Mysterious duality

Mysterious duality

For a generally accessible and less technical introduction to the topic, see Introduction to M-theory.

In theoretical physics, M-theory is an extension of string theory in which 11 dimensions of spacetime are identified as seven higher-dimensions plus the four common dimensions (11D st = 7 hd + 4D). Proponents believe that the 11-dimensional theory unites all five 10 dimensional string theories and supersedes them. Though a full description of the theory is not known, the low-entropy dynamics are known to be supergravity interacting with 2- and 5-dimensional membranes.

This idea is the unique supersymmetric theory in 11 dimensions, with its low-entropy matter content and interactions fully determined, and can be obtained as the strong coupling limit of type IIA string theory because a new dimension of space emerges as the coupling constant increases.

Drawing on the work of a number of string theorists (including Ashoke Sen, Chris Hull, Paul Townsend, Michael Duff and John Schwarz), Edward Witten of the Institute for Advanced Study suggested its existence at a conference at USC in 1995, and used M-theory to explain a number of previously observed dualities, initiating a flurry of new research in string theory called the second superstring revolution.

In the early 1990s, it was shown that the various superstring theories were related by dualities which allow the description of an object in one super string theory to be related to the description of a different object in another super string theory. These relationships imply that each of the super string theories is a different aspect of a single underlying theory, proposed by Witten, and named "M-theory".

Originally the letter M in M-theory was taken from membrane, a construct designed to generalize the strings of string theory. However, as Witten was more skeptical about membranes than his colleagues, he opted for "M-theory" rather than "Membrane theory". Witten has since stated that the different interpretations of the M can be a matter of taste for the user, such as magic, mystery, and mother theory.[1]

M-theory (and string theory) has been criticized for lacking predictive power or being untestable. Further work continues to find mathematical constructs that join various surrounding theories. However, the tangible success of M-theory can be questioned, given its current incompleteness and limited predictive power.

History and development

Prior to May 1995

Before 1995 there were five known consistent superstring theories (henceforth referred to as string theories), which were given the names Type I string theory, Type IIA string theory, Type IIB string theory, heterotic SO(32) (the HO string) theory, and heterotic E8×E8 (the HE string) theory. The five theories all share essential features that relate them to the name of string theory. Each theory is fundamentally based on vibrating, one-dimensional strings at approximately the length of the Planck length. Calculations have also shown that each theory requires more than the normal four spacetime dimensions (although all extra dimensions are in fact spatial). When the theories are analyzed in detail, significant differences appear.

Type I string theory and supplements

The Type I string theory has vibrating strings like the rest of the string theories. These strings vibrate both in closed loops, so that the strings have no ends, and as open strings with two loose ends. The open loose strings are what separates the Type I string theory from the other four string theories. This was a feature that the other string theories did not contain.

String vibrational patterns

Calculations of the String Vibrational Patterns show that the list of string vibrational patterns and the way each pattern interacts and influences others vary from one theory to another. These and other differences hindered the development of the string theory as being the theory that united quantum mechanics and general relativity successfully. Attempts by the physics community to eliminate four of the theories, leaving only one string theory, have not been successful.


M-theory attempts to unify the five string theories by examining certain identifications and dualities. Thus each of the five string theories become special cases of M-theory.

As the names suggest, some of these string theories were thought to be related to each other. In the early 1990s, string theorists discovered that some relations were so strong that they could be thought of as an identification.

Type IIA and Type IIB

The Type IIA string theory and the Type IIB string theory were known to be connected by T-duality; this essentially meant that the IIA string theory description of a circle of radius R is exactly the same as the IIB description of a circle of radius P/R, where P is the Planck length.

This was a profound result. First, this was an intrinsically quantum mechanical result; the identification did not hold in the realm of classical physics. Second, because it is possible to build up any space by gluing circles together in various ways, it would seem that any space described by the IIA string theory can also be seen as a different space described by the IIB theory. This implies that the IIA string theory can identify with the IIB string theory: any object which can be described with the IIA theory has an equivalent, although seemingly different, description in terms of the IIB theory. This suggests that the IIA string theory and the IIB string theory are really aspects of the same underlying theory.

