

(70) ∂x

(71) x=x2 If we have a dilatation transformation, that is, x → λx hφ1 (x1 )φ2 (x2 )i = λ∆1 +∆2 hφ1 (λx1 )φ2 (λx2 )i . For rotation and translation invariance hφ1 (x1 )φ2 (x2 )i = f (|x1 − x2 |), f (x) . → f (λx) = λ∆1 +∆2. Therefore hφ1 (x1 )φ2 (x2 )i =. C12 |x1 − x2 |∆1 +∆2. .. (2.29). Recall that the Jacobian for dilations and special conformal transformations is given by

(72) 0

(73)

(74) ∂x

(75) 1 1

(76)

(77)

(78) ∂x

(79) = (1 − 2b · x + b2 x2 )d = γ d , 1 hφ1 (x1 )φ2 (x2 )i = ∆1 ∆2 hφ1 (x01 )φ2 (x02 )i , γ γ C12 |x1 − x2 |∆1 +∆2. ∆1 +∆2. C12 (γ1 γ2 ) 2 = ∆1 ∆2 , γ1 γ2 |x1 − x2 |∆1 +∆2. if ∆1 = ∆2 = ∆, or must be 0 if ∆1 6= ∆2 . C12 is a constant that depends on the normalization of the fields. In other words, two quasi-primary fields are correlated only if they have the same scaling dimension. The 3-point functions are as follows, hφ1 (x1 )φ2 (x2 )φ3 (x3 )i =. C123 , a b x12 x23 xc13. where a, b, c is restricted such that a + b + c = ∆1 + ∆2 + ∆3 . As in the previous case, we have now C123 C123 (γ1 γ2 )a/2 (γ2 γ3 )b/2 (γ1 γ3 )c/2 = , xa12 xb23 xc13 xa12 xb23 xc13 γ1∆1 γ2∆2 γ3∆3.

(80) CHAPTER 2. CONFORMAL FIELD THEORIES. 15. the covariance under the special conformal transformations requires that a + c = 2∆1 , a + b = 2∆2 , b + c = 2∆3 . Therefore the 3-point function depends only on C123 : hφ1 (x1 )φ2 (x2 )φ3 (x3 )i =. C123 ∆1 +∆2 −∆3. |x1 − x2 |. |x2 − x3 |∆2 +∆3 −∆1 |x1 − x3 |∆1 +∆3 −∆2. . (2.30). But (n ≥ 4)-functions are not completely determined by conformal invariance. We know them up to a factor depending on the cross-ratios (2.26). In general, the 4-point functions are hφ1 (x1 )...φ4 (x4 )i = f. |x1 − x2 | |x3 − x4 | |x1 − x2 | |x3 − x4 | , |x1 − x3 || x2 − x4 | |x2 − x3 | |x1 − x4 |. Y 4. |xi − xj |∆/3−∆i −∆j ,. i<j. (2.31) where ∆ =. 2.4. P4. i=1. ∆i and f is an arbitrary function of cross ratios.. Conformal theories in 2-dimensions. The scale invariance in d = 2 is equivalent to conformal invariance at the classical level. But for a theory to be conformally invariant at the quantum level the integration measure of path integrals must be also invariant. Furthermore, if the theory is to be used in string theory, conformal invariance must be preserved by integrating over every two dimensional manifold. Considering conformally invariant theory at the quantum level, we analyze the restrictions to the correlation functions. Under a conformal map z → w(z), z¯ → w(¯ ¯ z ), the line element ds2 = dzd¯ z transforms as 2. ds →. . ∂f ∂z. ¯ ∂f ds2 . ∂ z¯. We can generalize this transformation to the form h ¯h¯ ∂f ∂f Φ(z, z¯) → Φ(f (z), f¯(¯ z )). ∂z ∂ z¯. (2.32).

