CLOSED AND OPEN STRING THEORIES IN NON-CRITICAL BACKGROUNDS Sameer Murthy A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS November 2004 c Copyright by Sameer Murthy, 2004. All rights reserved. Abstract This thesis is a study of closed and open string theories in low dimensional space- times, and the various relations between these theories. In particular, we focus on the theory of the two-dimensional black hole. We first study closed strings in the background of the Euclidean two- dimensional black hole (SL2(IR)/U(1)) tensored with flat space, using the duality relating these theories to non-critical superstrings described by the supersymmetric sine-Liouville interaction on the worldsheet. We point out a subtlety in their geo- metric interpretation, and clarify the symmetry structure of the theories based on the understanding of these theories as near horizon limits of wrapped NS5-branes. In one such example (cigar IR6), we use the brane description to understand the × enhancement of the global symmetry in the coset theory from U(1) to SO(3) under which the sine-Liouville and cigar interactions are related. In the same example, a 4 worldsheet description of the moduli space IR /ZZ2 is presented. We then study open strings in the topologically twisted Euclidean two- dimensional black hole which is equivalent to noncritical c = 1 bosonic string theory compactified on a circle at self-dual radius. These strings live on D-branes that are extended along the Liouville direction. We present explicit expressions for the disc two- and three-point functions of boundary operators in this theory, as well as the bulk-boundary two-point function. The expressions obtained are divergent because of resonant behavior at self-dual radius. However, these can be regularised and renormalized in a precise way to get finite results. The boundary correlators are found to depend only on the differences of boundary cosmological constants, sug- gesting a fermionic behaviour. We initiate a study of the open-string field theory localized to the physical states, which leads to an interesting matrix model. Finally, we present evidence that the worldvolume theory of N unstable D- particles in type IIB superstring theory in two-dimensions is represented by the supersymmetric matrix model of Marinari and Parisi. This identification suggests that the matrix model gives a holographic descriptions of superstrings in a two- dimensional black hole geometry. i Acknowledgements First, I would like to acknowledge my advisor Nati Seiberg for his constant presence, guidance and help over the last few years. I am grateful to him for teaching me, in what measure I have learnt, a way of doing physics - to strive for clarity of thought, simplicity of expression, and to think for myself. I would like to thank him especially for extending support when the going got tough, and to encourage me to be more independent. I would like to thank Debashis Ghoshal, John McGreevy, Sunil Mukhi and Her- man Verlinde for enjoyable collaborations on projects. I would like to acknowledge and thank the members of the high energy group in Jadwin Hall, IAS and Rutgers and my fellow graduate students for many enjoyable discussions - I have learnt a lot of physics from talking to you, and it has made the experience much more fun: Sujay Ashok, Chris Beasley, David Berenstein, Juan Burwell, Shoibal Chakravarty, Chang Chan, Sergey Cherkis, Keshav Dasgupta, Eleonora Dell’Aquila, Oliver De- Wolfe, Tamar Friedmann, Davide Gaiotto, Aki Hashimoto, Sunny Itzhaki, Joanna Karczmarek, Igor Klebanov, Jack Laiho, Wei-Li Lee, Oleg Lunin, Wei-Ji Ma, Juan Maldacena, Liat Maoz, Peter Ouyang, John Pearson, Leonardo Rastelli, Natalia Saulina, Ivo Savonije, David Shih, Kostas Skenderis, Peter Svrcek, Herman Ver- linde, Weida Wu, Emil Yuzbashyan; and especially Mukund Rangamani and John McGreevy. I would like to thank the Indian string theory community for enjoyable dis- cussions, and support and friendly advice on various occassions - Atish Dabholkar, Rajesh Gopakumar, Gautam Mandal, Sunil Mukhi, Ashoke Sen, Sandip Trivedi, Pattabhi Yogendran, and especially Shiraz Minwalla and Spenta Wadia. I would like to record here the quiet, but immense contribution of my parents, and my sister Sahana to what I am now - their unflinching support, encouragement, and belief in me throughout has been a solid rock and a constant source of self- confidence for me, especially through these years of graduate school. I would like to thank my friends Abhra, Ali, Arudra, Indrani, Ivo, Kunal, Liat, Rosalba, Sarin, Shoibal, Spyros, Sujay, Yeong for many wonderful moments spent together during the last five years. Indeed, it is you with who I have shared much of my joys and stresses of the day, and who have made life in Princeton fun. ii Finally, I would like to thank the members of the Argentine Tango community at Princeton for a very enjoyable time. Many a time have I walked out after dancing on Thursdays feeling much lighter and more energetic than when I walked in. Special thanks to Lesley Mitchell for introducing me to the dance and Juan-Carlos and Teresa Figueroa for teaching me to walk straight. iii Contents 1. Introduction, Summary and Discussion ............. 