Abstract of Solitons and Spin Chains in Gauge/String Dualities by Georgios Papathanasiou, Ph.D., Brown University, May 2011.

Transcription

1 Abstract of Solitons and Spin Chains in Gauge/String Dualities by Georgios Papathanasiou, Ph.D., Brown University, May The AdS/CFT correspondence offers a unique possibility for the nonperturbative investigation of gauge theories, through their conjectured equivalence to string theories. In the past years, there has been tremendous progress in the computation of the anomalous dimensions of gauge theory operators, or equivalently energies of string states, thanks to the integrable structures that emerge in the t Hooft limit. In this thesis, we study the two main manifestations of integrability in gauge/string dualities, namely solitons in string theory and spin chains in gauge theory. We first review the emergence of an integrable spin chain in maximally supersymmetric Yang-Mills theory, and the identification of the string dual to a single excitation in a certain limit, the giant magnon solution. We then semiclassicaly quantize the solution and obtain the first quantum correction to its energy, in agreement with expectations based on symmetry. The fact that the string configuration in question amounts to a single solitary wave of the string sigma model, subsequently allows us to use a nonlinear superposition method known as dressing, in order to construct a general formula describing the scattering of an arbitrary number of solitons. We further generalize our solutions for the most recent instance of a gauge/string duality, involving the 3-dimensional super Chern-Simons theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We also study the symmetry of the integrable spin chain arising in the latter duality. In particular we compute how the Hilbert space of spin chain states of length up to 4, decomposes into irreducible representations of the superconformal group, for which we also obtain the characters. This structural information enables us to calculate the leading correction to the Hagedorn temperature of weakly coupled planar ABJM theory, and more importantly serves as a tool for the detailed spectroscopic analysis of anomalous dimensions, which we turn to next. Employing a combination of Bethe Ansatz and Hamiltonian diagonalization techniques, we find the spectrum of low-lying states of length 4 at 2 loops, and identify three new sequences of rational eigenvalues.

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3 Solitons and Spin Chains in Gauge/String Dualities by Georgios Papathanasiou M. Sc., Imperial College London, 2005 B. Sc., National and Kapodistrian University of Athens, 2004 Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Department of Physics at Brown University Providence, Rhode Island May 2011

4 c Copyright 2011 by Georgios Papathanasiou

5 This dissertation by Georgios Papathanasiou is accepted in its present form by the Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Marcus Spradlin, Director Recommended to the Graduate Council Date Antal Jevicki, Reader Date Chung-I Tan, Reader Approved by the Graduate Council Date Peter M. Weber Dean of the Graduate School

6 Curriculum Vitæ Personal Information Born on September 25, 1981, in Athens, Greece Education Brown University, Providence, RI, USA Ph.D. in Physics, 2011 Research Group: High Energy Theory Advisor: Professor Marcus Spradlin Imperial College London, UK M.Sc. in Quantum Fields and Fundamental Forces, 2005 Classification: Distinction Thesis Topic: AdS/CF T Correspondence Advisor: Professor Kellogg Stelle University of Athens, Greece Ptychio (4-year Bachelor) in Physics, 2004 Grade: 8.04/10 Specialization: Nuclear and Particle Physics Thesis Topic: Canonical Quantization of the Electromagnetic Theory Advisor: Professor Alexandros Karanikas University of Bonn, Germany Exchange Student, Spring 2001 Bonn International Physics Programme iv

7 Awards & Distinctions Brown University, Department of Physics Dissertation Fellowship, Award for Excellence as a Graduate Teaching Assistant, 2008 University of Athens Scholarship from Ioannis Varykas Foundation, Admissions Ranking: 11/255 Publications [1] Giant Magnons in Symmetric Spaces: Explicit N-soliton solutions for CP n, SU(n) and S n, JHEP 1007, 068 (2010) [arxiv: [hep-th]], with C. Kalousios. [2] Two-Loop Spectroscopy of Short ABJM Operators, JHEP 1002, 072 (2010) [arxiv: [hep-th]], with M. Spradlin. [3] The Morphology of N = 6 Chern-Simons Theory, JHEP (2009) [arxiv: [hep-th]], with M. Spradlin. [4] Exact solutions for N-magnon scattering, JHEP 0808, 095 (2008) [arxiv: [hep-th]], with C. Kalousios and A. Volovich. [5] Semiclassical Quantization of the Giant Magnon, JHEP 0706, 032 (2007) [arxiv: [hep-th]], with M. Spradlin. Talks & Presentations Multi-Magnons in Symmetric Spaces, University of Porto & University of Santiago de Compostela, January 2011 Giant Magnons in Symmetric Spaces (Poster Presentation), Integrability in Gauge and String Theory, NORDITA Stockholm, July 2010 Two-Loop Spectroscopy of Short ABJM Operators, Brown University, February 2010 Integrability and gauge/string duality, University of Athens, December 2009 The Morphology of N = 6 Chern-Simons Theory (Poster Presentation), Integrability in Gauge and String Theory, AEI Potsdam, July 2009 The Morphology of N = 6 Chern-Simons Theory, Brown University, March 2009 Semiclassical Quantization of the Giant Magnon, Brown University, May 2007 v