(81) CHAPTER 2. CONFORMAL FIELD THEORIES. 16. This property under conformal transformations defines the primary fields Φ of con¯ Fields not transforming in this way are known as secondary fields. formal weight (h, h). A primary field is always a quasi-primary field because it satisfies (2.27) under global transformations. A secondary field may or may not be a quasi-primary field. Conformal invariance forces the n-point functions of n primary fields to transform as −hi ¯−h¯ i n . Y df df Φ1 (f1 , f¯1 )...Φn (fn , f¯n ) = hΦ1 (z1 , z¯1 )...Φn (zn , z¯n )i . dz f =fi d¯ z f¯=f¯i i=1. (2.33). This relation fixes the form of 2 and 3-point functions. In contrast to the previous section, ¯ i. primary fields can have spin. The spin value is incorporated in the difference s = hi − h Therefore, 2-point functions are hΦ1 (z1 , z¯1 )Φ2 (z2 , z¯2 )i =. (z1 −. C12 2h z2 ) (¯ z1. − z¯2 )2h¯. ,. (2.34). ¯1 = h ¯ 2 = h. ¯ In any other case it is zero. The sum of the spins within if h1 = h2 = h and h the correlation function must be zero. The 3-point functions are hΦ1 (z1 , z¯1 )Φ2 (z2 , z¯2 )Φ3 (z3 , z¯3 )i = C123. 1. 1. h1 +h2 −h3 h2 +h3 −h1 h3 +h1 −h2 z12 z23 z13. ¯ 1 +h ¯ 2 −h ¯3 h ¯ ¯ ¯ h ¯ 3 +h ¯ 1 −h ¯2 , h z¯12 z¯232 +h3 −h1 z¯13. (2.35) where zij = zi − zj . Constants C12 and C123 can also be determined due to the conformal invariance. The 4-point functions are not completely fixed. In d = 2 there are only three independent cross-ratios invariant under global conformal transformations. They can be written as x=. z12 z34 , z13 z24. 1−x=. z14 z23 , z13 z24. x z12 z34 = . 1−x z14 z23. (2.36). Therefore, the functional form of a 4-point function is hΦ1 (x1 )...Φ4 (x4 )i = f (x, x¯). 4 Y. h. zij3. −hi −hj. ¯ h. z¯ij3. ¯ i −h ¯j −h. ,. (2.37). i<j. where h =. P4. i=1. ¯ = P4 h ¯ hi and h i=1 i .. Radial quantization To probe carefully the consequences of conformal invariance in a two dimensional quantum field theory, we explain some of the details of the quantization procedure. We.

(82) CHAPTER 2. CONFORMAL FIELD THEORIES. 17. can consider it either on a plane or on a cylinder with coordinates σ ∈ [0, 2π] and τ ∈ [−∞, +∞]. The first one is a Lorentzian variety R2 and the second one is an Euclidean variety R × U (1). The first step is to perform a Wick rotation σ ± = τ ± σ → −i(τ ± iσ) where τ and σ are two space-time coordinates. Next step is to define complex coordinates on the cylinder z 0 = τ − iσ, z¯0 = τ + iσ.. (2.38). These coordinate transformations applied respectively to the left and the right moving fields in two-dimensional Minkowski space-time transform them into Euclidean fields depending on holomorphic or anti-holomorphic coordinates. 0. z = ez = eτ −iσ , 0. z¯ = ez¯ = eτ +iσ .. (2.39). The infinite past and future in the cylinder (τ = ∓∞) are mapped to the points |z| = 0, ∞ on the plane. Equal time lines in the cylinder (τ = cte) correspond to circles with center at the origin in the plane, and the time inversion in the cylinder (τ → −τ ) correspond to the map z → 1/¯ z in the complex plane. It is important to realize that dilatations in the complex plane correspond to temporal translations in the cylinder. Consequently, the generator of dilatations in the plane can be thought as the Hamiltonian of the system and the Hilbert space is built from concentric circles. This method of defining a quantum theory in the plane is called radial quantization.. 2.5. The stress-tensor and the Virasoro algebra. The Noether theorem establishes that local transformations of coordinates are generated by charges built from the energy-momentum tensor Tµν , and in d = 2 every such transformation is a conformal transformation. In this case the energy-momentum tensor is not only symmetric but also traceless and it results in a two component tensor that can be.

