String theory requires supersymmetry why

Strings and Branen Worlds: Some Aspects of a Unified Theory of All Interactions

Research report 2004 - Max Planck Institute for Physics

Lüst, Dieter; Blumenhagen, Ralph; Erdmenger, Johanna
Theoretical Physics - Mathematical Physics, String Theory (Prof. Lüst) (Prof. Dr. Dieter Lüst) MPI for Physics, Munich
In this article we cover some aspects of superstring theory. After an introduction to string theory as a unified quantum theory of all interactions, we introduce the so-called brane worlds. These models represent the universe as a three- or higher-dimensional membrane that is embedded in the 9-dimensional space of string theory and open up many interesting possibilities to derive the standard model of elementary particle physics from string theory.
In this article we discuss some aspects of superstring theory. After a short introduction of string theory as unifying quantum theory of all interactions, we introduce the socalled brane world models. These models describe the universe as 3- or higher dimensional brane, embedded into the 9-dimensional space of string theory. They offer many interesting possibilities to derive the standard models of particle physics from string theory.

Unification of the interactions

Theoretical physics describes the building blocks of matter and the forces that act between them. These forces are also known as interactions. They ensure that the building blocks or particles attract or repel each other. An example of such an interaction is the electromagnetic force that acts on electrically charged particles. Further forces are the so-called weak and strong nuclear forces, which play an important role in binding the atomic nuclei and in other subatomic structures such as quarks. All three named forces play an important role in elementary particle physics and can be described with a uniform physical theory, the so-called standard model of elementary particles. This theory is a quantum theory, i.e. there is a probability interpretation for the particles described in this way, which leads to the well-known wave-particle dualism.

The fourth and last of the known interactions is gravity or gravitation. It causes the attraction between masses and is described with Einstein's general theory of relativity, according to which the attraction of masses is caused by the curvature of space. The theory of relativity is not a quantum theory, but a classical theory in which the position and speed of the particles can be given for all times at the same time.

Seventy years ago Einstein himself named the description of all four fundamental interactions in a single unified theory as a fundamental task of physics. This task has not yet been completely solved. A major problem is that the theory of relativity - in contrast to the theory of the other three interactions - cannot be written in the conventional way as a quantum theory. Therefore it is difficult to integrate gravity into the theory of the three remaining interactions. However, there has already been significant progress in this direction: String theory, in particular, is a possible candidate for a unified theory.

String Theory and Membranes

In string theory, the problem of quantizing gravity is solved by the fact that the fundamental particles are no longer viewed as point-shaped as in quantum and relativity theory, but have an extension. Due to the finite expansion of the strings, the behavior of quantum mechanical scattering rungs with spin 2 gravitons, which occur as force particles in quantum gravity, is influenced to the effect that all infinities are absent, which made the quantisability of the gravitational force with point particles impossible in the context of perturbation theory Has.
This is an important advance in unifying the interactions. First, within the framework of string theory, the particles were described by the excitation modes of one-dimensional extended objects, that is, by threads or strings that are extended in one spatial direction. Since 1995, string theory has evolved to include multi-dimensional, extended objects, membranes, or also called p-branes. 0-branes are ordinary particles, 1-branes are one-dimensional threads, 2-branes are two-dimensional surfaces, which are also called membranes. There are also 3-branes, 4-branes, etc. The theoretical developments of the last ten years strongly suggest that there is a previously unknown unified theory in eleven space-time dimensions, which all these objects in a mathematically consistent manner describes.

In string theory, the conventional point particles correspond to the lowest harmonic oscillation modes of the string, whereby two different types of strings are essentially considered, namely the closed string and the open string, which has a starting point and also an end point. In this way, the variety of elementary particles is reduced to two fundamental degrees of freedom, so first of all an enormous simplification compared to the standard model of elementary particle physics. With high energies, of course, more and more vibration modes of a string can be excited, which correspond to very heavy, i.e. massive particles. The experimental proof of these heavy string particles would to a certain extent represent the "proof" of string theory, but it is experimentally very difficult or even practically impossible, since the so-called string mass scale must be beyond the energy scale of the Standard Model (several 100 GeV), and possibly with Planck -Scale of 1019 GeV is to be identified. These gigantic energies are far beyond the possibilities of current (and probably future) particle accelerators. That is why one also looks for indirect traces of strings, especially for the so-called supersymmetry, which postulates a supersymmetric partner particle with the same quantum numbers but different spin for every known elementary particle. The search for supersymmetry will be a main focus of research at the LHC (Large Hadron Collider) at CERN, which will start its work in 2008, and at which the Max Planck Institute for Physics is part of the ATLAS experiment ( A Toroidal LHC Apparatus) is instrumental in this.

