Chapter 1

1

The one of the most promising candidates for such an extension of the SM and GR to one unified formalism is supersymmetric string/M-theory (next to caninical or loop quantum gravity).^B.3 Historically string theory was discovered by accident within the attempts to describe the strong nuclear force in the late 1960s, known under the names of dual models or Veneziano models (1968). In 1970 Nambu, Nielsen and Susskind realized that the equations of this theory in fact described interactions of extended 1-dimensional objects, called strings. The advent of QCD in 1973 as an alternative and very successful theory of the strong interaction let the string model become an oddity within theoretical physics. However, in 1974 Schreck and Schwarz proposed string theory as a possible candidate for a unified theory of fundamental forces (including gravity) and elementary particles. The first incarnation of this idea was known under quantum bosonic string theory. Later, incorporating the concept of space-time supersymmetry, superstring theory was born. It is interesting to note that the notion of supersymmetry was in fact rediscovered in 1971 from two entirely different approaches. In the first, 2-dimensional world-sheet supersymmetry arose within the context of string theory as discovered by Neveu, Ramond and Schwarz in that year. The second approach was that of Gol'fand and Likhtman, who found the super Poincaré algebra. Apparently, the first known mention of a supersymmetric group was by Myazawa in 1966. Wess and Zumino, inspired in part by the Neveu-Ramond-Schwarz string model, conceived the idea of space-time supersymmetry in 1974.

Until 1984 superstring theory was only pursued by very few physicists. However, during that year many discoveries transformed the subject into one of the most active areas of theoretical physics. The names associated with this `Superstring Revolution' were among many others Green, Schwarz and Witten and from 1984 through 1986 more than 1000 research papers were published. There emerged five totally self-consistent superstring theories, namely: type I with open and closed strings and $SO(32)$ gauge symmetry, type IIA and IIB with closed strings and no gauge symmetry, heterotic theory with closed strings and $SO(32)$ gauge symmetry (HS) and heterotic theory with closed strings and $E_8 \times E_8$ gauge symmetry (HE). All five string theories are weakly coupled and hence perturbative theories. The `Second Superstring Revolution' started in 1995. On a string conference in that year, Witten introduced new symmetry concepts (called dualities, where a perturbative expansion in one theory contains information about non-perturbative effects in the dual theory) taking string theory to a powerful non-perturbative level. As the name suggests this ignited a whole new industry in string theory which eventually led to the unification of all five 10-dimensional superstring theories into a new and only vaguely understood 11-dimensional theory, called M-theory. Most prominently Witten and Horava showed that within IIA and HE string theory a constant increase of the coupling constant results in the appearance of a new space dimension and in the stretching of the string to a 2-dimensional membrane living in this 11-dimensional space-time.^B.4 Hence M-theory can be understood as strongly coupled or non-perturbative string theory, although it has the richer structure of not only describing 1-dimensional objects. In effect I, IIA, IIB, HS, HE and M-theory are unified in a web of dualities. It is understood that all six theories represent six special vacua (quantum solutions) within the framework of a single overarching and fundamental new theory. To this day the nature and meaning of this theory is obscured. There is not even a consent in its name. Sometimes it is called U-theory (Sen) but mostly the name M-theory is also used for this underlying theory thus giving the term an ambiguous meaning. To summarize: U-theory has many degenerate vacua labeled by a set of parameters. Special regions (limits) of the parameter space correspond to the five weakly coupled 10- dimensional string theories and 11-dimensional M-theory. In the low energy limit all five string theories corresponds to 10-dimensional supergravity. M-theory is the high-energy limit of 11-dimensional supergravity. The term string/M-theory will denote the six known solutions of U-theory. As noted above, string/M-theory also contains a generalization of the concept of strings to new fundamental objects, so-called $p$-dimensional membranes^B.5 or $p$-branes. Thus point particles and strings are just two possibilities of the plethora of $p$-branes. Technically these membranes have two origins. One is the already stated extension of the fundamental strings where the most prominent are 2- and 5-branes in 11-dimensional space-time. The second origin is related to the fact that type I theory has open strings. The end points of the open strings can obey so-called Dirichlet boundary conditions, where they can lie on definite surfaces in space. The dimensionality $p$ of this surface defines a membrane, called D-$p$-brane (a solitonic object). In 1997 a new proposal was put forward for defining M-theory beyond its low energy approximation, called Matrix theory. The idea is that in the infinite momentum frame M-theory is equivalent to a quantum mechanical system with matrices as dynamical variables, i.e. position coordinates of D-branes. Thus Matrix theory can be associated with non-commutative geometry; see appendix A.6. Although this theory appears to be a step in the right direction, there is still no conclusive result to be stated. In 1998 D-branes were also brought into context with non-commutative field theories.

There appears to be an inexhaustible amount of literature on the subject. Here is a small selection (in order of technicality) of papers: [#!Du98!#], [#!Mal00!#], [#!Yo00!#], [#!Se00!#], [#!Va98!#], [#!Gi98!#], [#!Gre97!#], [#!Li98!#], [#!Ban97!#], [#!Po96!#] and text books: [#!Ka93!#], [#!CMS89!#], [#!Ha92!#], [#!GSW87!#] and [#!Po98!#].