(83) CHAPTER 2. CONFORMAL FIELD THEORIES. 18. written as T (z) ≡ Tzz (z),. T (¯ z ) ≡ T¯z¯z¯(¯ z ),. (2.40). where the holomorphic and the anti-holomorphic parts are separated. The operator T (z) is related to the trace of the energy-momentum tensor. The following expansion of the energy-momentum tensor T (z) =. X. z −n−2 Ln ,. T¯(¯ z) =. n∈Z. X. ¯ n, z¯−n−2 L. (2.41). n∈Z. is important since the modes Ln do generate the local conformal transformations at the quantum level in the equivalence to what the generators (2.23) do at the classical level. They satisfy the famous Virasoro algebra , c [Ln , Lm ] = (n − m)Ln+m + n(n2 − 1)δn+m,0 , 12 ¯ Ln , Lm = 0, ¯ n, L ¯ m = (n − m)L ¯ n+m + c n(n2 − 1)δn+m,0 , L 12. (2.42). where the parameter c is the central charge. The correlation functions can have singularities when the positions of two or more fields coincide. The operator product expansion (OPE) is the representation of a product of two operators inserted in points z and w given by a finite sum of terms, each being a single operator, well defined as z → w, multiplied by a function of z − w. The OPE between the energy-momentum tensor and a primary field φ of conformal dimension h is h 1 φ(w, w) ¯ + ∂w φ(w, w) ¯ + ... (2.43) 2 (z − w) z−w ¯ and w and w¯ interand similarly for the anti-holomorphic part with h substituted by h T (z)φ(w, w) ¯ =. changed. The dots ... express that regular terms have been ignored in the right side of the equation. One last advantage of radial quantization to be mentioned is the relation that it establishes between commutators and OPEs. Let us consider two operators A and B that are integrals over space at fixed time of the fields a(z) and b(z), respectively, I I A = a(z)dz, B = b(z)dz,. (2.44).

(84) CHAPTER 2. CONFORMAL FIELD THEORIES. 19. where the contours of integration are circles centered around the origin. If we perform an equal time commutator between A and B we can express it in terms of both integrations. This calculation imposes the circles to be, one infinitesimally bigger than the other. Operating with these contours we end up with the following expression, I I [A, B] = dw dz a(z)b(w). 0. (2.45). w. In this way, only the term in 1/(z − w) of the OPE between a(z) and b(z) contributes to the commutator, by the theorem of residues. Therefore, OPEs establish equal time commutators.. An example of the above are the modes of the energy-momentum tensor, I 1 Ln = dz z n+1 T (z). 2πi. (2.46). We can deduce the Virasoro algebra if we consider the OPE between the energy-momentum tensor and itself. The central charge Not all fields satisfy the transformation law (2.32) under conformal transformations. For example a derivative of a primary field, in general have more complicated transformation properties. They are called secondary fields. Another example is the energy-momentum tensor. Its OPE with itself has the form T (z)T (w) =. c/2 2 1 + T (w) + ∂T (w) + ..., 4 2 (z − w) (z − w) z−w. (2.47). where the factor present in the (z−w)−4 term is the central charge c and its value in general will depend on the particular theory under consideration. This parameter cannot be determined by symmetry considerations. It is imposed by the behavior at short distances and it represent somehow an extensive measure of the number of degrees of freedom of the system. It is also related with the soft breaking of conformal symmetry under the introduction of a macroscopic scale via a local conformal transformation (it is trivial to see that restricting to just the global conformal group corresponding to n = −1, 0, 1 in (2.42) the central charge does not appear)..

(85) Chapter 3 Liouville Theory and Correlation Functions In 1981 A. Polyakov studied a string theory equivalent to Feynman diagram summation [40], instead of summing over line diagrams with an increasing number of loops. His idea consists in summing over closed (two-dimensional) Riemann surfaces with an increasing genus. In this context (uniformization theorem) the Liouville theory was introduced as an attempt to discover the proper measure for the path integral in the closed bosonic string formulation in a non-critical dimension. This leads to a Lagrangian that looks like Z p 1 SL = d2 ξ g˜ ∂a φ∂b φ˜ g ab + QRφ + 4πµe2bφ . (3.1) 4π We identify the following terms: kinetic term for the free scalar field, the metric g˜ is referred to as the ”reference” metric (R is its scalar curvature with coupling constant 2. Q = b + 1/b). The quantity gab = e Q φ gãb is referred to as the ”physical” metric and an exponential potential term. We begin by calculating the stress tensor T for the free field component and will complete the other terms of the Liouville Lagrangian, so we notice how it changes and interpret the terms we have added. We will also derive the form of the central charge and of the conformal dimensions of primary fields. Finally, we present the correlation functions for three- and four-points.. 20.

