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On String Integrability !"# $%&'!%! #!& & !' ( )) )(()* + !&," u qrt rrs9p sQuvyu ISBN 789778 : ; < 9 = <<> < ? @ = @ ? ? :A;C @A ? D C = 9E E << 9 @ = E > ?= < ? ! "# "$ #Uur rvphy "#% &'()# Vvr v T@&$ ' Vhyh Trqr F <A9 <>A789778 7E: C G H 7E; To Dagmar List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I V. Giangreco Marotta Puletti, “Operator product expansion for 5 pure spinor superstring on AdS5 × S ”, Journal of High Energy Physics, 0610:057, 2006 [arXiv:hep-th/0607076]. II V. Giangreco Marotta Puletti, T. Klose and O. Ohlsson Sax, “Fac- 5 torized world-sheet scattering in near-flat AdS5 × S ”, Nuclear Physics B792:228-256, 2008 [arXiv:0707.2082 [hep-th]]. III D. Astolfi, V. Giangreco Marotta Puletti, G. Grignani, T. Harmark and M. Orselli, “Finite-size corrections in the SU(2)×SU(2) sec- 3 tor of type IIA string theory on AdS4 × CP ”, Nuclear Physics B810:150-173, 2009 [arXiv:0807.1527 [hep-th]]. IV V. Giangreco Marotta Puletti, “Aspects of quantum integrability 5 for pure spinor superstring in AdS5 ×S ”, Journal of High Energy Physics, 0809:070, 2008 [arXiv:0808.0282 [hep-th]]. Reprints were made with permission from the publishers. To them, I said, the Truth would be literally nothing but the shadow of the images Plato, The Republic If you look only in one direction your neck will become stiff Chinua Achebe, A man of the people Contents 1 Introduction: Motivations, Overview and Outline . 1 2 The AdS5/CFT4 duality . 7 2.1 Introduction . 7 2.2 N = 4 super Yang-Mills theory in 4d . 9 2.3 The algebra . 11 2.4 Anomalous dimension and spin chains . 14 2.4.1 The Coordinate Bethe Ansatz for the su(2) sector . 15 3 Classical vs. Quantum Integrability . 25 3.1 Principal Chiral Model . 25 3.2 Coset model . 30 3.3 The magic of (1+1)-dimensional theories . 33 3.4 Quantum Integrability . 40 4 Green-Schwarz-Metsaev-Tseytlin superstring . 43 4.1 Green-Schwarz action in flat space . 43 5 4.2 Type IIB superstring on AdS5 × S : GSMT action . 45 4.3 Classical integrability for the GSMT superstring action . 49 5 5 The Pure Spinor AdS5 × S superstring . 51 5.1 Motivations . 51 5.2 The Berkovits formalism: basic review . 53 5 5.3 Type IIB superstring on AdS5 × S : PS action . 58 5 5.4 Classical integrability of the AdS5 × S PS superstring action . 66 5 5.5 Quantum integrability of the AdS5 × S PS superstring action . 69 5.6 Quantum Integrability: Papers I and IV . 70 5.6.1 Absence of anomaly: Paper IV . 70 5.6.2 The operator algebra: Papers I and IV . 72 5.6.3 The field strength: Paper IV . 75 6 AdS/CFT as a 2d particle model and the near-flat-space limit . 77 6.1 Introduction . 77 6.2 Light-cone gauge, BMN limit and decompactification limit . 79 6.2.1 Light-cone gauge . 79 6.2.2 Decompactification limit . 83 6.2.3 The fields . 84 6.2.4 BMN limit . 85 6.3 Near flat space limit . 89 6.4 The world-sheet S-matrix in the NFS limit . 92 6.4.1 The dressing phase . 98 6.5 Paper II . 100 7 The AdS4/CFT3 duality . 105 7.1 Introduction . 105 7.2 The field theory . 107 7.3 The algebra . 108 7.4 Spin chains and anomalous dimension . 110 7.4.1 The SU(2) × SU(2) spin chain . 111 7.5 Integrability on the string theory side . 113 7.5.1 The BMN limit . 114 7.6 Paper III . 118 8 Epilogue . 123 9 Summary in Swedish Aspekter på integrabilitet av strängteori . 127 10 Appendix . 131 10.1 Notation . 131 10.2 PS formalism: BRST invariant charges . 131 10.3 The AdS4/CFT3 duality: Preliminaries . 133 Bibliography . 137 1 Introduction: Motivations, Overview and Outline The main purpose of this thesis is to explain the results which are contained in the papers I, II, III and IV. Also I would like to illustrate which are the main motivations which pushed my research in such directions, the context and to give at least the flavor of the incredible “hidden” beauty which is in the gauge/string dualities. In order to understand the results and the techniques used in the works, I will need to introduce certain topics and formalisms as well as some back- ground material. As this is the typical readership, I have chosen the level of the presentation such as to address a Ph.D. student who works on String The- ory, but not necessarily on the AdS/CFT correspondence or on integrability. My task is to give the reader the possibility to be able to autonomously under- stand and go through the papers at the end of this thesis. Nevertheless, I will assume that the reader is familiar with supersymmetric strings, in particular with type IIB/A superstrings. The order used to illustrate the various subjects does not strictly reflect how they have been historically developed, but rather the necessity to