Scuola Internazionale Superiore di Studi Avanzati Doctoral Thesis Conformal symmetry in String Field Theory and 4D Field Theories Author: Supervisor: Stefano G. Giaccari Prof. Loriano Bonora A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Elementary Particle Physics Trieste - September 2013 Acknowledgements I would like to thank my supervisor Prof. Loriano Bonora for introducing me to the research topics appearing in this thesis, for giving me the possibility of working with him and taking advantage of his huge experience and insight in many problems of current research in physics, for devoting me so much time and care and giving me invaluable help during all our work together. On a more personal level, I'd like to thank him for his kindness, for his patience in tolerating my shortcomings and defects, for his unwavering enthusiasm for research and tireless application to our projects which so many times helped me out of difficult moments. I'd also like to thank Driba Demissie Tolla for collaborating with me, for sharing a lot of enlightening discussions with me and for giving me important support and advice in the first steps of my research career. I wish to express my gratitude to all the High Energy Sector professors and staff for their teachings, advices and help along these years in SISSA. In particular I'd like to thank my fellow students who started this exciting challenging experience with me and shared so much of the toil and fun of PhD time, Dinh, Hani, Les law, Michele, Francesco, Aurora, Dario, Giulio, Jacopo, Xiao Quan, Jun. I wish also to thank my old good friends, Domenico, Giuseppe, Marco and Valentina, without whom my present and past life would have been so much harder and whose support and encouragement have been fundamental to me. Finally I want to thank my parents Ugo and Wilma who have always supported me unconditionally and let me free to choose my way. ii Contents Acknowledgements ii 1 Introduction1 2 Analytic solutions in COSFT5 2.1 Introduction...................................5 2.2 Witten's Cubic Open String Field Theory..................7 2.3 The CFT formulation of SFT......................... 11 2.3.1 Evaluation of correlation functions.................. 11 2.3.2 Surface states.............................. 13 2.3.3 Projectors................................ 14 2.3.4 Wedge states subalgebra........................ 18 2.4 Solutions in split SFT............................. 21 2.4.1 Erler-Schanbl Tachyon condensation solution............ 24 2.4.2 Solutions from singular gauge transfromations............ 25 2.4.3 Relevant deformations and solutions in OSFT............ 27 3 The energy of the analytic lump solutions 35 3.1 Introduction................................... 35 3.1.1 Witten's boundary deformation.................... 36 3.2 The energy functional............................. 39 3.3 Behaviour near s = 0.............................. 42 3.4 The behavior near s = 1 ........................... 43 3.4.1 The quadratic term as s ! 1 ..................... 43 3.4.2 The cubic term as s ! 1 ....................... 45 3.5 Numerical evaluation.............................. 48 3.5.1 The cubic term............................. 49 3.5.2 The quadratic term.......................... 50 3.5.3 Last contribution............................ 51 3.5.4 Overall contribution.......................... 51 3.5.5 Error estimate............................. 51 3.6 The Tachyon Vacuum Solution in KBcφ algebra.............. 52 3.7 The energy of the tachyon vacuum...................... 54 3.7.1 The energy in the limit ! 0..................... 56 3.7.1.1 The -terms and the ! 0 limit in the UV........ 57 3.7.1.2 The -terms and the ! 0 limit in the IR......... 58 3.8 The lump and its energy............................ 61 iii Contents iv 3.9 A D23-brane solution.............................. 63 3.9.1 The IR and UV behaviour....................... 65 3.10 The -regularization.............................. 67 3.11 The energy of the D23{brane......................... 68 3.12 D(25-p) brane solutions............................ 72 4 Mathematical issues about solutions 73 4.1 Introduction................................... 73 4.2 Nature of the parameter........................... 75 4.3 The problem with the Schwinger representation............... 78 4.4 A new (formal) representation for 1 ................... 80 K+φu 4.4.1 The energy for the lump solution u, ................ 82 4.5 A distribution theory inspired approach to the problem.......... 86 4.6 The space of test string fields......................... 90 4.6.1 Good test string fields......................... 90 4.7 The topological vector space of test states.................. 94 4.7.1 Seminorm topology........................... 