(86) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 3.1. 21. Free Field Theory. We consider the action of a free bosonic field, invariant under a translation φ 7→ φ + a, Z p 1 S= (3.2) d2 ξ g˜ gãb ∂a φ∂b φ. 4π The n primary fields correlator is given by the expression. 2α1 φ 2αn φ e ...e .. (3.3). In order to achieve invariance under translation (since this is one of the conformal symmetries), we demand the condition X. αi = 0,. (3.4). i. P displaying, under translation, a phase exp { i αi }. We use also the Ward Identity * + * + Z Y Y 1 2 a d ξ ∂a J (ξ) Oi (ξi ) = δ Oi (ξi ) , (3.5) − 2π i i where J a is the current associated with a transformation, Oi are a set of operators, is an infinitesimal transformation. This general relation can be applied to our case by making two simplifications. First, using the fact that, for any vector J a , Z I I I a a 2 1 ∂a J = Ja n ˆ = (J1 dξ − J2 dξ ) = −i (Jz dz − Jz¯d¯ z ), . ∂. ∂. (3.6). ∂. where Jz = 21 (J1 − iJ2 ) and Jz¯ = 21 (J1 + iJ2 ). We get, * + * + * + I I Y Y Y i i dz Jz (z, z¯) Oi (ξi ) − d¯ z Jz¯(z, z¯) Oi (ξi ) = δ Oi (ξi ) . (3.7) 2π ∂ 2π ∂ i i i Second, since we consider only conformal transformations; one can make Jz holomorphic and Jz¯ anti-holomorphic, then the contour integrals only pick up the residues of the product of the J’s with the first operator. Thus, for our case, a translation induces a variation in the vertex operator δV = 2αaV , and we have * + * + * + ! I I Y Y Y X i i ¯ dz ∂φ Oi (ξi ) − d¯ z ∂φ Oi (ξi ) = 2 Oi (ξi ) αi . (3.8) 2π ∂ 2π ∂ i i i i.

(87) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 22. for infinitesimal a. Since the contours enclose all of space, and there are no operators inserted at infinity, the LHS in (3.8) is zero and for non-zero correlators, we obtain X. αi = 0.. (3.9). i. The stress-energy tensor is defined as usual, 4π δS Tαβ = − √ |η . g δg αβ αβ Let us consider the following action for free field in flat space-time Z Z 1 1 2 a d ξ ∂a φ∂ φ = − d2 ξ φ∂ 2 φ. S= 4π 4π. (3.10). (3.11). The propagator is easily obtained, and is given by the expression 1 hφ(ξ)φ(ξ 0 )i = − ln(ξ − ξ 0 )2 . 2 Using complex coordinates, the action is Z 1 ¯ S= dz d¯ z ∂φ∂φ, 4π. (3.12). (3.13). ¯ = 0. This allows us to split φ into left- and rightso that the equation of motion is ∂ ∂φ ¯ z ). The product of φ0 s can be written as moving pieces: φ(z, z¯) = φ(z) + φ(¯ ¯ z ))(φ(w) + φ( ¯ w)) ¯ z )φ( ¯ w), φ(ξ)φ(ξ 0 ) = (φ(z) + φ(¯ ¯ = φ(z)φ(w) + φ(¯ ¯. (3.14). where the mixed terms vanish in the correlation function. We also have that log(ξ − ρ)2 = log (ξ 1 − ρ1 )2 + (ξ 2 − ρ2 )2 2 2 ! 1 1 1 1 (z + z¯) − (w + w) ¯ + (z − z¯) − (w − w) ¯ = log 2 2 2i 2i = log(z − w) + log(¯ z − w). ¯. (3.15). We see that, in complex coordinates, our left-mover propagator is 1 hφ(z)φ(w)i = − log(z − w). 2. (3.16).

(88) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 23. The stress-energy tensor is 4π δS 1 Tαβ = − √ |ηαβ = ηαβ (∂φ)2 − ∂α φ∂β φ. αβ g δg 2. (3.17). Therefore, in flat space with complex coordinates we find (remembering that ds2 = (dξ 1 )2 + (dξ 2 )2 = dzd¯ z ) that Tαβ can be expressed as Tzz¯ = 0, Tzz = T (z) = −∂φ∂φ, ¯ ∂φ. ¯ Tz¯z¯ = T¯(¯ z ) = −∂φ. 3.2. (3.18). Coupling the Curvature. We can modify the above action by adding a term that couples to the curvature, Z p ab 1 S= d2 ξ g˜(˜ g ∂a φ∂b φ + QRφ). 4π. (3.19). where R is the Ricci scalar and Q is a coupling parameter. We look for a momentum conservation rule similar to (3.9). Now the action is no longer invariant under translation; it instead changes as Qa δS = 4π. Z. d2 ξ. √ gR.. (3.20). It is known that for two-dimensional compact, boundaryless, orientable manifolds, the Gauss-Bonnet theorem states that 1 4π. √ d2 ξ gR = χ,. Z. (3.21). where ξ is the Euler characteristic of the surface under consideration, which is 2 in the case of the sphere. So, we have a variation of the action equal to 2aQ, which for infinitesimal Q a adds a term 2Q h i Oi (ξi )i to the LHS of (3.8). Then the condition (3.9) becomes X. αi = Q.. i. This result can be interpreted as a background charge −Q at ”infinity”.. (3.22).