Even if string theory solves the problem of quantizing gravity in principle, there are still many questions to be explored. On the one hand, string theory can so far only be formulated using certain approximation methods; on the other hand, it still has to be clarified how string theory relates to both the quantum theories of elementary particle physics mentioned at the beginning and to the theory of relativity. String theory describes physics at very high energies, so high that today it is hardly conceivable to ever achieve it in an experiment. However, in the limit case of lower energies that can be generated in the particle accelerator, i.e. when the resolution becomes so coarse that the threads only appear point-like, string theory has to merge with the theories of quantum and relativity that are known today. It is expected that string theory will point to new physical phenomena in this energy range and create new relationships between previously independent parameters. Current research results indicate that membrane theory will play a central role both in the exact formulation of string theory at high energies and in answering the question about the low-energy limit case.

The Department of Quantum Field Theory and String Theory at the Max Planck Institute for Physics in Munich also deals with various aspects of string theory. In particular, the following two main topics are the focus of interest.

1.) String compactifications and membrane worlds

One of the most important findings in string theory is that a string needs nine spatial directions in order to carry out its oscillations in a mathematically consistent manner. In superstring theory, space-time is not four-dimensional, as in conventional quantum field theory or as in general relativity, but string theory must be embedded in a 10-dimensional space-time with nine spatial directions. From a mathematical point of view, a higher-dimensional space is nothing out of the ordinary.

In three-dimensional Euclidean space, every movement can be divided into north-south, west-east and, in the vertical direction, upwards and downwards. This means that a third, vertical spatial direction can be spanned at every point on the two-dimensional plane. It is the same with a four-dimensional space, for example: Above every three-dimensional point in space there is a fourth direction of movement. In order to explain in string theory why our observed universe only has three spatial dimensions, one uses an analogous way of describing nine-dimensional space: one assumes that there is a six-dimensional space above every point in the four-dimensional space-time continuum which in itself represents a compact manifold.

This six-dimensional space can therefore be viewed as a generalized circle or also as a generalized torus, since its directions represent periodic structures. This process is called compactification of six spatial directions. A special case of six-dimensional, compact manifolds are the Calabi-Yau spaces, which have proven to be particularly important in string theory. The physical question that arises immediately is why the additional six dimensions in string theory have not yet been discovered. There are essentially two possible answers for this:

First, the extra dimensions are less than about 10{-16}cm. This means that, according to Heisenberg's uncertainty principle, energies are required that are higher than approx. 100 GeV in order to use particle accelerators to resolve the extra dimensions, i.e. to make them visible.

This fact can also be described a little differently: if a particle can move in higher-dimensional space, its wave function always has a four-dimensional part, which is multiplied by a wave function in the extra spatial directions. With periodic boundary conditions in the extra spatial directions, the corresponding energy operator has a discrete spectrum, a fact that implies that every known elementary particle, e.g. a quark, an electron or a photon, is accompanied by an infinite number of excited particles that differ from the known particles differ only by their higher masses, but otherwise have identical properties, e.g. with regard to their electrical charge. According to Theodor Kaluza and Oskar Klein, these particles are called Kaluza-Klein particles (KK particles), whereby the mass of the KK particles is always given by a multiple of the inverse radius of the extra dimensions, i.e. by a multiple of at least 100 GeV / c2. A possible detection of the KK particles in future accelerator experiments (LHC in Geneva or Linear Collider) would give a direct indication of the existence of extra dimensions, and thus also the existence of strings.

But there is a second logical possibility why extra space dimensions have so far eluded us. This is related to the already mentioned presence of higher-dimensional objects in string theory, namely the p-branes. In string models with open strings and p-branes, the particles of the standard model of elementary particle physics, such as electrons, muons, neutrinos, quarks, photons, gluons, W and Z bosons, can only be spatially p-dimensional, in the simplest case move three-dimensional membrane, which is embedded in the nine-dimensional space given by string theory. This membrane also represents our observed universe in which the processes of elementary particle physics take place.

Plato's allegory of the cave can be used as an analogy to this scenario. Here some prisoners are firmly chained to a stone bench in a cave, so that the prisoners can only move along the bench (i.e. in the x-direction) and also vertically (i.e. in the z-direction), but not in the transverse direction (y-direction) ) perpendicular to the bench.

The cave is illuminated by a candle that projects the movement of the prisoners or the objects behind them onto a screen in front of them. It is clear that the prisoners indulge in the illusion that they only live in a two-dimensional space, since the third spatial direction remains closed to them. It is very similar in the membrane world of string theory. The particles of the Standard Model are the lowest excitation modes of an open string, the ends of which, for reasons of the mathematical consistency of the theory, are glued to a three-dimensional or more generally to a p-dimensional space in the simplest case. (In string theory, this is sometimes referred to as the holographic principle - see also the next chapter.)

