Light Cone Gauge Quantization of Strings, Dynamics of D-brane and String dualities. Muhammad Ilyas1 Department of Physics Government College University Lahore, Pakistan Abstract This review aims to show the Light cone gauge quantization of strings. It is divided up into three parts. The first consists of an introduction to bosonic and superstring theories and a brief discussion of Type II superstring theories. The second part deals with different configurations of D-branes, their charges and tachyon condensation. The third part contains the compactification of an extra dimension, the dual picture of D-branes having electric as well as magnetic field and the different dualities in string theories. In ten dimensions, there exist five consistent string theories and in eleven dimensions there is a unique M-Theory under these dualities, the different superstring theoies are the same underlying M-Theory. 1ilyas Contents Abstract i INTRODUCTION 1 BOSONIC STRING THEORY 6 2.1 The relativistic string action . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 The Nambu-Goto string action . . . . . . . . . . . . . . . . . . 7 2.1.2 Equation of motions, boundary conditions and D-branes . . . . 7 2.2 Constraints and wave equations . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Open string Mode expansions . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Closed string Mode expansions . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Light cone solution and Transverse Virasoro Modes . . . . . . . . . . . 14 2.6 Quantization and Commutations relations . . . . . . . . . . . . . . . . 18 2.7 Transverse Verasoro operators . . . . . . . . . . . . . . . . . . . . . . . 22 2.8 Lorentz generators and critical dimensions . . . . . . . . . . . . . . . . 27 2.9 State space and mass spectrum . . . . . . . . . . . . . . . . . . . . . . 29 SUPERSTRINGS 32 3.1 The super string Action . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Equations of motion and Boundary conditions . . . . . . . . . . 33 3.2 Neveu Schwarz sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Ramond sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Super transverse Virasoro operators . . . . . . . . . . . . . . . . . . . . 40 3.5 Counting states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 Open superstrings and the GSO projection . . . . . . . . . . . . . . . . 43 3.7 Closed string theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ii 3.7.1 Type IIA Superstring Theory . . . . . . . . . . . . . . . . . . . 45 3.7.2 Type IIB Superstring Theory . . . . . . . . . . . . . . . . . . . 46 3.7.3 Heterotic superstring theories . . . . . . . . . . . . . . . . . . . 47 3.8 Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.9 Critical dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 D-BRANES 50 4.1 Tachyons and D-brane decay . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Quantization of open strings in the presence of various kinds of D-branes 52 4.2.1 Dp-branes and boundary conditions . . . . . . . . . . . . . . . . 52 4.2.2 Quantizing open strings on Dp-branes . . . . . . . . . . . . . . 53 4.2.3 Open string between parallel Dp-branes . . . . . . . . . . . . . 55 4.2.4 Strings between parallel Dp and Dq-branes . . . . . . . . . . . . 58 4.3 String charge and electric charge . . . . . . . . . . . . . . . . . . . . . . 60 4.3.1 Fundamental string charge . . . . . . . . . . . . . . . . . . . . . 60 4.3.2 Visualizing string charge . . . . . . . . . . . . . . . . . . . . . . 62 4.3.3 Strings ending on D-branes . . . . . . . . . . . . . . . . . . . . 63 4.4 D-brane charges and stable D-branes in Type II . . . . . . . . . . . . . 65 STRING DUALITIES 69 5.1 T-duality and closed strings . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1.1 5.1.1 Mode expansions for compact dimension . . . . . . . . . . 69 5.1.2 Quantization and commutation relations . . . . . . . . . . . . . 71 5.1.3 Constraint and mass spectrum . . . . . . . . . . . . . . . . . . . 72 5.1.4 State Space of compactified closed strings . . . . . . . . . . . . 73 5.2 T-Duality for Closed Strings . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.1 Type II superstrings and T-duality . . . . . . . . . . . . . . . . 76 5.3 T-duality of open strings . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3.1 T-duality and open strings . . . . . . . . . . . . . . . . . . . . . 76 5.3.2 Open strings and Wilson lines . . . . . . . . . . . . . . . . . . . 78 5.4 Electromagnetic fields on D-branes and T-duality . . . . . . . . . . . . 80 5.4.1 Maxwell fields coupling to open strings . . . . . . . . . . . . . . 80 5.4.2 D-branes with Electric fields and T-dualities . . . . . . . . . . . 80 5.4.3 D-branes with Magnetic fields and T-dualities . . . . . . . . . . 82 5.5 String coupling and the dilaton . . . . . . . . . . . . . . . . . . . . . . 83 5.6 S-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 A Appendix A The Massless States Of Closed String 86 B Appendix B The Spinors Algebra In 2D 88 INTRODUCTION General relativity and quantum mechanics were the two major breakthroughs that revolutionized theoretical physics in the twentieth century. General relativity gives the idea to understand of the large-scale expansion of the Universe and gives a small correction to the predictions of Newtonian gravity for the motion of planets and the deflection of light rays, and it predicts the existence of gravitational radiation and black holes. It describes the gravitational force in terms of the curvature of spacetime which has fundamentally changed our view of space and time i.e. they are now viewed as dynamical [1]. Quantum mechanics, on the other hand, is the essential tool for understanding the sub atomic particles and microscopic physics. The evidence continues to build that it is an exact property of Nature [2]. The fundamental law of Nature is surely incomplete until general relativity and quantum mechanics are successfully reconciled and unified. String theory is a candidate which resolves this problem and a straightforward attempt to combine the General relativity and Quantum mechanics. String theory is based on the idea that particles are not point-like, but rather tiny loops (i.e. closed strings) or (open) pieces of string i.e. the 0-dimensional point particle is replaced by a 1- dimensional string [3]. This assumption leads to some features which are 1 2 General features Even though string theory is not yet fully formulated, and we cannot yet understand the detailed description of how the standard model of elementary particles should emerge at low energies, or how the Universe originated, there are some general features of the theory that have been well understood. Vibrating string String theory predicts that all objects in our universe are composed of Vibrating strings and different vibrational modes of the strings represent different kinds of particles. Since there is just one type of string, and all particles arise from string vibrations, all particles are naturally incorporated into a single theory [3]. Gravity String theory attempts to reconcile the General relativity and Quantum mechanics. One of the vibrational modes of strings is the graviton particle, the quantum version of the gravity, so string theory has the remarkable property of predicting gravity [3]. Unification of forces There are four fundamental forces that had been recognized to exist in nature. 