8 Conferences & Schools The Structure of Local Quantum Fields Ecole de Physique, Les Houches, June 2010 Strings Rome, June 2009 Mathematica School on Theoretical Physics University of Porto, June 2009 Integrability in Gauge and String Theory University of Utrecht, August 2008 AEI Potsdam, July 2009 NORDITA Stockholm, July 2010 Simons Workshop in Mathematics and Physics S.U.N.Y. Stony Brook, June 2008, August 2010 Prospects in Theoretical Physics Institute of Advanced Study, Princeton, July 2007 New England String Meeting Brown University, November 2006, November 2007, October 2008, April 2010 RTN Winter School on Strings, Supergravity and Gauge Theories CERN, January 2006, January 2007 Teaching & Appointments Brown University, Department of Physics Graduate Research Assistant, Graduate Teaching Assistant, Collaborated with Professor Leon Cooper, Nobel Laureate in Physics, 1972, on the design and teaching of the course Images from Science, Images for Science for two consecutive years Taught conference sections in Modern Physics and Classical Mechanics, and instructed labs for Electricity & Magnetism and Introductory Mechanics Developed a reflexive teaching practice through the successful completion of the Teaching Certificates I & III at the Sheridan Center for Teaching & Learning vi

9 Service Brown University, Department of Physics Co-organizer of the weekly High Energy Theory Journal Club, Science Club leader in the University s outreach effort to encourage the students of the Community Preparatory School to appreciate science, 2009 Discussion leader at the International Teaching Assistant Orientation, 2010 President of the Hellenic Student Association, with significant participation in the organization of campus-wide cultural activities, Imperial College London, Department of Physics Study group and bibliography database organizer among M.Sc. students University of Athens, Department of Physics Student Representative at the Departmental Board Languages Greek (native), English (fluent), German (proficient), French (basic) References Marcus Spradlin (Ph.D Advisor) High Energy Theory Group Brown University Physics Providence, RI 02912, USA Marcus_Spradlin@brown.edu Antal Jevicki High Energy Theory Group Brown University Physics Providence, RI 02912, USA Antal_Jevicki@brown.edu Anastasia Volovich High Energy Theory Group Brown University Physics Providence, RI 02912, USA Anastasia_Volovich@brown.edu Leon Cooper Brown University Physics Providence, RI 02912, USA Leon_Cooper@brown.edu vii

10 Acknowledgements I am grateful to my advisor Marcus Spradlin for all his support during my Ph.D., providing me both guidance and encouragement for independent work. I would also like to thank Chrysostomos Kalousios and Anastasia Volovich for our fruitful research collaboration, and Antal Jevicki and Chung-I Tan for reading this dissertation. I am indebted to all members of the High Energy Theory Group with whom I have interacted during the course of my graduate studies, and especially Ines Aniceto, Pawel Caputa and Cristian Vergu for many valuable physics discussions. I am also thankful to Benjamin Basso, Matteo Beccaria, Alexander V. Mikhailov and Joseph Minahan for insightful comments and very helpful correspondence. I would like to acknowledge financial support provided by the US National Science Foundation and Department of Energy. Last but not least, none of this would have been possible without the love and support of my family and friends, and especially my parents Maria and Sotiris, my brother Andreas, my grandmother Dimitra, as well as Maria Stournara, Arjun Bansal and Babis Papamanthou. Thank you so much for everything you ve done for me! viii

11 This thesis is dedicated to my grandmother Dimitra. ix

12 Contents List of Tables xv List of Figures xvi I AdS 5 /CFT 4 Correspondence 1 1 Introduction Gauge/String Dualities The AdS/CFT Spectral Problem Integrability in Gauge and String Theory Overview Integrable Spin Chains R-matrix & Yang-Baxter Equation The XXX 1/2 Heisenberg Spin Chain SU(2) Bethe Ansatz Semiclassical Quantization of the Giant Magnon Abstract Introduction Bosonic Sector The Giant Magnon x

13 3.3.2 AdS 5 Fluctuation Spectrum S 5 Fluctuation Spectrum Fermionic Sector The Fluctuation Equations Solving the Equation for ϑ The κ-fixed Solutions The 1-loop Functional Determinant Exact Solutions for N-magnon Scattering Abstract Introduction Giant Magnons on R S Review of the Dressing Method Application and Recursion The N-magnon Solution Hirota Form of the Solution Determinant Form for Z Asymptotic Behavior II AdS 4 /CFT 3 Correspondence 64 5 The Morphology of N = 6 Chern-Simons Theory Abstract Introduction Oscillator Construction for OSp(6 4) Sp(4, R) SO(3, 2) SO(6) SU(4) OSp(6 4) and Super-Young Tableaux xi

14 5.3.4 Serre-Chevalley Basis Representations and their Partition Functions Notational Conventions Calculating Characters: An Example All Irreducible Multiplets with up to 4 Sites Tensor Product Decompositions Digraphs and Syllables of the ABJM Language Four-Fold Tensor Products Four-Letter Words of the ABJM Language The OSp(4 2) Subsector Partition Functions Tensor Products A First Peek at the Two-Loop Dilatation Operator The Trace D 2 (x) of the Hamiltonian Density The Two-Loop Hagedorn Temperature Two-Loop Spectroscopy of Short ABJM Operators Abstract Introduction Preliminaries Unpaired OSp(6 4) Multiplets Proof of the V 4 2m,2 Eigenvalue Sequence Proof of the V 4 4m+1,3 Eigenvalue Sequence Numerical Methods for Other Eigenvalues Shortening Conditions and Multiplet Splitting Multiplets in Various Subsectors OSp(4 2) Sector SL(2 1) Sector xii

15 6.6.3 SL(2) (-like) Sectors Some Comments on Length-6 Operators fold Tensor Product Decomposition in the SL(2 1) Sector Sums of Unpaired Eigenvalues Giant Magnons in Symmetric Spaces: Explicit N-soliton Solutions for CP n, SU(n) and S n Abstract Introduction Sigma Models and Dressing Single Soliton Two Solitons Building Up Recursion N-soliton Solutions CP n Case SU(n) Case S n Case Classical Time Delay Conclusions and Outlook 174 A Construction Rules & Examples for N-magnons on R S B Decomposition of OSp(6 4) Super-Young Tableaux 183 C Fermionic Root Dualization for the ABJM Bethe Ansatz 188 C.1 A Useful Example C.2 General Considerations D Symmetric Spaces 194 xiii