Other dualities

There are other dualities between the other string theories. The heterotic SO(32) and the heterotic E8×E8 theories[2][3] are also related by T-duality; the heterotic SO(32) description of a circle of radius R is exactly the same as the heterotic E8×E8 description of a circle of radius P/R. This implies that there are really only three superstring theories, which might be called (for discussion) the Type I theory, the Type II theory, and the heterotic theory.

There are still more dualities, however. The Type I string theory is related to the heterotic SO(32) theory by S-duality; this means that the Type I description of weakly interacting particles can also be seen as the heterotic SO(32) description of very strongly interacting particles. This identification is somewhat more subtle, in that it identifies only extreme limits of the respective theories. String theorists have found strong evidence that the two theories are really the same, even away from the extremely strong and extremely weak limits, but they do not yet have a proof strong enough to satisfy mathematicians. However, it has become clear that the two theories are related in some fashion; they appear as different limits of a single underlying theory.

Only two string theories

Given the above commonalities there appear to be only two string theories: the heterotic string theory (which is also the type I string theory) and the type II theory. There are relations between these two theories as well, and these relations are in fact strong enough to allow them to be identified.

Last step

This last step is best explained first in a certain limit. In order to describe our world, strings must be extremely tiny objects. So when one studies string theory at low energies, it becomes difficult to see that strings are extended objects—they become effectively zero-dimensional (pointlike). Consequently, the quantum theory describing the low energy limit is a theory that describes the dynamics of these points moving in spacetime, rather than strings. Such theories are called quantum field theories. However, since string theory also describes gravitational interactions, one expects the low-energy theory to describe particles moving in gravitational backgrounds. Finally, since superstring string theories are supersymmetric for supersymmetry is needed for consistency, one expects to see supersymmetry appearing in the low-energy approximation. These three facts imply that the low-energy approximation to a superstring theory is a supergravity theory.

Supergravity theories

The possible supergravity theories were classified by Werner Nahm in the 1970s. In 10 dimensions, there are only two supergravity theories, which are denoted Type IIA and Type IIB. This similar denomination is not a coincidence; the Type IIA string theory has the Type IIA supergravity theory as its low-energy limit and the Type IIB string theory gives rise to Type IIB supergravity. The heterotic SO(32) and heterotic E8×E8 string theories also reduce to Type IIA and Type IIB supergravity in the low-energy limit. This suggests that there may indeed be a relation between the heterotic/Type I theories and the Type II theories.

In 1994, Edward Witten outlined the following relationship: The Type IIA supergravity (corresponding to the heterotic SO(32) and Type IIA string theories) can be obtained by dimensional reduction from the single unique eleven-dimensional supergravity theory. This means that if one studied supergravity on an eleven-dimensional spacetime that looks like the product of a ten-dimensional spacetime with another very small one-dimensional manifold, one gets the Type IIA supergravity theory (while the Type IIB supergravity theory can be obtained by using T-duality). However, eleven-dimensional supergravity is not consistent on its own — it does not make sense at extremely high energy, and likely requires some form of completion. It seems plausible, then, that there is some quantum theory — which Witten dubbed M-theory — in eleven dimensions which gives rise at low energies to eleven-dimensional supergravity, and is related to ten-dimensional string theory by dimensional reduction. Dimensional reduction to a circle yields the Type IIA string theory, and dimensional reduction to a line segment yields the heterotic SO(32) string theory.

Same underlying theory

M-theory would implement the notion that all of the different string theories are different special cases.

Recent developments

In late 2007, Bagger and Lambert set off renewed interest in M-theory with the discovery of a candidate Lagrangian description of coincident M2-branes, based on a non-associative generalization of Lie Algebra, Nambu 3-algebra or Filippov 3-algebra. Practitioners hope the Bagger–Lambert–Gustavsson action will provide the long-sought microscopic description of M-theory.