follow the contents of the papers closely. As opening the Russian Matryoshka dolls, I will start from the biggest doll (the String Theory) to the smallest one (the integrability in AdS/CFT) to illus- trate, in this introduction, the contents which the thesis is focused on. With the exception of gravity all the other three fundamental forces which are present in Nature (electromagnetic, weak and strong nuclear interactions) are unified in the Standard Model. They are derived from the same first prin- ciple, which is a (local) symmetry principle: the gauge symmetry. For this reason, these theories are defined as gauge theories. The Standard Model is based on the fundamental concept of point-like particles and the interactions are described in terms of mediators (photons, W ± and Z0 bosons and gluons respectively). I will think of such a model as describing particle physics, as something distinguished by gravity in the traditional approach. A revolution- ary point of view is adopted in String Theory. String Theory provides us with an elegant framework, where all the four interactions are joined together. The string is a one-dimensional object and its spectrum, namely the collection of frequencies and masses that the string produces by vibrating, naturally con- tains the mediator for the gravitational force, the graviton, treating gravity on 1 equal footing with the other fundamental interactions. The concept of replac- ing the point-particles with an extended fundamental object (the string) can be generalized: one can construct surfaces of higher dimensions (the branes) which replace the strings. These also are important building-blocks of String Theory. The word “framework” used to define String Theory might seem reductive, but it is the correct one: String Theory is not a complete and fully understood theory, but it is more a “structure”, an incredible rich one, where different types of string theories live1. They are related by dualities, a very special kind of symmetry which relates two apparently different physical systems. I will come back on the topic of dualities below. However, all these pluralities of string theories should be a special limit, or at least they should be contained, in a more general and yet quite mysterious (including its name!) theory, the M-theory. A part from the hope to see the Standard Model emerge from String Theory one day, there is another way in which the fate of the string is tied to parti- cle field theories. In 1997, Maldacena conjectured that certain closed super- strings in a ten-dimensional curved background describe the same physics of the gauge theory of point-particles in four-dimensions (AdS5/CFT4) (another smaller Russian shell). In particular, on one side we have the type IIB super- 5 string on AdS5 ×S , and on the other side the supersymmetric N = 4SU(N) Yang-Mills theory in four dimensions. The backgrounds where the string lives 5 (AdS5 × S ) is built of a five-dimensional anti-De Sitter space (AdS), a space with constant negative curvature, times a five-dimensional sphere (S), cf. fig- ure 1.1. In 2008 Aharony, Bergman, Jafferis and Maldacena proposed the existence of a further gauge/gravity duality between a theory of M2-branes (three-dimensional membranes) in eleven dimensions and a certain gauge the- ory in three dimensions (AdS4/CFT3). The eleven-dimensional M2-theory can be effectively described by type IIA superstrings when the string cou- pling constant is very small. For a reason that will be clear later, I will con- sider only the type IIA as the gravitational dual in the AdS4/CFT3 correspon- dence, but the reader should keep in mind that this is just a particular regime of the full correspondence. The background where the type IIA strings live is a four-dimensional anti-De Sitter space times a six-dimensional projective space (CP3). Hence, we have seen that the gravity side in the dualities is associated with the word “AdS”. What is CFT? They are the initials of Conformal Field Theory. The dual 1The different string theories I am referring here in the Introduction are the type IIA superstring, the type IIB superstring, the type I superstring, the Heterotic SO(32) string, the Heterotic E8 × E8 string, and finally I should also include the eleven-dimensional supergravity theory. In the rest of the thesis we will consider only the type IIB superstring, cf. 2 – 6, and the type IIA superstring in chapter 7. 2 5 Figure 1.1: AdS5 × S . The five-dimensional anti-De Sitter space is represented as a hyperboloid on the right hand side, while the five-dimensional sphere is drawn on the left hand side.
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