String Theory Limits and Dualities Voor Heit en Mem. Omslagillustratie : bewerkte foto van de passage van boei nummer D24 tijdens een zeiltocht naar het Waddeneiland Texel, augustus 1999. Rijksuniversiteit Groningen STRING THEORY LIMITS AND DUALITIES PROEFSCHRIFT ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, Dr. D.F.J. Bosscher, in het openbaar te verdedigen op vrijdag 30 juni 2000 om 14.15 uur door Jan Pieter van der Schaar geboren op 28 december 1972 te Heerenveen Promotor: Prof. Dr. D. Atkinson Referenten: Dr. E.A. Bergshoeff Dr. M. de Roo Beoordelingscommissie: Prof. Dr. P. Di Vecchia Prof. Dr. P.K. Townsend Prof. Dr. E.P. Verlinde ISBN 90-367-1254-8 Contents 1 Introduction 7 2 String Theory 13 2.1 The bosonic string .......................... 13 2.1.1 Closed string spectrum ................... 18 2.1.2 Open string spectrum .................... 24 2.1.3 String interactions . .................... 26 2.1.4 T-duality and the appearance of D–branes . ....... 28 2.1.5 D–branes .......................... 32 2.1.6 Strings in general background fields ............ 35 2.2 The superstring . .......................... 38 2.2.1 The GSO–projection .................... 44 2.2.2 Closed Type IIA and Type IIB superstrings . ....... 45 2.2.3 The Heterotic superstring .................. 47 2.2.4 The Type I superstring ................... 48 2.2.5 Superstring T–duality and D–branes ............ 49 3 Superstring low energy limits and dualities 51 3.1 Supersymmetry algebras and BPS states . ............. 51 3.1.1 The D = 11 supersymmetry algebra ............ 52 3.1.2 Supersymmetry algebras in ten dimensions . ....... 57 3.2 Supergravities . .......................... 61 3.2.1 Supergravity in eleven dimensions ............. 62 3.2.2 Supergravities in ten dimensions . ............. 65 3.3 Dualities, M–theory and p–branes ................. 72 3.3.1 BPS p–brane solutions ................... 72 3.3.2 Dualities and effective worldvolume theories ....... 79 5 CONTENTS 4 Matrix theory 83 4.1 The Matrix model conjecture .................... 83 4.1.1 D0–branes and DLCQ M–theory .............. 84 4.2 BPS objects in Matrix theory .................... 90 4.2.1 Basic Matrix theory extended objects . ........ 91 4.2.2 Intersecting BPS objects in Matrix theory . ........ 99 4.2.3 Matrix BPS states and the M–theory algebra .......103 4.3 The status of Matrix theory .....................107 5 String solitons and the field theory limit 109 5.1 String soliton geometries . .....................109 5.1.1 Near–horizon geometries of p–branes . ........110 5.1.2 Domain–walls and Anti–de Sitter spacetimes .......116 5.2 The field theory limit . .....................126 5.2.1 The general setup . .....................126 5.2.2 The AdS/CFT examples . ...............134 5.2.3 Non–trivial dilaton Dp–branes in D = 10.........138 5.2.4 Non–trivial dilaton dp–branes in D = 6 ..........146 6 Emerging structure and discussion 155 A String theory units and charge conventions 159 Bibliography 163 Samenvatting 173 Dankwoord 181 6 Chapter 1 Introduction At this moment the physics of elementary particles is well described by the Stan- dard Model. At the smallest scales we can experimentally probe ( 100 GeV), the Standard Model predicts the outcomes of (scattering) experiments with incredi- ble accuracy. The Standard Model accommodates the (observed) constituents of matter, the quarks and leptons, and the vector particles responsible for the medi- ation of the strong and electroweak forces. The only ingredient of the Standard Model which still lacks experimental support is the Higgs boson particle, which is thought to be responsible for the breaking of the electroweak gauge symmetry and the masses of the different particles. One can therefore conclude that at this mo- ment there is no direct experimental need to construct and investigate theoretical models that go beyond the Standard Model1. On the other hand there are many theoretical reasons to go beyond the Stan- dard Model. The pillars of contemporary theoretical physics are quantum me- chanics and general relativity. The Standard Model is a collection of quantum (gauge) field theories which can be considered to be a merger of quantum me- chanics and special relativity, describing physics at small scales and relativistic energies. General relativity has proven its accuracy on large scales describing very massive objects. When we keep increasing the energy scales and at the same time keep decreasing our length scales, we expect new physics which is not described by either general relativity or the Standard Model. General relativity breaks down at short distances and in the Standard Model or in quantum field theory we should incorporate the effects of (quantum) gravitational interactions. The typical energy 1We have to remark here that recent experiments have most probably excluded the possibility that all neutrinos are massless, which requires a modification of the Standard Model. Still, in the context of this thesis we would like to consider this a minor modification. 