NONPERTURBATIVE FORMULATIONS OF SUPERSTRING THEORY BY LUBOSˇ MOTL A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Physics and Astronomy Written under the direction of Thomas Banks and approved by arXiv:hep-th/0109149v1 19 Sep 2001 New Brunswick, New Jersey October, 2001 ABSTRACT OF THE DISSERTATION Nonperturbative Formulations of Superstring Theory by LuboˇsMotl Dissertation Director: Thomas Banks After a short introduction to Matrix theory, we explain how can one generalize matrix models to describe toroidal compactifications of M-theory and the heterotic vacua with 16 supercharges. This allows us, for the first time in history, to derive the conventional perturbative type IIA string theory known in the 80s within a complete and consistent non- perturbative framework, using the language of orbifold conformal field theory and conformal perturbation methods. A separate chapter is dedicated to the vacua with Hoˇrava-Witten domain walls that carry E8 gauge supermultiplets. Those reduce the gauge symmetry of the matrix model from U(N) to O(N). We also explain why these models contain open membranes. The compactification of M-theory on T 4 involves the so-called (2, 0) supercon- formal field theory in six dimensions, compactified on T 5. A separate chapter describes an interesting topological contribution to the low energy equations of motion on the Coulomb branch of the (2, 0) theory that admits a skyrmionic solution that we call “knitting five- branes”. Then we return to the orbifolds of Matrix theory and construct a formal classical matrix model of the Scherk-Schwarz compactification of M-theory and type IIA string theory as well as type 0 theories. We show some disastrous consequences of the broken supersymmetry. Last two chapters describe a hyperbolic structure of the moduli spaces of one-dimensional M-theory. ii Acknowledgements I am grateful to the many people who helped me along the way, explained many things to me and to those that have shared their friendship and love. For example, I am grateful to Micha Berkooz, Michael Dine, Rami Entin, Michal Fabinger, Tomeu Fiol, Willy Fischler, Ori Ganor, David Gross, Petr Hoˇrava, Marcos Mari˜no, Greg Moore, Arvind Rajaraman, Christian R¨omelsberger, Moshe Rozali, Steve Shenker, Andy Strominger, Lenny Susskind, Herman Verlinde, Edward Witten and many others. A lot of thanks goes to my advisor Tom Banks who not only taught me a lot of things about physics but who was also a significant source of my support in the USA. My work was supported in part by the Department Of Energy under grant number DE- FG02-96ER40559. I would like to thank the ITP in Santa Barbara for hospitality during the final stages of this work. iii Dedication For the victims of 20th century totalitarian regimes, those who have been executed, jailed or dismissed from schools by fascists and communists. iv Table of Contents Abstract .......................................... ii Acknowledgements ................................... iii Dedication ......................................... iv List of Tables ....................................... ix List of Figures ...................................... x 1. Introduction to M(atrix) theory ......................... 1 1.1. BasicsofM(atrix)theory . ... 2 1.1.1. Membranes ................................ 3 2. Matrix string theory ................................ 6 2.1. Reflections on the second quantization and its matrix competitor . 6 2.2. M(atrix) theory in 0 + 0 and 0 + 1 dimensions . ..... 9 2.2.1. An old-fashioned approach to the compactification of one spatial co- ordinate.................................. 11 2.2.2. Limits of the resulting type IIA theory . ..... 13 2.3. Representation of the “long” strings, screwing strings to matrices . 14 2.4. The origin of the level matching conditions and the interactions . 16 2.4.1. The DVV interaction vertex . 17 2.4.2. Compactification to nine dimensions . 19 3. Heterotic matrix models .............................. 21 3.1. Introduction................................... 21 3.2. Heterotic matrix models in ten and eleven dimensions . .......... 23 v 3.2.1. Heterotic string theory as an orientifold of M-theory......... 23 3.2.2. Open membranes in M(atrix) theory . 24 3.2.3. Additional degrees of freedom . 25 3.2.4. Screwing heterotic strings . 29 3.3. Multidimensionalcylinders . ..... 31 3.3.1. TypeIAstrings.............................. 33 3.4. Conclusions .................................... 35 3.4.1. Cosmological constant problem . 36 4. Knitted fivebranes in the (2,0) theory ..................... 38 4.1. Introduction................................... 38 4.2. Reviewofthe(2,0)theory. 39 4.2.1. Realization ................................ 