(89) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 24. The contribution to the stress tensor of this new term, QRφ, can be calculate. Since we are going to take the flat-metric limit, at the end, we drop any term that will lead to a derivative or variation of gµν . Now we calculate the contribution to the variation of the action, that is δS. 0. = = = = =. Z Q √ d2 ξ gφδR, 4π Z Q √ d2 ξ gφg µν δRµν , 4π Z. Q √ d2 ξ gφ∂µ g αβ δΓµαβ − g αµ Γτατ , 4π Z Q 1 √ µτ √ τµ 2 √ g) + g ∂τ (log g) , d ξ g(∂µ φ)δ √ ∂τ (g 4π g Z Q d2 ξ(∂µ ∂τ φ)δg µτ . − 4π (3.23). As a consequence, the contribution to Tαβ will be Tαβ 0 = Q∂α ∂β φ. In complex coordinates, it reads as 0 Tzz = Q∂ 2 φ.. (3.24). Adding (3.18) and (3.24), we find that the total stress-energy tensor is Tzz¯ = 0, Tzz = T (z) = −(∂φ)2 + Q∂ 2 φ, ¯ 2 + Q∂¯2 φ. Tz¯z¯ = T¯(¯ z ) = −(∂φ). 3.3. (3.25). Computing the Central Charge. With the previous results, we can now calculate the central charge, from (3.12) 1 + ... z−w. 2. 1 ∂ φ(z)φ(w) = − + ... (z − w)2 h∂φ(z)φ(w)i =. 1 + ... (z − w)2. 2. 6 ∂ φ(z)∂ 2 φ(w) = − + ... (z − w)4 h∂φ(z)∂φ(w)i =. (3.26).

(90) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 25. We see now from (2.47), that the central charge is the coefficient in front of the (z − w)−4 term in the OPE of T (z)T (w), −(∂φ(w))2 + Q∂ 2 φ(w). = 2 h∂φ(z)∂φ(w)i2 + Q2 ∂ 2 φ(z)∂ 2 φ(w) + ... (1 + 6Q2 )/2 = + ... (z − w)−4. T (z)T (w) =. −(∂φ(z))2 + Q∂ 2 φ(z). . (3.27). We can read off that the central charge is c = 1 + 6Q2 .. 3.4. (3.28). Primary Fields and their Conformal Dimension. Now we derive the primary fields in this theory. Using the complex euclidean coordinates z = ξ 1 + iξ 2 . The field φ(z, z¯) varies under holomorphic coordinate transformations z → w(z) as

(91)

(92) 2

(93) ∂w

(94) Q φ(w, w) ¯ = φ(z, z¯) − log

(95)

(96)

(97)

(98) . 2 ∂z. (3.29). We can see that the field φ is not scalar. Given the transformation (3.29) for φ, we construct the primary field as the exponential of φ. We have (. ¯ Vα = e2αφ(w,w). −Q !) 1 ∂w ∂ w¯ = exp 2α φ(z, z¯) + log 2 ∂z ∂ z¯ −αQ −αQ ∂w ∂ w¯ = e2αφ(z,¯z) , ∂z ∂ z¯. (3.30). and, recalling our transformation law for primary field (2.32), we identify the conformal ¯ = (αQ, αQ) of Vα . To compute the quantum conformal dimension of dimension (∆, ∆) our primaries Vα , we write our primary as Vα (z) =: e2αφ(z) :. (3.31). The OPE between the stress tensor (3.18) and primary field (3.30) is T (z)Vα (w) =. ∆α 1 Vα (w) + (L−1 Vα )(w) + ... 2 (z − w) z−w. (3.32).