7 Chapter 1. Introduction 19 scale at which this happens is called the Planck mass ( 10 GeV). This scale is way beyond the experimental energies currently accessible and one might wonder whether it will ever be possible to attain these energies in a controlled experiment. This does not mean, however, that this problem is of purely academic interest. 43 Immediately after the big bang ( 10 s), from which our observable universe evolved, the energies were of the order of the Planck scale and the physics during that (short) time determined the further development of the universe. It is therefore of direct interest in cosmology to search for a theory that is able to unify or merge quantum mechanics and general relativity. Other arguments are of a more theoretical and/or aesthetic nature. The Stan- dard Model contains a huge amount of parameters which have to be determined by experiments, e.g. masses, coupling constants, angles and so on. One would expect (or perhaps one likes to expect) that a fundamental theory of nature will not allow too many adjustable parameters. In the best case scenario, we would like our the- ory to be unique. Many people therefore like to think of the standard model as an effective theory, only applicable at our current available energy scales [1]. This point of view is backed up by considering the running of the coupling constants of the different gauge theories that are part of the Standard Model. When plotting the coupling constants as a function of the energy scale, one finds that at a particular high energy scale, referred to as the Grand Unified Theory (GUT) energy scale 14 ( 10 GeV), the three coupling constants all seem to meet in (approximately) the same point. This suggests a possible unified description at and above the GUT energy scale. The GUT energy scale lies several orders of magnitude below the Planck scale, so this unified theory would not involve quantum gravity. However, for the three gauge theory coupling constants to meet at the same point, the theoretical concept of supersymmetry improves on the approximate result without supersymmetry. Supersymmetry is a symmetry that connects bosons and fermions: starting with a bosonic particle one can perform a supersymmetry transformation and end up with a fermionic particle. In our world supersymmetry, if it exists, must be a broken symmetry, because we have not detected any supersymmetric partners of the Stan- dard Model particles. Global supersymmetric quantum field theories have slightly different properties than their non–supersymmetric counterparts. Most impor- tantly, in the perturbative expansion cancellations take place between bosons and fermions, generically making supersymmetric quantum theories better behaved. Local supersymmetric theories automatically include supergravity, the supersym- metric version of general relativity. So Grand Unified Theory, supersymmetry and supergravity are intimately linked, which again suggests a possible unified 8 description of gauge field theories and quantum gravity at the Planck scale. From another point of view the basic fact that general relativity is a classical field theory is unsatisfactory. At scales around the Planck length, quantum gravi- tational effects are bound to become important and we will need a quantum theory of gravity. However, so far general relativity has resisted all standard methods of quantization and is said to be non–renormalizable (for an overview see [2]). One may think this is just a technical problem, but there are fundamental interpreta- tional problems as well when trying to quantize general relativity because we are trying to quantize space and time. Many theoretical physicists agree that in order to deal with the problem of quantization of space and time a radically new ap- proach is called for. This is also emphasized by the confusing properties of black holes in general relativity, which in a semiclassical approach are not black at all and emit black body Hawking radiation. Studying these objects in general relativ- ity, it turns out that one can formulate black hole laws that are strikingly similar to the laws of thermodynamics. For example, one can assign a temperature and an entropy to a black hole. At this moment one of the key questions in theoretical physics is to try to understand what the fundamental degrees of freedom are that make up the entropy of the black hole and how these thermodynamic degrees of freedom arise from (quantum) general relativity. At this moment, string theory is the only theoretical construction that can deal with quantum gravity2 albeit in a perturbative, background dependent way. String theory needs supersymmetry and extra spacetime dimensions to be set up consis- tently (free of anomalies). In fact, there exist five different superstring theories which are all living in ten spacetime dimensions and which are distinguished by the number of supersymmetries and by the kind of strings (open and/or closed). The construction of these five different anomaly free string theories is referred to as the first string revolution (1984 1985).