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M(Embrane)-Theory Master of science thesis in physics M(embrane)-Theory Viktor Bengtsson Department of Theoretical Physics Chalmers University of Technology and GÄoteborg University Winter 2003 M(embrane)-Theory Viktor Bengtsson Department of Theoretical Physics Chalmers University of Technology and GÄoteborg University SE-412 96 GÄoteborg, Sweden Abstract We investigate the uses of membranes in theoretical physics. Starting with the bosonic membrane and the formulation of its dynamics we then move forward in time to the introduction of supersymmetry. Matrix theory is introduced and a full proof of the continuous spectrum of the supermembrane is given. After this we deal with various concepts in M-theory (BPS-states, Matrix Theory, torodial compacti¯cations etc.) that are of special importance when motivating the algebraic approach to M-theoretic caluclations. This approach is then dealt with by ¯rst reviewing the prototypical example of the Type IIB R4 amplitude and then the various issues of microscopic derivations of the corresponding results through ¯rst-principle computations in M-theory. This leads us to the mathematics of automorphic forms and the main result of this thesis, a calculation of the p-adic spherical vector in a minimal representation of SO(4; 4; Z) Acknowledgments I would like to extend the warmest thanks to my supervisor Prof. Bengt E.W. Nilsson for his unwavering patience with me during the last year. Many thanks also to my friend and collaborator Dr. Hegarty. I am most grateful to Dr. Anders Wall and the Wall Foundation for funding during the last year. I would like to thank Prof. Seif Randjbar-Daemi and the ICTP, Trieste, for their hospitality during this summer as well as Dr. Pierre Vanhove and SPhT in Saclay for their kind welcome. I thank the organizers of TH-2002 for their excellent work. I have also had fruitful discussions with Prof. Boris Pioline on the more recent algebraic approaches to problems in M-theory, Robert Berman on appli- cations of geometrical quantization and Ronnie Jansson on the dynamics of (super)membranes. I thank my friends at the Department of Theoret- ical Physics at GÄoteborg University, the young brats as well as the old geezers. And last but not in anyway least I thank the friends who need not be named since they know all too well who they are and how much they mean to me. iv Preface The thesis now in your hand has undergone several revisions before as- suming this, its ¯nal form. Upon commencing work on this version, in the late summer of 2002, I set out a number of goals for myself. One of these goals was to try to write an as self-contained thesis as possible, the amount of paper in your hand at this very moment is a direct conse- quence of this goal. Another goal was to create a ¯rm base of knowledge to stand on in my future research, so I traced back to the very birth of the membrane, an event taking place in an ancient era shrouded in mystery and known to some as 'the sixties'. This is where the journey taking place in this thesis begins. It then spans an interval of some forty of the lord's years, a period that saw the birth and death of many excel- lent attempts in physics (and in membrane theory). Roaming across this vastness of publications was a very humbling task, as I have come across many seminal ideas, but also some that make me wonder whether future generations will say about these times that\Some things that should not have been forgotten, were lost". The early parts of this thesis can best be described as a collection of reviews, and though I present no new results I have tried to collect these reviews in a manner which I have found in no other publication so far. In the latter part it becomes easier to add at least some insight and new ideas to the presented material as it is both incomplete and something that I have spent a great deal of time working on myself. I have also taken the risk of including some of my own thoughts and ideas in the last chapter in hope of at best awake some interest or at least amusement. With these words I leave the prospective reader to walk the path which I have cut through the wilderness of actions and symmetries, God- speed. -TÄank dig hur enkelt det var, kunde han sucka. TÄank dig 1900- talets lilla universum, en liten hemtrevlig rymd med nagr_ a mil- jarder vintergator, nagr_ a miljoner ljusar_ ifran_ varandra. Man kunde sitta sa_ trygg vid sitt teleskop och nÄastan kÄanna hÄodoften och hÄora fagelkvittr_ et utifran_ rymderna . - Peter Nilson v vi Contents 1 Introduction 1 1.1 History . 