A movement transverse to the p-brane is impossible for the particles of the Standard Model. Because of this, the extra dimensions in the transverse directions can also be much larger than just 10-16 cm, the experimental limits are only a few micrometers here (see below). Nevertheless, the extra dimensions do not completely lose their influence on the three-dimensional world, since the force particles of gravitation, namely the gravitons, appear as the lowest excitation mode of the closed string. In contrast to the open string, the closed string can propagate in all nine spatial directions. Experimentally, this means that in the case of very short distances in the range of distances shorter than approx-5 Meters, could result in deviations from the Newton potential of the theory of gravity, which result from the entry of the gravitons into the extra dimensions.

In the String Theory Department at the Max Planck Institute for Physics, various aspects of Brane worlds are now being investigated. In the foreground of the discussion is the question of whether the supersymmetric standard model of elementary particle physics (referred to as MSSM for short) can be derived from string theory in this way. It turned out [1] that brane worlds in which the p-branes on the one hand completely fill the entire three-dimensional universe, but on the other hand extend into part of the additional, compact 6-dimensional space and can also intersect there , are particularly well suited to reproducing the MSSM. This is why these string models are also called "intersecting branes".

The quarks and leptons of the MSSM correspond to open strings that are located at the intersections of the p-branes in the inner space. In addition to these brane world models, other, often dual string compactifications with so-called
studied magnetic fluxes [2].

An interesting aspect of the intersecting brane world models is that in them one can also specifically calculate the phenomenon of supersymmetry breaking [3], which is responsible for the masses of the supersymmetric partner particles. This is certainly also important for future experiments at the LHC in Geneva. The supersymmetry is broken by the fact that the supersymmetry is broken by certain “magnetic” flux field strengths that lie in the internal space. These so-called background flows also have the further important property that they freeze many of the otherwise undefined geometric parameters (so-called module fields) to a fixed value [4]. In this way, the inner compaction space is given a rigid, solid appearance, which can no longer be changed by deforming the geometric module fields. The "freezing" of the module fields has numerous phenomenologically desirable advantages, such as the absence of additional forces (so-called 5th. Force) in nature, which are caused by massless module fields, or the basic predictability of numerous couplings in the standard model, as well as the masses of the supersymmetric partner particles [5]. Furthermore, the mechanism of modulus stabilization also has drastic consequences in cosmology, such as a possible string-theoretical explanation of the dark energy of the universe, which Einstein had introduced as a cosmological constant in his gravitational formula. After all, as has long been known, the number of possible string compatifications is huge, on the order of 10500-1000 or more. That is why one speaks in this context of the so-called string landscape, and one tries to obtain statistical statements about the distribution of the physical parameters in the string landscape [6].

These and many other physical and phenomenological aspects were the subject of the international conference "String Phenomenology 2005", which was jointly organized by the Max Planck Institute for Physics and the Arnold Sommerfeld Center for Theoretical Physics from 13.-18. June 2005 at the LMU-Munich as well as the workshop "The string vacuum workshop", 22.-24. Nov. 2004 at the MPI for Physics.

It must be emphasized that there are still a large number of unsolved problems in string theory, which are both of a principle nature, and in particular with regard to the structure of space and time at very short intervals near the Planck scale as regards the derivation of the MSSM from string theory. In particular, there is still no string model that completely correctly describes and explains all the properties of the MSSM.

2.) Equivalence of quantum and relativity theory - the AdS / CFT-

Conformal field theories (CFT) are important in theoretical physics. Point-like particles and their interactions are generally characterized by Fields described. Conformal field theories are special field theories that have a particularly high degree of symmetry, since they are invariant under conformal coordinate transformations. They can be used as illustrative examples in particular in connection with the issues of standardizing interactions and the effects of string theory on elementary particle physics.

With a conformal coordinate transformation, not only can the position of an area in space change, as is the case with rotation, but also its size and shape, albeit in a very specific way: the symmetry transformation is angularly true at every single point.

Conforming field theories describe the models of elementary particle physics only in certain borderline cases. Due to their solvable mathematical structure, however, questions can be answered for these theories that would initially be too difficult to answer for realistic models. The strategy is to then generalize the results found for conformal field theories to the realistic models.

In 1997, Juan Maldacena discovered an important equivalence between a conformal quantum field theory and a theory of gravity, i.e. a specific model of the theory of relativity. He obtained this equivalence, known as the dS / CFT correspondence, from the investigation of the low-energy limit case of a model of membrane theory. “AdS” stands for the anti-de-Sitter space, named after the Dutch physicist Willem de Sitter, ie for the theory of gravity. "CFT" is the abbreviation for conformal field theory. What is remarkable about the AdS / CFT correspondence is that for the first time a connection is established between quantum theory on the one hand and classical relativity theory on the other. In the AdS / CFT correspondence there are two different theories that describe the same physical phenomenon. This enables the computation of identical physical observables in two different ways. Therefore, it is hoped that using the AdS / CFT duality, non-perturbation-theoretical phenomena in gauge theories, such as confinement in QCD as the theory of strong interaction, can be calculated by considering corresponding quantities in the dual, but classical theory of gravity .