(99) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 26. and we find ! ∞ X 1 j T (z)Vα (w) = −(∂φ(z)) + Q∂ φ(z) ((2α)φ(w)) j! j=0 1 1 2 2 2 3 = − 0 + 0 + (2α) · 2 h∂φ(z)φ(w)i + (2α) · 3 · 2 h∂φ(z)φ(w)i φ(w) + ... 2 6 . 2. 2 1 2 +Q 0 + (2α) ∂ φ(z)φ(w) + (2α) · 2 ∂ φ(z)φ(w) φ(w) + ... + ... 2 " ! 2 # X ∞ 1 1 1/2 1 = −(2α)2 − + Q(2α) (2αφ(w))j + ... 2z −w (z − w)2 j! j=0 2. = (−α(α − Q)). 2. 1 Vα (w) + ... (z − w)2. (3.33). We thus arrive at the expression ∆α = α(Q − α).. 3.5. (3.34). Adding the Liouville Exponential. Adding an exponential term to the action (3.19), we get Z p 1 d2 ξ g˜ gãd ∂a φ∂d φ + QRφ + 4πµe2bφ , S= 4π. (3.35). with an arbitrary parameter b. In order to preserve conformal invariance, the extra term ¯ = (1, 1). must be a marginal deformation with conformal dimensions (∆, ∆) We see that this is true if and only if b(Q − b) = 1, or Q = b + 1/b, moreover, we find that our primary conformal dimensions, stress-energy tensor, field transformation law, and central charge do not change with the addition of this extra term. The reason for this is, because our theory does not depend on the particular value of the field φ, but rather only on the form of the action. By making φ 0, the Liouville interaction term vanishes, and we find that our results are those from before the addition of the potential term. Let us summarize our results in the Table 3.1 (3.1)..

(100) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS Background Charge. Q = b + 1/b. Central Charge. c = 1 + 6Q2. Primary Field. : exp2αφ(z, z¯) :. Conformal Dimension. α(Q − α). 27. Table 3.1: Important elements from Liouville CFT. We will study this theory on a two-sphere S 2 . We take the reference metric to be the flat metric ds2 = dzd¯ z with φ = −2Qlog(r) + O(1) as r → ∞, r = |z| ,. (3.36). then the physical metric is smooth on S 2 . This ensures that φ is nonsingular at infinity with respect (3.29). The motivation for the condition (3.36) is that there is an operator insertion at infinity representing the curvature of S 2 , which has been suppressed in taking the reference metric to be flat. Though the use of a flat reference metric is convenient, with this choice there is some subtlety in computing the action; we need to introduce a boundary term. Let D be a disk of radius R, the action for large R is of the form Z I Q 1 2 2bφ d ξ ∂a φ∂a φ + 4πµe φdθ + 2Q2 log(R). SL = + 4π D π ∂D. (3.37). The last two terms ensure finiteness of the action and also invariance under R → ∞.. 3.6. The semi-classical limit. In the semi-classical limit b → 0, we study the theory (3.35) on flat space (i.e. R = 0) with the rescaled field φc = 2bφ, in terms of which the action becomes Z 1 2 SLiouville [φ] = b SL = d2 ξ ∂a φc ∂a φc + 16πµb2 eφc 16π I 1 + φc dθ + 2log(R) + O(b2 ). 2π ∂D. (3.38). The boundary condition will be φc (z, z¯) = −2log(z z¯) + O(1), |z| → ∞.. (3.39).

(101) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 28. The field φc (z, z¯) satisfies the classical Liouville equation, ¯ c = 2πµb2 eφc ∂ ∂φ. (3.40). and locally describes a surface of constant negative curvature −8πµb2 .. 3.7. Semiclassical Correlators. Based on the articles of Dorn, Otto [12] and Al.B., A.B.Zamolodchikov [55], in this section we present some details that led to calculate the well-known DOZZ formula. Now, we consider the correlation functions of primary fields Vαi , Z hVα1 (z1 , z¯1 )...Vαn (zn − z¯n )i ≡. −SL. Dφc e. n Y. exp. i=1. αi φc (zi , z¯i ) b. .. (3.41). We see that the action (3.38) scales like b−2 . This is an important detail because we want to use the method of steepest descent to approximate the path integral in (3.41) for small b, so we need to know how the αi ’s scale with b. Thus, the insertion of any Vαi affects the classical field dynamics saddle point, only if the Liouville momentum α scales as b−1 , i.e., if α = η/b and keeping η fixed for b → 0. This is called ”the heavy” Liouville primary field. Also, we define ”light” operators with α = bσ. Now σ is kept fixed for b → 0, and the insertion of such an operator has no effect on the saddle point φc . When we insert a heavy operator an additional delta function term appears in the equation of motion, ¯ c = 2πµb2 eφc − 2π ∂ ∂φ. X. ηi δ 2 (ξ − ξi ).. (3.42). i. If we assume that in the vicinity of one of the operator insertions the exponential term can be ignored, this equation then becomes Poisson’s equation, O2 φc = −8πηi δ 2 (ξ − ξi ).. (3.43). φc (z, z¯) = C − 4ηi log |z − zi | ,. (3.44). The solution will be.