16 E N-soliton to N-magnon reduction 196 E.1 CP E.2 SU(2) E.3 S xiv

17 List of Tables 5.1 OSp(6 4) representations of length OSp(6 4) representations of length OSp(6 4) representations of length Unpaired eigenvalues of ABJM Bethe Ansatz for states of length Classification of OSp(6 4) multiplets with respect to shortening conditions Anomalous dimensions for low-lying states of length 4 in the OSp(4 2) sector Anomalous dimensions for low-lying states of length 6 in the SL(2 1) sector. 139 xv

18 List of Figures 4.1 Plot of N-magnon solution for N = 1, 2, 3, The OSp(6 4) Dynkin diagram The OSp(6 4) distinguished Dynkin diagram Super-Dynkin diagrams for (a) OSp(6 4) (b) SL(2 1) C.1 The OSp(6 4) Dynkin diagram after one fermionic root dualization xvi

19 Part I AdS 5 /CFT 4 Correspondence 1

20 Chapter 1 Introduction 2

21 3 Gauge theories such as Quantum Chromodynamics (QCD), the theory of the strong force binding quarks and gluons inside protons and neutrons, have been instrumental in our quest to understand what the elementary constituents are, and how they interact. In particular, gauge theories form the basis of the Standard Model of particle physics, the most accurate description of nature at the microscopic level. The main tool for calculating physical quantities in any quantum theory is perturbation theory, whereby one approaches more accurate values for the quantity by iteratively including corrections controlled by increasing powers of a small dimensionless parameter. This parameter, called the coupling constant, is a measure of the strength of the interaction between particles, and in fact it varies with their energies. For most gauge theories such as Quantum Electrodynamics (QED), describing the interaction of light and matter, this is not a problem, since the coupling constant remains small for all observable energy scales. However QCD has the very special property of asymptotic freedom, where the coupling is small for high energies, but becomes large at low energies. This renders perturbation theory inapplicable for studying low energy phenomena such as quark confinement and chiral symmetry breaking, and developing an alternative analytical framework for their description remains a very important task. Remarkable progress in the investigation of nonperturbative aspects of certain gauge theories has been made through the well-supported conjecture of their equivalence to string theories, namely theories where the elementary objects are extended, 1-dimensional objects (strings), rather than point-like particles. The beauty and utility of this conjectured equivalence, or duality, termed Anti-de Sitter/Conformal Field Theory (AdS/CF T) correspondence, is that when the gauge theory is strongly coupled and cannot be treated directly, the dual string theory is weakly coupled and hence amenable to perturbative computations. In this thesis, we will study the spectral problem of AdS/CFT, and in particular how the main manifestations of the underlying integrable structure, spin chains and solitons, can be successfully employed towards its resolution. For the rest of this introductory chapter,

22 4 let us present the key concepts of our topic, before closing with a brief overview. 1.1 Gauge/String Dualities The first hint that gauge theories and string theories are somehow related emerged in the study of Yang-Mills theories with gauge group SU(N), for which the number of colors N becomes large [1]. The motivation originated from the fact that in QCD the dimensionless coupling constant g Y M becomes replaced by the characteristic scale Λ QCD due to renormalization, and so the only other possibility for a perturbative expansion would be in terms of the parameter 1/N. Indeed it was shown that an 1/N expansion for any gauge theory amplitude is possible if one maintains the t Hooft coupling λ gy 2 MN constant as N becomes large, in which case it takes the same form as the perturbative expansion of a closed string theory with coupling constant g s 1/N. From the form of the expansion it also follows that in the t Hooft limit, where we strictly have N as λ remains constant, a remarkable simplification occurs: Only a small fraction of planar diagrams, namely Feynman diagrams which can be drawn on a plane without any crossover of lines, contributes to the expansion. The AdS/CF T correspondence [2] is the framework in which the relation between these seemingly different types of theories was made precise, see [3] and [4] for reviews. It conjectures the mathematical equivalence, or duality, between certain pairs of string and gauge theories. In all cases, the string theories are defined on a AdS d M geometric background, where AdS d+1 is (d + 1)-dimensional Anti-de Sitter space and M a compact manifold, and the gauge theories are d-dimensional and conformal, namely scale invariant. Due to scale invariance the gauge theory has no asymptotic states, so the natural objects to consider are local gauge invariant operators, and the quantitative nature of the duality lies in the fact that it offers a prescription on how to map them to states on the string theory side. The original, and most well-studied example of the AdS/CF T correspondence identifies

23 type IIB string theory on AdS 5 S 5, where S 5 is the 5-dimensional sphere, to 4-dimensional super Yang-Mills (SYM) theory with maximal N = 4 supersymmetry and gauge group SU(N). The parameters of the gauge theory are again the number of colors N and coupling g Y M, which we can equivalently trade for λ gy 2 MN. In the string side apart from the respective coupling g s we also have the effective string tension R/α as an overall factor in the action, where R is the common radius of the AdS 5 and S 5 spaces and α is related to the string tension T by T = 1/(2πα ). The heuristic derivation of the duality identifies the two pairs of parameters as 4πg s = gy 2 M = λ N, R 2 = α g 2 Y M N = λ, (1.1.1) 5 and from the above relations, we can see that in the t Hooft limit g s 0 and strings will propagate freely, without any splitting and joining interactions. The example of the correspondence we have just described will be the subject of study in Part I of this thesis, and we will refer to it as AdS 5 /CFT 4. Another more recent instance of the duality, which will be the focus of Part II of this thesis, is related to the conformal N = 6 supersymmetric Chern-Simons matter theory proposed by Aharony, Bergman, Jafferis and Maldacena (ABJM) [5]. The latter is a 3-dimensional U(N) U(N) gauge theory with four complex scalars and their fermionic partners in the bi-fundamental representation and gauge fields with Chern-Simons levels +k and k. It has a t Hooft limit with N, k and λ = N/k fixed, and it is in this limit that it admits a dual description in terms of type IIA string theory on AdS 4 CP 3. We will similarly denote this duality as AdS 4 /CFT The AdS/CF T Spectral Problem A necessary condition for the validity of any gauge/string duality, is that the global symmetries on both sides should match. In AdS 5 /CFT 4, the common symmetry group is