An essential feature of the AdS / CFT correspondence is that a model of quantum theory in four space-time dimensions and a model of classical relativity theory in five dimensions, i.e. in one more dimension, are related to each other. This is also known as the holographic principle.

The research projects at the Max Planck Institute aim to expand and generalize the AdS / CFT correspondence in various ways. Research is being carried out to expand the correspondence in such a way that it applies not only to quantum theoretical models with conformal symmetry, but also to the three interactions in elementary particle physics. For this, the symmetry content in particular must be reduced. On the quantum theoretical side of the correspondence one therefore deviates from the conformal field theories and turns to more general quantum field theories (QFT), which are related to those of elementary particle physics.

As a further step with regard to elementary particle physics, we recently published results with which the AdS / CFT correspondence in quantum theory can be used to describe quarks, the building blocks of protons, neutrons and mesons. For this purpose, p-branes are placed in the anti-de-sitter space. We have thus succeeded in providing a gravitational description of low-energy phenomena in the theory of strong interaction. An example of this are the chiral symmetry breaking and the light masses of some mesons [7]. Furthermore, with similar generalizations of the AdS / CFT correspondence, we were able to predict a new phase transition in quantum field theories at finite temperature [8]. Another aspect is the classification of the generalized AdS / CFT correspondence in string theory - so far this correspondence has only been formulated for the low-energy borderline case of string theory. In this context we were able to clarify some open questions [9].

Answering these fascinating questions in basic research will go a long way towards a better understanding of matter and its interactions. However, it should also be mentioned that the methods developed in this context, in particular the use of conformal symmetry, can also be used in other areas of physics, for example in solid-state physics when describing magnetic systems. This is another example of how basic research is an important engine of research as a whole.

Original publications

Blumenhagen, R., L. Görlich, B. Körs and D. Lüst; Blumenhagen, R., B. Körs, D. Lüst and T. Ott; Blumehagen, R., M. Cvetic, P. Langacker and G. Shiu:
Noncommutative Compactifications of Type I Strings on Tori with Magnetic Background Flux; The Standard model from stabel intersecting brane world orbifolds; Toward Realistic Intersecting D-Brane Models.
Preprint hep-th / 0007024, Journal of High Energy Physics 0010, 006-030 (2000); Preprint hep-th / 0107138, Nuclear Physics B616, 3-33 (2001); Preprint hep-th / 0502005.
Blumenhagen, R., G. Honecker and T. Weigand:
Loop-Corrected ompactifications of the Heterotic String with Line Bundles,
hep-th / 0504232. Journal of High Energy Physics, 0506, 020-059 (2000).
Lüst, D., S. Reffert, W. Schulgin and S. Stieberger:
Flux-induced soft supersymmetry breaking in chiral type IIB orientifolds with D3 / D7-branes,
hep-th / 0406092, Nucl. Phys. B706, 3-52 (2005).
Lüst, D., S. Reffert, W. Schulgin and S. Stieberger:
Moduli stabilization in type IIB orientifolds (I): orbifold limits,
Lüst, D., S. Reffert and S. Stieberger; Blumenhagen, R., M. Cvetic, F. Marchasano, and G. Shio.
MSSM with soft SUSY breaking terms from D7-branes with fluxes; Chiral D-brane Models with Frozen Open String Moduli.
hep-th / 0410074; hep-th / 0502095, Journal of High Energy Physics 0503, 050 (2005).
Blumenhagen, R., F. Gmeiner, G. Honecker, D. Lüst and T. Weigand:
The statistics of supersymmetric D-brane models,
hep-th / 0411173, Nuclear Physics B713, 83-135 (2005).
Babington, J., J. Erdmenger, N.J. Evans, Z. Guralnik and I. Kirsch:
Chiral symmetry breaking and pions in non-supersymmetric gauge / gravity duals,
hep-th / 0306018, Physical Review, D 69 066007-066020 (2004).
Apreda, R., J. Erdmenger, N. Evans and Z. Guralnik:
Strong coupling effective Higgs potential and a first order thermal phase transition from AdS / CFT duality,
hep-th / 0504151, Physical Review D 71, 126002-126013 (2005).
Erdmenger, J. and I. Kirsch:
Mesons in gauge / gravity dual with large number of fundamental fields; Spectral flow on the Higgs branch and AdS / CFT duality,
hep-th / 0502224, Journal of High Energy Physics, 0506, 052-067 (2004).