When Edward Witten named M-theory, he did not specify what the M stood for—perhaps because the nascent theory was not fully defined. Some, including Sheldon Glashow, speculate that Witten chose the letter because it resembles an inverted W. According to Witten, "M can stand variously for 'magic', 'mystery', or 'matrix', according to one's taste."[4]

Faced with this ambiguous initial, countless scientists and commentators have offered their own expansions of the M—some sincere, others facetious. M should stand for membrane, say some. Meanwhile, Michio Kaku, Michael Duff, Neil Turok, and others suggest mother or master (i.e., the "mother of all theories" or the "master theory").[5]

Although Witten coined the term M-theory to refer to his model of an eleven-dimensional universe, other scientists have generalized the moniker for application to any of various meta-theories involving string theory and brane cosmology. (Ashoke Sen proposed u-theory (ur, 'über', 'ultimate', 'underlying', or perhaps 'unified') as a more distinctive appellation.) When unqualified, M-theory now usually denotes this more general definition, rather than the one Witten originally advanced.

M-theory and membranes

In the standard string theories, strings are assumed to be the single fundamental constituent of the universe. M-theory adds another fundamental constituent - membranes. Like the tenth spatial dimension, the approximate equations in the original five superstring models proved too weak to reveal membranes.


A membrane, or brane, is a multidimensional object, usually called a P-brane, with P referring to the number of dimensions in which it exists. The value of 'P' can range from zero to nine, thus giving branes dimensions from zero (0-brane point particle) to nine - five more than the world we are accustomed to inhabiting. The inclusion of p-branes does not render previous work in string theory wrong on account of not taking note of these P-branes. P-branes are much more massive ("heavier") than strings, and when all higher-dimensional P-branes are much more massive than strings, they can be ignored, as researchers had done unknowingly in the 1970s.

Strings with "loose ends"

Shortly after Witten's breakthrough in 1995, Joseph Polchinski of the University of California, Santa Barbara discovered a fairly obscure feature of string theory. He found that in certain situations the endpoints of strings (strings with "loose ends") would not be able to move with complete freedom as they were attached, or stuck within certain regions of space. Polchinski then reasoned that if the endpoints of open strings are restricted to move within some p-dimensional region of space, then that region of space must be occupied by a p-brane. These type of "sticky" branes are called Dirichlet-P-branes, or D-P-branes. His calculations showed that the newly discovered D-P-branes had exactly the right properties to be the objects that exert a tight grip on the open string endpoints, thus holding down these strings within the p-dimensional region of space they fill.

Strings with closed loops

Not all strings are confined to p-branes. Strings with closed loops, like the graviton, are completely free to move from membrane to membrane. Of the four force carrier particles, the graviton is unique in this way. Researchers speculate that this is the reason why investigation through the weak force, the strong force, and the electromagnetic force have not hinted at the possibility of extra dimensions. These force carrier particles are strings with endpoints that confine them to their p-branes. Further testing is needed in order to show that extra spatial dimensions indeed exist through experimentation with gravity.

Membrane interactions

One of the reasons M-theory is so difficult to formulate is that the numbers of different types of membranes in the various dimensions increases exponentially. For example once one gets to 3-dimensional surfaces, one has to deal with solid objects with knot-shaped holes, and then one needs the whole of knot theory just to classify them. Since M-theory is thought to operate in 11 dimensions this problem then becomes very difficult. But just like string theory, in order for the theory to satisfy causality, the theory must be local, and so the topology changing must occur at a single point. The basic orientable 2-brane interactions are easy to show. Orientable 2-branes are tori with multiple holes cut out of them.

Matrix theory

The original formulation of M-theory was in terms of a (relatively) low-energy effective field theory, called 11-dimensional Supergravity. Though this formulation provided a key link to the low-energy limits of string theories, it was recognized that a full high-energy formulation (or "UV-completion") of M-theory was needed.