(102) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 29. so we find that in a neighborhood of a heavy operator we have φc (z, z¯) = −4ηi log |z − zi | + O(1) as z → zi .. (3.45). The physical metric in this region has the form ds2 =. 1 r4ηi. (dr2 + r2 dθ2 ).. (3.46). Substituting the solution (3.44) in the equation of motion, we find the following condition, if we consider that the exponential term is subleading, 1 Re(ηi ) < . 2. (3.47). When this inequality is not satisfied, the interactions affect the behavior of the field arbitrarily close to the operator. This condition is referred to as Seiberg bound for ”good” Liouville operators [42]. We can notice that α and Q − α correspond to the same quantum operator and has the same dimension α(Q − α), VQ−α = R(α)Vα ,. (3.48). where R(α) is called the reflection coefficient [42]. Then α or Q − α will always obey the Seiberg bound. We consider ηi <. 1 2. and doing a simple change of variables to find the metric, we have, ds2 = dr02 + r02 dθ02 ,. (3.49). where r0 ∈ (0, ∞) and θ0 ∈ (0, (1 − 2ηi )2π). Thus we can notice that the operator produces a conical singularity in the physical metric. Now, we have the following conditions: 0 < ηi < 21 , when we have a conical deficit and ηi < 0 when we have a conical excess. Finding real solutions of the equation of motion in presence of heavy operators with real η’s is equivalent to finding metrics of constant negative curvature on the sphere punctured by conical singularities because of the presence of ηi . However, if we remember the Gauss-Bonet theorem, in order to have positive Euler character, we require that the integrated curvature is positive. Then, in the case of our punctured.

(103) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 30. sphere of constant negative curvature we must introduce enough positive curvature to cancel the negative curvature we have. Thereby we obtain an extra condition for the existence of real solutions φc that together with the Seiberg bound restrict the Liouville momentum α. By integrating (3.42) and using (3.39), we get X. ηi > 1.. (3.50). i. In this way, the inequalities (3.47) and (3.50) define the so-called physical region. We say that the condition (3.50) will imply that a product of light fields on S2 will not be a real solutions φc , and in this case ηi = 0. At this point we can make some comments, since in chapter four we will be interested in complex solutions φc . In general, for complex η’s, the saddle points φc will be complex and the condition (3.50) will not be valid. We can see that when we insert the solution (3.45) inside the action (3.38), both the kinetic and the source terms are divergent. To solve this, we follow [55] and perform the action integral only over the part of the disk D that excludes a disk di of radius about each of the heavy operators. Then the ”semiclassically renormalized” operators are I 2ηi2 ηi 2 η b V i (zi , z¯i ) ≈ exp φc dθ . (3.51) b 2π ∂di 2ηi2. The prefactor b2 in (3.51) contributes a term −2ηi2 /b2 to the scaling dimension of the ¯ consistent with the operator Vηi /b ; this is a contribution of −ηi2 /b2 to both ∆ and ∆, quantum shift −αi2 of the operator weights. We can define the regularized Liouville action on the remaining part D of the complex plane incorporating the effects of all the heavy operators b S˜L 2. Z I 1 1 2 φc = d ξ(∂a φc ∂a φc + 16λe ) + φc dθ + 2 log R 16π D−∪i di 2π ∂D X ηi I 2 − φc dθi + 2ηi log i . 2π ∂di i. (3.52). The equations of motion for this action now include both Liouville’s equation (3.40) and the boundary condition (3.39) and (3.45). The semiclassical expression for the expectation.