2 1.2 Outline . 4 1.3 Notation and Conventions . 7 2 The Supermembrane I: General Theory and Problems 9 2.1 The Bosonic Membrane . 9 2.2 Adding Supersymmetry . 18 3 The Supermembrane II: M-theory 39 3.1 Five String Theories . 39 3.2 Dualities, D-branes and Moduli . 45 3.3 BPS States . 48 3.4 Representation Theory of Duality Groups . 49 3.5 The BFSS-Conjecture . 52 4 The Supermembrane III: An Algebraic Approach 59 4.1 Using Exact Symmetries in M-Theory . 59 4.2 Membrane Amplitudes as Automorphic Forms . 70 4.3 Calculation of Automorphic Forms . 75 5 Conclusion 95 5.1 The Bosonic Membrane . 95 5.2 Membranes, Supersymmetry and Matrices . 96 5.3 p-adic Numbers in Physics . 97 5.4 The Algebraic Approach . 99 A Brief Introduction to Supersymmetry and Supergravity 103 A.1 The Wess-Zumino Model . 103 A.2 Supersymmetry Multiplets and BPS States . 108 A.3 Instanton Solutions in Type IIB Supergravity . 111 A.4 Type IIB and (p; q)-strings . 113 vii B The Field of p-adic Numbers 117 C A Touch of Number Theory 125 C.1 Eisenstein Series . 125 C.2 Additions concerning Sl(2;Z)(¿) . 130 E2;s Bibliography 133 Notation 140 viii List of Figures 2.1 The potential x2y2 . 31 3.1 The moduli space of M-theory . 46 3.2 Object correspondence between Type IIA and M-theory . 49 3.3 The Cremmer-Julia groups and their maximal compact subgroups . 51 3.4 The U-duality groups . 51 4.1 Tadpole diagram with closed string stated and fermionic zero modes coupling to a open string worldsheet. 61 4.2 R4 tadpole diagram . 62 4.3 Dual pairs related to various level of compacti¯cation. 75 4.4 Dynkin diagram of D4 . 83 4.5 Groups and their corresponding cubic form I3 as well as subgroup H0. 84 ix x 1 Introduction String theory has been said to be\21th century physics cast into the 20th century" and the same thing can undoubtedly be said about membrane theory (or M-theory). The many fundamental questions that have yet to be answered can best be summed up in the single question \What is M-theory?". Being less general we can also ask the question, what is the membrane? The theory of membranes has been envisaged to describe a multitude of physical systems, none of which have been completely suc- cessful or adequately investigated. From electron models to bag-models onto relativistic surfaces and more recently fundamental degrees of free- dom the fundamental problems essentially remains the same. This thesis is an attempt to review all of these attempts to some extent, highlight- ing the problems and noting the di®erent attempts at solving them. The thread running through each and every chapter is the desire to gain un- derstanding about the aforementioned question regarding the true nature of the membrane and it is this question that drives us through several decades of physics and a body of material so vast that no one can claim to have overlooked it all. The content of this thesis rests upon the shoul- ders of those who have gone before us and to which we should be ever so grateful. Are we ahead of our times? Are we presumptive in thinking that we can answer the questions that stand before us? Perhaps we are. But in science there is no way to go but forward and boldly so. It is our obligated duty as physicists, scientists, humans and inhabitants of the ever-expanding entity that is our universe to grab a pick-ax and hack away at the mountain of unresolved issues1. 1But also to drink loads of co®ee and show the world how intellect enables us to pick up beautiful women in spite of looking like road-kill ourselves. 1 2 Chapter 1 Introduction 1.1 History The history of membranes in theoretical physics is a long and complicated one. The ¯rst real attempt to use membranes to construct a fundamen- tal theory was [1] where Dirac considered the electron to be a charged conducting surface, \a bubble in the electromagnetic ¯eld". This the- ory never really became a hit and it su®ers from a multitude of prob- lems, some that are general membrane-problems that we will study here. Membranes became popular again with the rising interest in string theory during the 70's. The reason for this was quite natural, after all, if one con- siders extended objects of one dimension why then not consider extended object of two, three, four and generally d dimensions. The ¯rst thorough analysis of the dynamics of classical and quantum (bosonic) membranes was done by Collins and Tucker in [2]. This was before string theory was regarded as a TOE and one of the chief motivations for studying mem- branes was to describe the dynamics of quarks.
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