24 6 P SU(2, 2 4), whose bosonic subgroup is SU(2, 2) SU(4) SO(4, 2) SO(6), corresponding to the (conformal spacetime) (R-symmetry) group on the gauge side, and to the AdS 5 S 5 isometry group on the string side. The AdS 4 /CFT 3 symmetry group is in turn OSp(6 4), and its bosonic subgroup Sp(4, R) SO(6) SO(3, 2) SU(4) can be interpreted in a similar fashion. Symmetry in particular identifies the generator of scale transformations in gauge theory to D to the generator t of shifts in the AdS global time coordinate t. As a consequence, the two generators eigenvalues, scaling dimensions A and energies with respect to global time E A respectively, should be equal for operators and states which are dual to each other. In more detail, if we focus on local gauge invariant single trace operators O A = Tr (φ i1 (x)φ i2 (x)...φ in (x)), (1.2.1) where φ i are the elementary fields of the gauge theory and their derivatives, then classically the effect of scaling transformations or dilatations is x µx φ i µ (0) i φ i and O A µ i (0) i O A = µ (0) A OA. (1.2.2) However the operators O A get renormalized order by order in perturbation theory, and hence obtain anomalous dimensions as quantum corrections on top of their classical scaling dimensions (0) A. The AdS/CF T spectral problem is the task of determining the exact scaling dimensions for all operators in the gauge theory, or equivalently the full quantum energy spectrum for all string states, and verifying that they match. The solution of this problem would not only essentially amount to a proof of the AdS/CF T correspondence, but would also be the first step towards fully solving the duality in the following sense: Knowledge of the scaling dimensions completely determines the two-point correlators of the gauge theory (in

25 7 the appropriate normalization and assuming for simplicity that they are scalar), O A (x)o B (y) = δ AB, (1.2.3) x y 2 A and fixes the three-point functions up to a set of structure constants a ABC, O A (x)o B (y)o C (z) = a ABC x y A+ B C y z B + C A z x C + A B. (1.2.4) Then all higher point functions may in principle be obtained from the above with the help of the operator product expansion (OPE). As ambitious as the goal of the spectral problem may be, tremendous progress towards its realization has been achieved in the t Hooft or planar limit, where the only remaining parameter is the t Hooft coupling λ. On the one hand, only the set of single trace operators (1.2.3) survives in this limit, as multi-trace operators are suppressed by powers of 1/N. And on the other hand, even though the regimes where perturbation theory is applicable are incompatible in the two theories, λ 1 on the gauge side and λ 1 on the string side, the missing link between the two opposite regimes has been provided by integrability. 1.3 Integrability in Gauge and String Theory The role of integrability in the AdS/CFT spectral problem was realized after the key observation that the planar one-loop generator of scale transformations, dilatation operator D (1), takes the form of an SO(6) integrable spin chain hamiltonian when acting on operators consisting of only the scalar fields φ i, i = 1,..., 6 of N = 4 SYM [6]. Let us for simplicity restrict to a smaller subset of states, made of two complex scalars Z φ 1 + iφ 2 and W = φ 3 + iφ 4. Then any local single trace operator may be mapped to a

26 8 state of a cyclic spin chain Tr[ZZZWZZZZ] cyclic, (1.3.1) where the positions of the fields correspond to sites of the spin chain, the types of fields at each site to different spin directions, and cyclicity is imposed by the trace. More importantly, D (1) acting on the operators becomes proportional to the to the Heisenberg XXX 1/2 spin chain hamiltonian, D (1) = λ 8π 2H XXX 1/2 = λ 8π 2 L (I i,i+1 P i,i+1 ) (1.3.2) i=1 where P i,j is the operator that permutes the fields in sites i and j, and we impose periodic boundary conditions by identifying site L + 1 with site 1. Since P 2 i,j = I, it follows that up to the overall λ/(8π 2 ) factor, the eigenvalues of D (1) will lie in the range 0 E 2L. In more detail, E = 0 when all spins are aligned (ferromagnetic spin chain), and we may pick the ground state as Excitations of the ground state are given by up-spins or magnons, and we can think of them as particles moving in a vacuum made from the down-spins. A defining property of an integrable system is that it has as many conserved charges, as degrees of freedom. In chapter 2 we will show that this is indeed the case for the spin chain Moreover we will see that this property implies that the diagonalization of the hamiltonian, and hence the solution of the spectral problem in this sector, reduces to solving a simpler set of algebraic equations, known as the Bethe Ansatz equations. Can one expect similar structures also in the string theory side of the correspondence? The integrability of the corresponding worldsheet sigma model [7, 8] certainly suggests so, and in a variety of simplifying limits, such as the plane wave/bmn [9] limit, it has been established that spin chain states get mapped to solitonic solutions of string theory. Ideally 1 We could alternatively choose... without changing our description. In all systems where the ground state breaks the symmetries of the hamiltonian, one has choose a particular direction, but all such choices are equivalent.