1 1.1.Whatisthisdissertationabout? . 1 1.2.Stringtheoryingeneralspacetimes . 4 1.3. Strings in low dimensions: Non-critical backgrounds . ....... 7 1.3.1.Supersymmetry . 8 1.4.Blackholesinlowdimensions . 9 1.5. Open-closed string duality, branes, holography . .. ..11 1.6.Discussion . .14 2. Non-Critical Superstrings in Various Dimensions ........16 2.1.Introductionandsummary . .16 2.2.SuperstringtheoriesontheCigar . 20 2.2.1. Type 0 theories on the cigar IRd ...............22 × 2.2.2. The chiral GSO projection and Type II theories . 23 2.2.3.Holographicinterpretation . 26 2.3. Specialfeaturesofthevarioustheories . 27 2.3.1. d =0 ...........................27 2.3.2. d =2 ...........................29 2.3.3. d =4 ...........................29 2.3.4. d =6 or k =2NS5-branes . .30 2.3.5. A note on the = (4, 4) algebra in the d =6theory. .33 N 2.4. The globalsymmetries ofthe varioustheories . .35 Appendix 2.A. Spectrum of the higher dimensional theories . .. ..39 Appendix 2.B. Partition functions of the various theories . .......41 2.B.1.d=0 ...........................42 2.B.2.d=2 ...........................43 2.B.3.d=4 ...........................43 2.B.4.d=6 ...........................45 Appendix 2.C. Quantization in the Green-Schwarz formalism (d =4) . .45 Appendix 2.D. Conformal invariance at second order in the d =6theory . 47 iv 3. Stable D-branes and Open String Field Theory in Two Dimensions 49 3.1.Introduction . .49 3.2. Two-dimensional Open String Theory and the FZZT branes . .52 3.3.BoundaryTwo-PointFunction . 55 3.4.BoundaryThree-PointFunction . 58 3.5. Bulk-BoundaryTwo-PointFunction . 61 3.6. Physical Correlators and Open String Field Theory . .65 3.7.Discussion . .69 Appendix 3.A. Special Functions at c =1...............71 Appendix 3.B. Evaluation of a Contour Integral for the Three-Point Function 72 4. Unstable D-branes in Two-dimensions and the Supersymmetric Ma- trix Model .............................76 4.1.Introduction . .76 4.2.TheMarinari-Parisimodel . .76 4.3.Two-dimensionalSuperstrings . 78 4.4.DualCorrespondence . .82 4.5.Concludingremarks . .87 Appendix 4.A. The gauged Marinari-Parisi model . 88 Appendix 4.B. Boundary states for =2Liouville . .91 N Appendix 4.C. Ground states of the harmonic superpotential for arbitrary fermionnumber . .97 v Chapter 1 Introduction, Summary and Discussion 1. What is this dissertation about? This dissertation is an investigation of a few related topics in the quantum theory of one-dimensional objects - strings - living in certain spacetimes having the feature that the strength of the string interactions - the string coupling - varies as a function of one of the spatial directions. These theories are also known as non- critical string theories, and there are many known examples with a small number of dimensions (two or less). In particular, this thesis shall focus on a solution of these theories which, from the spacetime point of view, looks like a black hole in two dimensions. Since string theory is a consistent theory of quantum gravity, this is exciting because it might throw some light on issues in the physics of black holes, like the problem of information loss, i.e. understanding unitarity of the quantum theory, and the origin of entropy. Following this introductory chapter, this thesis consists of three chapters based on the works [1–3]. The second chapter discusses the gravitational theory of closed superstrings in various spacetime backgrounds in less than ten dimensions which contain the two dimensional black hole geometry as the most important part. The main focus is on the geometry of these backgrounds, and where these theories fit into the space of solutions of superstring theory. The third and the fourth chapter, motivated by the concept of holography as manifested by open-closed string duality, study open strings in the black hole with the intention of getting information about the closed string gravitational theories. The third chapter sets forth the quantum field theory of open strings on D-branes in the topological Euclidean black hole. The fourth chapter studies the tachyonic open string theory in the physical black hole background and based on this, proposes a dual unitary quantum theory to describe the black hole. The current chapter has two goals. Firstly, it is meant to put this thesis in perspective for the reader with a general interest in particle physics, gravitation 1 and string theory by introducing the main ideas in the thesis along with some relevant background. Secondly, for the reader with more specific interest in the topics discussed – superstrings in low dimensions, the two dimensional black hole, open-closed string duality and matrix models – a brief summary of results and some discussion relating the various threads is provided towards the end of the chapter.