REVISTA MEXICANA DE FISICA´ S 53 (4) 66–77 AGOSTO 2007 String theory: a decade of progress A. Guijosa¨ Departamento de F´ısica de Altas Energ´ıas, Instituto de Ciencias Nucleares, Universidad Nacional Autonoma´ de Mexico,´ Apartado Postal 70-543, Mexico´ D.F. 04510, e-mail: [email protected] Recibido el 1 de mayo de 2006; aceptado el 1 de noviembre de 2006 String theory aims at providing a complete description of the fundamental structure of our universe. Independently of whether or not it is eventually able to achieve this ambitious goal, over the years it has already proven to be an enormously rich theoretical structure, with several points of contact with other problems of interest in modern theoretical physics. In this article we review the basic ideas of string theory, and provide a brief overview of the achievements of the past ten years, which have radically improved our understanding of the theory. Keywords: String theory; dualities; black holes. La teor´ıa de cuerdas busca dar una descripcion´ completa de la estructura fundamental de nuestro universo. Independientemente de si logra o no eventualmente alcanzar esta ambiciosa meta, en el transcurso de los anos˜ ha demostrado ya ser una estructura teorica´ enormemente rica, con varios puntos de contacto con otros problemas de interes´ en la f´ısica teorica´ moderna. Este art´ıculo describe las ideas basicas´ de la teor´ıa de cuerdas, y ofrece un breve panorama de los logros de los ultimos´ diez anos,˜ los cuales han mejorado radicalmente nuestra comprension´ de la teor´ıa. Descriptores: Teor´ıa de cuerdas; dualidades; dualidad norma-gravedad; agujeros negros. PACS: 11.25.-w; 11.25.Uv; 11.25.Tq; 04.70.Dy; 11.25.Yb; 04.60.-m 1. Basic String Theory The discovery that the graviton, in particular, emerges naturally as a mode of oscillation of the string provided for The basic proposal of string theory [1, 2] is to interpret the the first time a perturbative description of quantum gravity various elementary particles known to us not as distinct point- that is free of ultraviolet divergences. The fact that, at this like objects, but as different manifestations of a single one- level, the description is perturbative implies in particular that dimensional object, a string. Upon quantization, the differ- prior to any computation a background must be chosen, in or- ent modes of oscillation of the string give rise to an infi- der to consider the propagation of a small number of weakly- nite tower of states with progressively higher masses, each interacting strings on it. It is important, however, to keep state having the properties of a specific type of particle and in mind that, just like point particles represent small fluctua- thus being interpretable as a fluctuation in an associated tions about a chosen background value for the corresponding field. Forp a closedp string, one obtains a free mass spectrum field, strings can explicitly be seen to correspond to small m = 0; 4=ls; 8=ls;:::, with ls the only dimensionful pa- excitations of the selected background. What we are study- rameter of the theory, known as the string length. ing, then, is not ‘just’ a string moving on a fixed background, In conventional models ls is close to the Planck length, but the background itself (a rather drastic generalization of 10¡33 cm (corresponding to an energy scale » 1019 GeV); spacetime), undergoing small fluctuations. but in recent years alternative scenarios have been considered Starting from closed strings, (and, as we will see below, ¡17 in which ls could perhaps be as large as 10 cm (i.e., the much more naturally from open strings,) it is also possi- string scale be as low as » 103 GeV), on the verge of exper- ble to obtain non-Abelian gauge fields. A single basic ob- imental detection. Either way, up to now we would have ob- ject, the string, is thus capable of generating the basic in- served only the states that are (in a first approximation) mass- gredients of the Standard Model plus gravity. And, remark- less. Closed strings yield massless states that correspond to ably, the unique interaction of the theory, which allows a a metric g¹º , a scalar field ' known as the dilaton, and an- string to split into two (or the reverse), is able to repro- tisymmetric tensor fields (generalized gauge fields) B¹º and duce the gravitational, Yang-Mills and Yukawa interactions C¹1:::¹p+1 for various values of p, plus fermionic partners for needed to describe the world around us. Moreover, there all of these fields. This is precisely the content of the field- are no free parameters in the theory. In particular, the di- theoretic generalization of General Relativity known as su- mensionless