39 4.2.2. Equations of motion for a free tensor multiplet . ....... 40 4.3. Low-energy correction terms – derivation from SUGRA . ......... 42 4.4. Compactification ................................ 45 4.4.1. Dimensional reduction of the correction term . ....... 46 4.4.2. Interpolation between 3+1D and 5+1D . 47 4.4.3. A derivation from 4+1D SYM . 48 4.4.4. Componentform ............................. 49 4.5. Conservedquantities . 51 4.5.1. Traceless stress tensor and supercurrent . ....... 53 4.6. Speculations about the fundamental formulation . .......... 56 4.7. String-like solution of the (2,0) theory . ......... 57 4.7.1. Aroughpicture.............................. 58 4.7.2. TheAnsatz ................................ 58 4.7.3. Numerical solution, tension and speculations . ........ 59 4.8. Discussion..................................... 60 4.9. Appendix A: Formulae for SUSY transformations . ........ 61 vi 4.9.1. SUSYtransformation . .. .. .. .. .. .. .. 61 4.10. Appendix B: Quantization . 62 4.10.1.Commutators............................... 63 4.10.2. Correspondence to Super Yang Mills . 65 4.10.3. Normalization of the current . 67 4.11.AppendixC:Identities . 67 4.11.1. Identities for gamma matrices . 67 4.11.2.Fierzrearrangments . 68 4.11.3. A few notes about spin(5, 1) ...................... 70 5. Nonsupersymmetric matrix models ....................... 72 5.1. The conformal field theory description . ....... 72 5.2. Thematrixmodel ................................ 76 5.2.1. The U(N) U(N)formalism ...................... 78 × 5.2.2. Actions for the local and nonlocal formulations . ........ 79 5.3. Alternative derivation and connection with type 0 theories.......... 81 5.4. Thematrixstringlimits . 83 5.4.1. Rohm compactified type IIA strings . 83 5.4.2. Dependenceonthefluxes . 84 5.4.3. Type0matrixstrings .......................... 87 5.4.4. Breakdown of the cluster property . 88 5.5. Conclusions .................................... 92 6. Asymptotic limits in the moduli space of M-theory ............ 94 6.1. Introduction................................... 94 6.2. Moduli, vacua, quantum cosmology and singularities . ........... 96 6.2.1. Some idiosyncratic views on general wisdom . ...... 96 6.2.2. Quantumcosmology .. .. .. .. .. .. .. .. 98 6.2.3. Kasner singularities and U-duality . 100 6.3. The moduli space of M-theory on rectangular tori . ......... 103 vii 6.3.1. The2/5transformation . 104 6.3.2. Extrememoduli ............................. 105 6.3.3. Coveringthemodulispace. 108 6.4. ModulispaceswithlessSUSY. 112 6.5. Discussion..................................... 116 6.6. Appendix A: The appearance of Γ8 and Γ9,1 .................. 117 7. Hyperbolic structure of the heterotic moduli space ............. 120 7.1. Introduction................................... 120 7.1.1. Thebilinearform............................. 122 7.1.2. Moduli spaces and heterotic S-duality . 124 7.2. CoveringtheSO(32)modulispace . 127 7.2.1. Heterotic strings, type I, type IA and d 9.............. 127 ≥ 7.2.2. Type IA2 and d =8 ........................... 129 7.2.3. Type IA3, M-theory on K3 and d =7 ................. 130 7.2.4. Type IA4,5, type IIA/B on K3 and d = 6, 5 .............. 133 7.2.5. Type IA6 and S-duality in d =4 .................... 134 7.2.6. The infinite groups in d 2....................... 136 ≤ 7.2.7. Thelattices ................................ 137 7.3. Conclusions .................................... 138 References ......................................... 139 Vita ............................................. 146 viii List of Tables 2 2 3.1. Matrix representations of Z2 orbifolds of the sphere S and the torus T : here Ylm are the spherical harmonics and Jr,s are the Fourier modes on the torus. Both are represented by some matrix generators in the regularized case. 25 ix List of Figures 5.1. One-loopdiagrams.. .. .. .. .. .. .. .. .. 90 5.2. Non-vanishing two-loop contributions to the effective potential. Here V ′ 2 ≡ (ǫ(p) 1)V ..................................... 90 − 2 6.1. The structure of the moduli space for T 2. ................... 107 7.1. The limits in d =9................................. 128 x 1 Chapter 1 Introduction to M(atrix) theory Superstring theory has been known to lead to a consistent perturbative S-matrix for a decade or so, since the discoveries of Green and Schwarz in the middle 80s. Superstring the- ory is only consistent in ten spacetime dimensions and it can be approximated by supergrav- ity theories at low energies (or supergravities coupled to supersymmetric Yang-Mills theo- ries). However the most symmetric supergravity theory required eleven spacetime dimen- sions and seemed to be disconnected from superstring theory