27 9 one would like to make the relation between the spin chain and soliton language more precise, by identifying the duals of the elementary magnons and their boundstates. It turns out that this task can be achieved for the class of very long operators with few, well separated excitations having fixed momentum. If we assume that the down-spin state Z carries charge 1 under an angular momentum generator J of the SO(6) R-symmetry, then one restricts to the aforementioned class of operators by considering the Hofman-Maldacena limit [10],, J with J = constant, (1.3.3) λ, p = constant, (1.3.4) on top of the planar limit. The vacuum state of the spin chain Tr[Z J ] with J = 0 survives in this limit, and has a corresponding string solution known as the rotating point particle. More importantly, Hofman and Maldacena were able to find the string dual of a momentum eigenstate of single spin chain excitation, O p l e ipl( ZZZW l ZZZ ), (1.3.5) which was hence named the giant magnon solution. The identification was based on the known all-loop dispersion relation for the excitation, J = 1 + λ p π 2 sin2 2, (1.3.6) determined with the help of elaborate symmetry arguments 2 [11], which for semiclassical strings reduces to J = λ π sin p 2. (1.3.7) 2 More precisely, symmetry constrains the form of the dispersion relation up to a function of the t Hooft coupling f(λ), but for AdS 5 /CFT 4 all perturbative calculations suggest f(λ) = λ/π 2.

28 10 In chapter 3 we will verify this identification also at one loop in the strong coupling expansion. For any theory of scattering, there also exists the possibility of elementary excitations forming bound states, and this was indeed noticed to be the case already in Bethe s original paper [12], where the XXX 1/2 ferromagnet was solved (see also the excellent review [13]). In [14] it was argued that an infinite tower of bound states, made of J magnons, should also be present in the spectrum of the theory in the Hofman-Maldacena limit, having an exact dispersion relation J = J 2 + λ π 2 sin2 p 2. (1.3.8) The classical string solutions corresponding to these states were subsequently found in [15], where J becomes a second angular momentum charge of the S 5 isometry group, [J, J ] = 0. Because they carry two charges on the sphere, these solutions have been termed dyonic giant magnons, and we will have a closer look at them in chapter 4. Similar two-charge solutions of the AdS 4 /CFT 3 duality will also be considered in chapter 7. For a detailed review on the subject of AdS/CF T integrability, see [16] and references therein. 1.4 Overview With all the fundamental notions laid out in the preceding sections, we are now ready to give an overview of the main text of this thesis. Part I of the thesis focuses on the AdS 5 /CFT 4 example of the duality. In chapter 2 we review how the integrability of a given system can be proven with the help of the system s R-matrix satisfying the Yang-Baxter equation. We further specialize for the XXX 1/2 Heisenberg spin chain which arises in the SU(2) sector of SYM, and derive the corresponding Bethe Ansatz. In chapter 3 we move to the string theory side and focus on the dual of a single spin-chain excitation, the Hofman-Maldacena giant magnon. We perform its semiclassical quantization by obtaining the spectrum of bosonic and fermionic fluctuations around the classical solution, and calculate the first quantum correction to its energy to be zero, in

29 11 agreement with the gauge theory expectation. Chapter 4 deals with the dual of magnon bound states, the dyonic giant magnon. Since the latter is a single solitary wave living in an R S 3 subspace of the AdS 5 S 5 sigma model, we employ a nonlinear superposition method known as dressing, in order to construct a general formula describing the scattering of an arbitrary number of solitons. We further prove that our solution exhibits the property of factorized scattering as expected from integrability. In Part II we examine the more recent AdS 4 /CFT 3 instance of the correspondence, between ABJM theory and type IIA string theory on AdS 4 CP 3. In chapter 5 we develop group-theoretic tools for the detailed spectroscopic analysis of the integrable two-loop planar dilatation operator. In particular we compute how the Hilbert space of spin chain states of length up to 4, decomposes into irreducible representations of the superconformal group OSp(6 4), for which we also obtain the characters. As an application we compute the leading correction to the Hagedorn temperature of the weakly-coupled planar theory on R S 2. The structural information we obtain enables us to address the two-loop spectroscopy of anomalous dimensions for short operators in chapter 6. Employing a combination of Bethe Ansatz and Hamiltonian diagonalization techniques, we find the spectrum of low-lying states of length 4, and identify three new sequences of rational eigenvalues. Chapter 7 generalizes the analysis of chapter 4, by presenting and proving explicit multisoliton solutions for the sigma models with CP n, SU(n) and S n target spaces. A new feature is that we can choose the solitonic constituents to be giant magnons, spiky strings or other classes of solitonic string solutions. We further specialize our results for the CP 2 analogue of the dyonic magnon. A summary of our conclusions and open questions is contained in chapter 8, whereas appendix A contains information related to chapter 4, appendix B to chapter 5, appendix C to chapter 6, and finally appendices D and E to chapter 7.