(104) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 31. value of a product of heavy and light primary field is [22] D. m E Y ˜ eσi φη (xi ,¯xi ) . V ηb1 (z1 , z¯1 )...V ηbn (zn , z¯n )Vbσ1 (x1 , x¯1 )...Vbσm (xm , x¯m ) ≈ e−SL [φη ]. (3.53). i=1. Where we have n heavy operators and m light operators, and φη is the solution (3.42) obeying the correct boundary conditions.. 3.8. DOZZ Formula. The formula DOZZ is a proposal for the Liouville three-point correlation function on S2 , which is the fundamental building block of the Liouville theory as a CFT. We saw that the operators Vα are primaries of weight ∆ = α(Q − α), so their three-point function takes the general form hVα1 (z1 , z¯1 )Vα2 (z2 , z¯2 )Vα3 (z3 , z¯3 )i =. C(α1 , α2 , α3 ) 2(∆1 +∆2 −∆3 ). |z12 |. |z23 |2(∆1 +∆3 −∆2 ) |z31 |2(∆2 +∆3 −∆1 ). , (3.54). where zij = zi − zj . Then, the DOZZ formula is an analytic expression for C in Liouville theory. In particular, this solution is unique due to the recursion relation that were derived by Teschner in [48] [50], and this will not be the case when we make the analytic continuation later. The DOZZ formula is h i(Q−Pi=1 αi )/b 2 2−2b2 C(α1 , α2 , α3 ) = πµγ(b )b ×. Υ0 Υb (2α1 )Υb (2α2 )Υb (2α3 ) , Υb (α1 + α2 + α3 − Q)Υb (α1 + α2 − α3 ) Υb (α1 − α2 + α3 )Υb (−α1 + α2 + α3 ) (3.55). where Vα = e2αφ , γ(x) ≡ Γ(x)/Γ(1 − x) and Υb (x) is an function of x defined (for real and positive b) by Z logΥb (x) = 0. ∞. dt t. ". Q −x 2. 2. sinh2 e−t − sinh. #. Q − x 2t 2 bt sinh 2bt 2. ,. 0 < Re(x) < Q. (3.56). Though this integral representation is limited to the strip 0 < Re(x) < Q , Υb (x) has an analytic continuation to an entire function of x. From the definition of Υ function, we can.

(105) CHAPTER 3. LIOUVILLE THEORY AND CORRELATION FUNCTIONS. 32. see: Υ(x) = Υ(Q − x),. (3.57). Υ(Q/2) = 1, and we also use the notation: Υ0 =. (3.58). dΥ(x) |x=0 . dx. We can calculate the DOZZ-formula following the Dorn-Otto [12] and ZamolodchikovZamolodchikov [55] procedure or the Trick of Teschner [50]. Let us start by showing the original derivation that uses the free field approach. Using the Knizhnik-Polyakov-Zamolodchikov (KPZ) scaling law,. e2α1 φ e2α2 φ ...e2αn φ. where we denote s ≡ (Q −. P. i. g. ∝µ. P (1−g)Q− i αi b. ,. (3.59). αi ) /b. Using (3.59) we can determine the power of µ for. the general correlation function in the Liouville Theory. We need to adjust α or b to the power of µ to become an integer, when we consider the perturbative calculations. By assuming that the 3-point function can be computed in a perturbative series in the cosmological constant µ, we have ∞ X. Gα1 ,α2 ,α3 (z1 , z2 , z3 ) =. Gnα1 ,α2 ,α3 (z1 , z2 , z3 ),. (3.60). n=0. Z Gα1 ,α2 ,α3 (z1 , z2 , z3 ) = hVα1 (z1 )Vα2 (z2 )Vα3 (z3 )i =. Dφ. 3 Y. e2αi φ(zi ) e−SL. i=1. ∼. ∞ Z X. Dφ. n=0. 3 Y. e. 2αi φ(zi ) (−µ). n. Z. 2. n. d ze. n!. i=1. 2bφ. 1. e− 4π. R. (∂a φ)2 d2 z. .. (3.61) ¯ Separating the zero-mode of the path integration over φ(z) = φ0 + φ(z) and integrate over the zero mode first, Gα1 ,α2 ,α3 (z1 , z2 , z3 ) ∼ −. ∞ X (−µ)n n=0. Z. Dφ¯. n! 3 Y. 1 2b(s − n) Z. ¯ i) 2αi φ(z. e. 2. 2bφ¯. d ze. n. 1. e− 4π. R. ¯ 2 d2 z (∂a φ). .. i=1. (3.62).