Analogy with water

For an analogy, the supergravity description is like treating water as a continuous, incompressible fluid. This is effective for describing long-distance effects such as waves and currents, but inadequate to understand short-distance/high-energy phenomena such as evaporation, for which a description of the underlying molecules is needed. What, then, are the underlying degrees of freedom of M-theory?

BFSS model

Banks, Fischler, Shenker and Susskind (BFSS) conjectured that Matrix theory could provide the answer. They demonstrated that a theory of 9 very large matrices, evolving in time, could reproduce the supergravity description at low energy, but take over for it as it breaks down at high energy. While the supergravity description assumes a continuous space-time, Matrix theory predicts that, at short distances, non-commutative geometry takes over, somewhat similar to the way the continuum of water breaks down at short distances in favor of the graininess of molecules.

IKKT model

Another matrix string theory equivalent to Type IIB string theory was constructed in 1996 by Ishibashi, Kawai, Kitazawa, and Tsuchiya.

Mysterious duality

A conjecture developed by Cumrun Vafa, Amer Iqbal, and Andrew Neitzke in 2001,[6] called "mysterious duality", concerns a set of mathematical similarities between objects and laws describing M-theory on k-dimensional tori (i.e. type II superstring theory on Tk − 1 for k > 0) on one side, and geometry of del Pezzo surfaces (for example, the cubic surfaces) on the other side. The main observation is that the large diffeomorphisms of del Pezzo surfaces match the Weyl group of the U-duality group of the corresponding compactification of M-theory. The elements of the second homology of the del Pezzo surfaces are mapped to various BPS objects of different dimensions in M-theory.

The complex projective plane P2(C) is related to M-theory in 11 dimensions. When k points are blown-up, the del Pezzo surface describes M-theory on a k-torus, and the exceptional del Pezzo surface, namely P1(C) × P1(C), is connected with type IIB string theory in 10 dimensions.

See also



  • Banks, T., W. Fischler, S.H. Shenker, L. Susskind (1996). M Theory As A Matrix Model: A Conjecture.
  • De Witt, B.; Hoppe, J.; Nicolai, H. "On The Quantum Mechanics Of Supermembranes", Nucl.Phys. B305:545 (1988).
  • Duff, Michael J. arXiv ePrint server.
  • Duff, Michael J. "The Theory Formerly Known As Strings", Scientific American, February 1998, pp. 64–69.
  • Gribbin, John. The Search for Superstrings, Symmetry, and the Theory of Everything, Little, Brown & Company, 1st BACK B Edition, August 2000, pp. 177–180. ISBN 0-316-32975-4
  • Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, W.W. Norton & Company, February 1999. ISBN 0-393-04688-5
  • Kaku, Michio (December 2004). Parallel Worlds: A Journey Through Creation, Higher Dimensions, and the Future of the Cosmos. Doubleday. ISBN 0-385-50986-3, 448.
  • Smolin, Lee. The Trouble with Physics, Houghton Mifflin, Mariner 2007. ISBN 0-618-91868-X
  • Taubes, Gary. "String theorists find a Rosetta Stone." Science, v. 285, July 23, 1999: 512–515, 517. Q1.S35
  • Witten, Edward. "Magic, Mystery and Matrix", Notices of the AMS, October 1998, 1124–1129.
  • A critical look at strings", August 2011.

Further reading

  • Brian Greene has written books explaining string theory and M-theory for the layperson:

External links

  • Brian Greene on the series Nova (original PBS broadcast dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003). Various images, texts, videos and animations explaining string theory and M-theory.
  •—the "Official String Theory Web Site", created by Patricia Schwarz. Excellent references on string theory and M-theory for the layperson and expert.
  • Basics of M-Theory by A. Miemiec and I. Schnakenburg is a lecture note on M-Theory published in Fortsch.Phys. 54:5–72, 2006.
  • M-Theory-Cambridge
  • M-Theory-Caltech
  • The Science Channel
  • TED 2005
  • Peter Woit's blog on physics in general, and string theory in particular