30 Chapter 2 Integrable Spin Chains 12

31 13 As we briefly mentioned in the introduction, integrability refers to the property of a physical system to have as many conserved quantities, as degrees of freedom. In this review chapter, we will show that this is indeed the case for XXX 1/2 Heisenberg spin chain, which arises in the SU(2) sector of SYM, and whose hamiltonian is given by equation (1.3.2). From the hamiltonian, we will then proceed to derive the respective Bethe Ansatz equations, whose generalizations for larger algebras and higher loops have been so fruitful in the addressing the AdS/CFT spectral problem. A particular generalization we will see in chapter 6 is the 2-loop OSp(6 4) Bethe Ansatz for ABJM theory. In section 2.1 we follow [6] and develop the formalism for obtaining the set of commuting charges for a general spin chain, which is a variation of the quantum inverse scattering method developed by Faddeev (see [17] for a historical account, and [18] for an exhaustive reference). In section 2.2 we specialize to the case of the XXX 1/2 R-matrix, and we prove that the hamiltonian (1.3.2) is indeed among the set of mutually commuting charges it generates. The presentation here is also based on [19, 20]. Finally the derivation of the SU(2) Bethe Ansatz in section 2.3 is similar to that of [21, 22]. 2.1 R-matrix & Yang-Baxter Equation A systematic framework for constructing the conserved charges of integrable models, is by finding the corresponding R-matrix, an object that satisfies the so-called Yang-Baxter equation. In particular, for a spin chain of length L, where the states of each individual site transform in a representation V of a Lie algebra (e.g. vector of SO(6)), the corresponding Hilbert space will evidently be the L-fold tensor product V L and the R-matrix R ij (u) will act on the 2-fold tensor product between the representations of site i and site j, V 1... V i... V j... V L (2.1.1)

32 14 and will depend on the complex spectral parameter u. The R-matrix must satisfy the Yang-Baxter equation R ij (u v)r ik (u)r jk (v) = R jk (v)r ik (u)r ij (u v), (2.1.2) essentially a statement about the property of factorized scattering, which implies integrability. In order to show the existence of a set of commuting charges, we first need tensor the Hilbert space with two auxiliary spaces 1, V a V b V L, and then define the monodromy matrix M a (u) = R a1 (u)r a2 (u)... R al (u), (2.1.3) and similarly for a b, and we can think of it as matrix with elements M ia a j a (u) acting on the Hilbert space. From the last two equations, it is possible to prove the following relation between the R- and monodromy matrices, R ab (u v)m a (u)m b (v) = M b (v)m a (u)r ab (u). (2.1.4) This is done by first exploiting the property that R-matrices acting on different indices commute, in order to write the left-hand side of (2.1.4) as L R ab (u v) R ai (u)r bi (v), (2.1.5) i=1 and then using (2.1.2) in order to commute R ab through the pair R ai R bi, consecutively for each i = 1,..., L. If we multiply (2.1.4) on the left with Rab 1 (u), and employ matrix element form with 1 In general these auxiliary spaces can be different representations of the Lie algebra, however for our discussion we will assume that they coincide with V.

33 15 respect to the auxiliary spaces, we get M ia a j a (u)m i b b j b (v) = R 1iai b ab k ak b (u v)m k b b l b (v)m ka a l a (u)r lal b ab j aj b (u v). (2.1.6) Then taking the trace on the V a and V b space simply amounts to summing over diagonal elements, and yields Tr a [M a (u)]tr b [M b (v)] = Tr b [M b (v)]tr a [M a (u)]. (2.1.7) Since we are tracing over two auxiliary spaces that are in the same representation, we can drop the a, b indices without any ambiguity, and if we define the transfer matrix T(u) Tr[M(u)], (2.1.8) then the previous relation becomes [T(u), T(v)] = 0. (2.1.9) Namely the transfer matrix commutes with itself for different values of the spectral parameter, and this allows us to use it as a generating function for a set of mutually commuting charges Q r by expanding it around a point u 0 on the complex plane, T(u) = r (u u 0 ) r Q r [T (u),t (v)]=0 [Q r, Q s ] = 0. (2.1.10) It is hence clear, that if we wish to prove the integrability of a given hamiltonian, we have to construct an R-matrix satisfying the Yang-Baxter equation, such that one of the conserved charges coincides with the hamiltonian in question. In the next section we will illustrate this procedure for the particular case of an XXX 1/2 Heisenberg ferromagnet, with an SU(2) symmetry algebra.

34 The XXX 1/2 Heisenberg Spin Chain In section 1.3 we saw the emergence of the XXX 1/2 Heisenberg ferromagnet in the planar one-loop AdS 5 /CFT 4 spectral problem. Here we will show that its hamiltonian L H H XXX1/2 = (I i,i+1 P i,i+1 ). (2.2.1) i=1 with P i,j the permutation operator and site L + 1 is identified with site 1, belongs to a set of commuting charges generated by the R-matrix R ij (u) = ui i,j + ip i,j. (2.2.2) Let us first verify that (2.2.2) satisfies the Yang-Baxter equation. If we denote the identity operator acting on the entire spin chain as I, then clearly I i,j = I and the LHS of (2.1.2) reads R ij (u v)r ik (u)r jk (v) = (u v)uvi + iuvp i,j + i(u v)vp i,k + iu(u v)p j,k vp i,j P i,k up i,j P j,k (u v)p i,k P j,k ip i,j P i,k P j,k, (2.2.3) whereas the RHS may be obtained by exchanging i k and (u v) v. The first line of (2.2.3) immediately cancels between the left- and right- hand side, whereas for the second line we need to use the following permutation operator identities, P l,m P l,n = P l,n P m,n = P m,n P l,m, (2.2.4) for any three sites l, m, n, together with the obvious ordering symmetry P r,s = P s,r. These identities may be verified directly, for example for i < j < k, P i,k P j,k (a b c) = P i,k (a c b) = b c a = P i,j (c b a) = P i,j P a,c (a b c).