1. Electromagnetic force 2. Weak force 3. Strong force 4. Gravitational force As the quantum version of electromagnetism describes the photon (a massless particle) and its interactions with charged particles, while the Yang-Mills theory describes W 3 and Z bosons and gluons (the mediators of the weak and the strong nuclear forces) and their interactions. All of these theories make a single theory named the Standard Model of particle interactions, which is a gauge theory. The Standard Model of particle physics does not include the graviton particle and its interactions. The graviton which has spin 2 can not be described by the gauge theory. Since the standard model is believed to incomplete due to it does not incorporate the gravity forces. String theory is currently the most promising candidate to unify all the fundamental forces. This is a general feature of the string theory [3]. Yang-Mills gauge theory Standard model of particle physics describe the elementary particles in nature. It reconciles the special relativity and Quantum mechanics [4]. And is based on Yang- Mills theory having the gauge group SU(3) × SU(2) × U(1) However it has some shortcomings. It does not include Gravity and it has about 20 parameters that cannot be calculated and we use them as an input. While string theory predicts the gravity as well as describe all the elementary particles and has one parameter, the string length. Its value is roughly equal to the typical size of strings. Yang-Mills gauge theories arise very naturally in string theory. However, it is not yet fully understood why the gauge group SU(3) × SU(2) × U(1) Of the standard model with three generations of quarks and leptons should be singled out in nature [5, 6]. Supersymmetry A supersymmetry is a symmetry which relates bosons and fermions. There exists a non supersymmetric bosonic string theory which is an unrealistic theory due to the lack 4 of fermions. In the order to get a realistic string theory which explains the beauty of nature, we need a supersymmetry. Hence supersymmetry is a general feature of string theory [3]. Extra dimensions of space In quantum field theory, for point particle, we let the dimensions of the spacetime to be four while superstring theories predict some additional dimensions of the spacetime. The superstring theories are only able to work in a ten dimensions or eleventh (in some cases) dimensions of spacetime. To make an ordinary four dimensional space time, there is a straight forward possibility that is, the additional six or seven dimensions can be curled up and compactified on an internal manifold having the sufficient small size, which can not be detectable at the low energies. The idea of an extra dimension was first introduce by Kaluza and Klein in 1920s. Their aim was to unify the electro- magnetic force and the gravitational force. The compactification of an extra dimension can be imagined as, let us consider we have a cylinder having the radius R. when the cylinder is viewed from a very large distance or equivalently, when the radius of the cylinder R becomes too small then the two dimensional cylinder will look like a one dimensional line. Generalizing this idea by letting the cylinder as a four dimensional spacetime and replac- ing the short circle of radius R (compact space) by a six or seven dimensional manifold, hence at large distance or at the low energies, the additional dimensions (compact man- ifold) can not be visible. These additional dimensions or compact manifold are called Calabi-Yau manifolds [3]. The size of the strings Quantum field theory deals the particles as a mathematical zero dimensional point while in string theory the ordinary point particles are replaced by a one dimensional string. These one dimensional strings will have a characteristic length scale which is denoted 5 be and can estimated by the dimensionality analysis. As string theory is a relativistic quantum theory which also includes the force of gravity, since it must involves the fun- damental constants speed of light c Planck’s constant h¯, and the gravitational constant G From these, we can form a length, called the Planck length ( ) 1/2 h¯G −33 ls = = 1.6 × 10 cm , 3 c The Planck mass becomes ( ) 1/2 h¯c −5 mp = = 2.176 × 10 g , G And similarly the Planck time ( ) 1/2 h¯G −44 tp = = 5.391 × 10 s . 5 c The Planck length scale is a natural guess for a fundamental string length scale as well as the characteristic size of compact extra spatial dimensions. At low energies string can be approximated by the point particle that explains why the quantum field theory has been successfully in the describing our world. The relativistic quantum gravity effects can be important on the above three scales of planks length, planks mass and planks time [3]. BOSONIC STRING THEORY The bosonic string theory is the simplest string theory that predicts and describes only a certain set of boson. As the theory does not describe any fermions, so it is an unrealistic theory but this theory is a natural place to start, because the same techniques and structures, together with some additional terms are required for analysis of more realistic theory (super string theories). String can be regarded as 1-brane moving through a space-time which is a special case of a p-brane, p-dimensional extended object. Point particle corresponds to 0-brane. Similarly the two dimensional extended object or 2-brane are called membranes. 2.1 The relativistic string action In quantum field theory, the action for point particle (0-brane) is proportional to the invariant length of the word-line or particle trajectory. Similarly when we replace the point particle by 1 dimensional string (1-brane) then the action is proportional to the proper area of the word-sheet or the area swept out by the string in D dimensional space-time. ∫ SNG = − T dA (2.1) The word-sheet is parameterized by the two coordinates ξ0= τ which is time-like and ξ1= σ which is space-like. For close string the σ will be periodic while for open string σ will be have some finite value [3]. 6