string coupling constant gs, which controls the pergravity, which is then the lowest-energy approximation to strength of the string-splitting interaction (andp consequently string theory. There is a generalization of the Feynman dia- the gravitational and Yang-Mills couplings, G / g and p N s gram expansion which allows the computation of string scat- gYM / gs), is determined by the expectation value of the tering amplitudes, from which the interactions between the dilaton field: gs = exp '. massless (as well as massive) modes can be deduced. The For a string theory to be consistent (e.g., for it to predict basic string interaction is a sort of cubic vertex, which allows only non-negative probabilities), the strings must live in more a string to split into two (or the reverse). than the 3 + 1 dimensions x¹ (¹ = 0; 1; 2; 3) that are evident STRING THEORY: A DECADE OF PROGRESS 67 TABLE I. Properties of different string theories. Superstring Theory: I IIA IIB HO HE Can strings break open? Yes No No No No Are strings oriented? No Yes Yes Yes Yes How many (10-dim) supersymmetries? N = 1 N = 2 N = 2 N = 1 N = 1 Non-Abelian gauge group? SO(32) None None SO(32) E8 £ E8 Is the theory parity invariant? No Yes No No No to us. There are in fact additional dimensions of two different the states of a string; but for (closed) string states there is types. an additional source of multiplicity, as each string can wind First, there are six additional bosonic (i.e., ordinary) di- around compact direction xa an arbitrary integer number of mensions, ~x ´ fxag, with a = 4;:::; 9. These must be times. From the lower-dimensional perspective, these wind- hidden from us in some way, and the conventional proposal ing numbers wa 2 Z are simply perceived as charges under 0(a) is that they are compact and very small. Our universe would the additional gauge fields A¹ ´ B¹a. thus be somewhat analogous to the surface of a garden hose, Second, there exist N = 1 or 2 additional dimensions with three spatial dimensions that are of astronomical size that are fermionic, meaning in particular that the correspond- (say, » 1010 light years), and six dimensions whose size is of ing coordinates θA (A = 1 or A = 1; 2) anticommute: order the Planck length (» 10¡33 cm). θAθB = ¡θBθA. This implies that each of the coordinates We have known since the days of Kaluza and Klein that squares to zero, and therefore the Taylor expansion of a field under such circumstances each ten-dimensional field would that is a function of all the bosonic and fermionic coordinates A 1 2 be understood from the four-dimensional perspective as an terminates, ©(x; θ) » Á(1)(x) + θ Ã(A)(x) + θ θ Á(2)(x). infinite tower of fields with progressively higher masses If the ‘superfield’ © is bosonic, it follows that the two fields and charges. To illustrate the idea, consider the simplest Á(A) are bosonic and the Ã(A) fermionic. This one-to-one case, where the additional dimensions form a straight six- pairing of bosonic and fermionic states, together with the dimensional torus with radii Ra. For a field Á to be periodic invariance of the physics under transformations that ‘rotate’ along these directions, the corresponding momenta must then from one to the other, is what is known as supersymmetryi, a be discrete, pa = na=Ra, and we can Fourier decompose property that (in moderate amounts) is appealing from the X theoretical perspective and for which we might even have ¹ ¹ Á(x ; ~x) = exp(i~p ¢ ~x)Á~n(x ): some indirect phenomenological evidence. The search for ~n this (necessarily broken) symmetry will be one of the main Each component field Á~n, with definite momenta tasks of the Large Hadron Collider (LHC), currently under along the hidden dimensions, is interpreted from the construction at CERN. four-dimensional viewpoint as a field with a defi- Up to 1994, there were five known consistent string the- ii nite mass m~n. E.g., if Á is massless in the ten- ories, all ten-dimensional and supersymmetric . Their defi- ¹ 2 dimensional sense (p p¹ + ~p =0), then the mass of Á~n i nitions differ at a technical level; but they can be easily dis- p ¹ m~n ´ ¡p p¹ = j~pj. This implies that, for small enough tinguished from one another by listing a few of their physical Ra, only the ~n = 0 modes will be experimentally accessible. properties, as we do in Table I. Choosing a more complicated topology for the six hidden di- mensions, it is possible to obtain not one but, e.g., three light 2. Branes families of particles— the number of fermion generations can be determined topologically! Our understanding has improved dramatically over the past In addition to this decomposition according to momenta ten years.