35 17 From the form of the R-matrix, it is clear that both the monodromy and transfer matrix will be polynomials of degree L with respect to the spectral parameter, and hence () takes the form of a finite expansion around u 0. Clearly Q 0 may be obtained from M a (0) = i L M (0), L M (0) = P a,l = P L,L 1 P L 1,L 2... P 3,2 P 2,1 P 1,a, (2.2.5) l=1 where the first equality follows from the definition of the monodromy and R-matrices, and the second equality may be proven with the help of the identities (2.2.4), or directly by inspecting that the action of both expressions on the V a V L tensor product is M (0) (v a v 1 v 2... v L ) = (v 1 v 2... v L v a ). (2.2.6) The main advantage of the second expression for M (0), is that tracing over the auxiliary space reduces to the trace of P 1,a only. The latter can be computed by expressing the permutation operator in the index notation, (P i,j ) r ir j s i s j = δ r j s i δ r i s j, (2.2.7) where it is hopefully clear that s i labels the particular component of the representation V that is found in the i-th site before the action of P i,j, r i labels the component in the same site after the action of P i,j, and similarly for i j. In this notation, and using the Einstein convention for the summation of indices, Tr a [ (P1,a ) r 1r a s 1 s a ] = Tra [ δ r a s 1 δ r 1 s a ] = δ r a s 1 δ r 1 r a = δ s 1 r 1 = (I 1 ) s 1 r 1 Tr a [P 1,a ] = I 1, (2.2.8) namely the identity operator acting on site 1. From (2.2.5) and (2.2.8) we can now easily find that U i L Q 0 = Tr a [M (0) ] is given by U = P L,L 1 P L 1,L 2... P 3,2 P 2,1, (2.2.9)

36 18 namely it is the shift operator, since it acts on a spin chain site as U(v 1 v 2... v L ) = v 2 v 3... v L v 1. (2.2.10) Moving on to find Q 1, we have to take the trace of the term in the transfer matrix multiplying u, M (1). This corresponds to collecting the u from one of the R-matrices in the product, say R 0,i (u), and a product of permutation operators from the rest. We can do this for any i = 1,..., L, so up to powers of imaginary units, M (1) will be given by summing terms of the form M (1) i i 1 P 0,k L k=1 k=i+1 P 0,l = M (0) P i,i+1, (2.2.11) where the last equality can be established by inspecting that both expressions act on an arbitrary state as M (1) i (v a v 1 v 2... v L ) = (v 1... v i 1 v i+1 v i v i+2... v L v a ). Computing the trace now yields L L Q 1 = Tr[M (1) ] = i L 1 Tr[ M (1) i ] = i L 1 Tr[M (0) L ] P i,i+1 = iq 0 P i,i+1, (2.2.12) i=1 i=1 i=1 which clearly implies that our spin chain hamiltonian (2.2.1) belongs to the family of mutually commuting charges! Since each site of the spin chain can be attributed one degree of freedom, and there exist L commuting charges, we have thus proven the integrability of our system. 2.3 SU(2) Bethe Ansatz In the previous section we proved the integrability of the XXX 1/2 spin chain, namely the existence of as many commuting charges, as degrees of freedom. But how does this fact help in any way in solving the system, namely finding its eigenvalues and eigenvectors? The

37 19 answer is that it it reduces the problem of diagonalizing a matrix whose size increases very fast with L 2, to a much smaller set of algebraic equations, known as the Bethe Ansatz equations. This structural simplification has the additional advantage of allowing for the study L limits. There are at least two ways to derive the algebraic equations in question, one is by extending the approach of the previous section (Algebraic Bethe Ansatz), which is however quite technical, and another one is by Bethe s original method (Coordinate Bethe Ansatz). Here we will describe the latter procedure, due to its simplicity and intuitiveness. Recalling the discussion of section 1.3 about the ground state of the spin chain and its excitations, let us denote a state with magnons in positions x 1 < x 2 <... < x M as x 1, x 2,... x M =... x 1... x 2... x M.... (2.3.1) From the form of H (2.2.1) we can see that its action does not change the total number of excitations M. For M = 1, it s easy to verify that simply the fourier transform, is an eigenstate of the hamiltonian, L ψ(p 1 ) = e i p 1 x x (2.3.2) x=1 H ψ(p 1 ) = = 2 L x=1 L x=1 e i p 1 x ( x + x 1 + x + x + 1 ) e i p 1 x x + L 1 x =0 e i p 1 (x +1) x + L+1 x =2 e i p 1 (x 1) x (2.3.3) = (2 e i p 1 e i p 1 ) ψ(p 1 ) or H ψ(p 1 ) = 4 sin 2 ( p 1 2 ) ψ(p 1), (2.3.4) 2 For a spin chain where the dimensionality of the single-site representation V is d, the hamiltonian becomes a d L d L matrix. Particularly for the Heisenberg spin chain example we are considering here, d = 2.

38 20 where we used the periodic boundary conditions x + L x. Invariance under x x + L for (2.3.2) also quantizes p = 2πn/L, n integer. Moving on to the case M = 2, if we assume the general form ψ(p 1, p 2 ) = ψ(x 1, x 2 ) x 1, x 2, (2.3.5) 1 x 1 <x 2 L and look for eigenvalues H ψ(p 1, p 2 ) = E ψ(p 1, p 2 ), we obtain the following two equations depending on whether x 1, x 2 are next to each other or not, x 2 > x E ψ(x 1, x 2 ) = 2 ψ(x 1, x 2 ) ψ(x 1 1, x 2 ) ψ(x 1 + 1, x 2 ) (2.3.6) + 2 ψ(x 1, x 2 ) ψ(x 1, x 2 1) ψ(x 1, x 2 + 1) x 2 = x E ψ(x 1, x 2 ) = 2 ψ(x 1, x 2 ) ψ(x 1 1, x 2 ) ψ(x 1, x 2 + 1). (2.3.7) Bethe solved these equations by making the following guess (ansatz) for the form of ψ, ψ(x 1, x 2 ) = e i(p 1 x 1 +p 2 x 2 ) + S(p 2, p 1 ) e i(p 2 x 1 +p 1 x 2 ), (2.3.8) where the first term signifies an incoming plain wave and the second term an outgoing plane wave, where the particles have simply exchanged momenta, and the only effect of the scattering is a solely momentum-dependent change in the amplitude of the plane wave, given by the S-matrix S(p 2, p 1 ). As such, it can be considered as a generalization of the Fourier transform, which would correspond to S(p 2, p 1 ) = 1. It is straightforward to check that the ansatz (2.3.8) solves (2.3.6) for arbitrary S-matrix, with an energy eigenvalue E = 4 sin 2 ( p 1 2 ) + 4 sin2 ( p 2 2 ), (2.3.9) namely a sum of one-magnon energies! Plugging the ansatz and the energy eigenvalue in