94 4.7.2 The dual space............................. 97 4.7.3 The strong topology.......................... 97 4.7.4 `Richness' of the space of test states................. 98 4.8 Some conclusions and comments....................... 99 4.8.1 Final comments............................. 100 5 Trace anomalies in N = 1 4D supergravities 103 5.1 Introduction................................... 103 5.2 Supercurrent Multiplets............................ 105 5.3 N=1 minimal supergravity in D=4 and its superfields........... 109 5.3.1 Superconformal symmetry and (super)anomalies.......... 110 5.3.2 Meaning of superconformal transformations............. 111 5.3.3 Anomalies in components....................... 113 5.3.3.1 The square Weyl cocycle.................. 114 5.3.3.2 The Gauss-Bonnet cocycle................. 115 5.4 Non minimal supergravity........................... 117 5.4.1 Superconformal transformations in the non minimal model.... 119 5.4.2 Cocycles in non minimal SUGRA................... 119 5.5 The 16+16 nonminimal model......................... 120 5.5.1 Cocycles in new minimal 16+16 SUGRA.............. 121 5.6 Reduction to component form......................... 121 5.7 Mapping formulas between different supergravity models.......... 123 5.8 Cocycles from minimal supergravity..................... 126 5.8.1 From minimal to nonminimal cocycles................ 126 5.8.2 From minimal to 16 + 16 nonminimal cocycles........... 128 5.9 Conclusions................................... 129 A The angular integration 131 A.0.1 The term without Gs ......................... 132 Contents v A.0.2 The term quadratic in Gs ....................... 133 A.0.2.1 Performing one discrete summation............ 134 A.0.3 The term cubic in Gs ......................... 135 A.0.3.1 Performing one discrete summation............ 138 B Reduction formulae 141 Bibliography 143 Chapter 1 Introduction This thesis is intended as an overview of the two main research topics I have dealt with in the course of my PhD studies: the quest for exact analytic solutions in the context of Witten' s OSFT with the purpose of investigating the moduli space of open strings and the investigation of the structure of trace anomalies in superconformal field theories. String theory is a widely investigated framework in which it is possible to address the problem of giving a consistent and unified description of our universe. This has long been expected on the basis of the fact that the five known perturbative superstring theories (IIA, IIB, I, heterotic SO(32) and heterotic E8 × E8) provide a set of rules to calculate on-shell scattering amplitudes for the modes describing the fluctuations of the strings; such perturbative spectra include, apart from an infinite tower of massive particles, also massless quanta associated with supergravity and super Yang-Mills gauge theories in ten- dimensional space time. These supergravity theories will in general admit black-hole like solutions covering a region of space-time with 9 − p spatial coordinates. In particular in type IIA/IIB supergravity the black p-brane solutions with even/odd p carry charge un- der Ramond-Ramond (RR) (p+1)-form fields so that they are stable and have a definite tension (mass per unit volume). A fundamental breakthrough came from understanding that they may be viewed as the low-energy supergravity limit of the (p + 1)-dimensional hypersurfaces where open strings with Dirichlet boundary conditions can end and that therefore these D-branes are actually manifestations of the non perturbative dynamics of String theory. D-branes have actually played a fundamental role in understanding the web of duality symmetries relating the five superstring models to each other and to a yet not fully defined theory, dubbed M-theory, whose low-energy limit is eleven-dimensional supergravity. This picture seems to suggest the attractive possibility all these theories can be formulated as specific limits of a unique non-perturbative string (or M ) theory which is explicitly background-independent. The discovery of D-branes is also related to another fundamental achievement of string theory, the AdS/CFT correspondence, 1 Introduction 2 which in its best-known formulation states that quantum type IIB closed string theory 5 on AdS5 × S with N units of RR 5-form flux is dual to N = 4 U(N) SYM theory on the projective 4- dimensional boundary of AdS5. The D-brane perspective is that such a gauge theory is realized as the low energy limit of open string theory on ND3-branes in flat space-time. So the afore mentioned correspondence is establishing a correspondence between the quantum theory of gravity on a given background space-time and the open string dynamics of the D-brane configuration which is creating it by backreacting on the original