39 21 (2.3.7) then determines the form of the S-matrix to be S(p i, p k ) = u i u k + i u i u k i, i k, where u i = 1 2 cot(p i). (2.3.10) Finally we need to impose the periodic boundary condition (recall magnons are ordered by position on the chain) ψ(x 1, x 2 ) = ψ(x 2, x 1 + L), (2.3.11) which leads to the two-magnon Bethe equations, expressed in terms of the new rapidity variables u i we introduced as ( ) L ui + i/2 = u i u k + i, i, k = 1, 2, i k. (2.3.12) u i i/2 u i u k i Of course it is little surprise that two magnons scatter elastically, as this is the case for many systems with energy and momentum conservation. However scattering of three particles and up is generically inelastic, and it is from this point on that intregrable systems start exhibiting their remarkable properties. As we saw in the previous section, the Yang-Baxter equation implies factorized scattering, namely the reduction of multi-body scattering into a sequence of elastic two-body interactions where the change in amplitude is only a function of the exchanged momenta 3. Hence integrability constrains the M-magnon wavefunction to the form which Bethe guessed and verified that works, ψ(x 1,..., x M ) = [ M exp i p Pi x i + i ] θ Pi P P S M i=1 2 j i<j (2.3.13) where we sum over all M! permutations P of the labels {1, 2,..., M}, P i is the i-th ordered 3 We can heuristically understand how the existence of conserved charges implies conservation of momenta in 3-body scattering, with charges for example P = p i, E = p 2 i, Q 3 = p 3 i : The only solutions are permutations of the p i

40 22 element of a given permutation P 4, and the phase shifts θ ik parametrize the change in amplitude. Similarly to the two-magnon case, the diagonalization equation analogues of (2.3.6)- (2.3.7) are solved by e iθ ik = S(p i, p k ), (2.3.14) where S(p i, p k ) the two-magnon S-matrix 5 (2.3.10), and M 1 E = i=1 u i2 + 1, (2.3.15) 4 whereas imposing periodic boundary conditions leads to the Bethe equations for general M, ( ) L ui + i/2 = u i i/2 M k=1,k i u i u k + i, i = 1,..., M. (2.3.16) u i u k i These are the advertised algebraic equations, whose solutions determine the spin chain energies and eigenfunctions via (2.3.15) and (2.3.13). Solutions have to have no Bethe roots u i, u j being equal, since this would lead the wavefunction (2.3.13) to vanish due common factors of the form e iθ ij/2 + e iθ ij/2 = ±i i = 0, which is not physically meaningful. Finally, not all spin chain states will correspond to single trace operators of the gauge theory, since the latter are cyclically invariant, where as the former are generally not. In the spin chain, the trace condition means invariance under the discrete shift operator (2.2.10), and since the total momentum is the generator of such shifts, this translates into the zero momentum condition exp( p i ) = 1, or alternatively M i=1 u i + i/2 u i i/2 = 1. (2.3.17) When interested in the spectroscopy of anomalous dimensions, we will thus select only those 4 For example in the S 4 permutation P = {2, 1, 4, 3}, P 1 = 2, P 2 = 1 and so on. 5 Notice that the S(p i, p k ) 1 = S(p k, p i ) property of (2.3.10) implies θ ij = θ ji.

41 solutions to the Bethe equations, which additionally obey (2.3.17). 23

42 Chapter 3 Semiclassical Quantization of the Giant Magnon 24

43 Abstract Solitons in field theory provide a window into regimes not directly accessible by the fundamental perturbative degrees of freedom. Motivated by interest in the worldsheet S-matrix of string theory in AdS 5 S 5 in the limit of infinite worldsheet volume we consider the semiclassical quantization of a particular soliton of this theory: the Hofman-Maldacena giant magnon spinning string. We obtain explicit formulas for the complete spectrum of bosonic and fermionic fluctuations around the giant magnon. As an application of these results we confirm that the one-loop correction to the classical energy vanishes as expected. 3.2 Introduction It is difficult to overstate the important role that solitons play in quantum field theory, especially in integrable theories. Because of their particle-like nature their dynamics can be effectively captured by a small (in particular, finite) number of collective degrees of freedom. Quantization of these collective coordinates following the work of Gervais, Jevicki and Sakita [23, 24, 25, 26] provides a systematic framework for studying the theory in regimes not directly accessible by perturbation theory involving the original degrees of freedom. An interesting soliton to emerge in recent studies of the AdS/CFT correspondence is the giant magnon of Hofman and Maldacena [10]. This is a soliton of the integrable [7, 8] AdS 5 S 5 worldsheet sigma-model [27] (actually living inside an R S 2 subspace) whose image in spacetime is a stretched string that is pointlike in AdS 5 and rotates uniformly around an axis of an S 2 S 5. Its name derives from the fact that it is the dual description of an elementary excitation (magnon) in the spin chain description [6, 28, 29, 30, 31, 32, 11] of the dual N = 4 gauge theory. String theory on R S 2 is classically equivalent to sine-gordon theory [33] and the giant magnon is the image of the sine-gordon soliton under this equivalence. Classical aspects of the giant magnon, such as the phase shift for magnon scattering [10], can be understood in