Friday, January 07, 2005

Types and meaning of the branes

Peter Woit is trying to find out what a brane is, how does it differ from a string, and whether it is useful, disastrous or necessary for making predictions. Because it is an elementary but a very important question, it may be helpful to try to help him out of his confusion - a confusion that was magnified by Peter's reading of the comments by Warren Siegel that, as I believe, are also kind of confusing.

Da brane

First of all, let me start with a rather formal definition. The word "brane", created by removing "mem" from a "membrane", is a generalization of a "membrane" that can have an arbitrary number of spatial dimensions. A brane is allowed to be "thick"; it's important that it is localized in some directions, however. There are some dimensions parallel to the brane, and some dimensions that are transverse to the brane. Moreover, we require that the brane does not destroy any rotational symmetries between the dimensions transverse to the brane, and it also preserves the translational and Lorentz symmetries involving the dimensions along the brane worldvolume which is the subspace of spacetime spanned by time as well as the spatial dimensions of the brane. The fact that we require the Lorentz symmetries to be preserved means that we're talking about "relativistic branes".

The last condition implies that a sheet of paper is not quite a brane because the Lorentz symmetry is spontaneously broken by the sheet of paper: there is a preferred reference frame associated with the atoms of paper. The rubber band is another example of a non-relativistic brane. The real, relativistic branes don't have such a special reference frame.

Are there some simple examples of branes? Yes. All point-like particles are examples of a 0-brane, where the figure 0 counts the number of spatial dimensions. Black holes may also be treated as 0-branes. When string theory was getting started, people thought that no particle was really point-like - a statement that we will have to clarify later in this article. In a similar way, the fundamental strings in string theory are examples of (relativistic) 1-branes because they have 1 spatial dimension plus one time - these two dimensions together form the worldsheet, a history spanned by the oscillating strings in spacetime.

The fundamental strings are very special and they're the only objects that can describe spacetime interactions while avoiding the ultraviolet (short-distance) divergences. They have more dimensions than zero, so that the short-distance physics in spacetime is regulated by their finite extent - the interaction is smeared out over a finite region. But their worldsheet dimension is two, which turns out to be the maximal number of dimensions that avoids the ultraviolet divergences of the worldsheet (or worldvolume) itself. The internal dynamics of higher-dimensional branes has its own short-distance problems, analogous to the short-distance problems in spacetime.

This argument was known to pick the relativistic strings as a special kind of object - and the spacetime observables like the scattering amplitudes could have been calculated from the correlators of some operators in a two-dimensional conformal field theory, a special kind of quantum field theory defined on the two-dimensional worldsheet. (Conformal symmetry is another thing that has very special properties for the two dimensions of the worldsheet, but I don't want to discuss it in this text.)

Black p-branes

That was the situation in string theory before 1995. But nevertheless, there were always some other branes in the game. Recall that 0-branes are 0-dimensional objects, by which we really mean that they are localized in all spatial dimensions. Clearly, the black holes may be classified as a new type of 0-branes. Moreover, there exist other solutions for the so-called black p-branes. Note that the letter p is a mathematical variable counting the number of their spatial dimensions. These black branes generalize the Schwarzschild solution - and other solutions (we should be really talking about the Reissner-Nordstrom solution) - but they are not localized in all spatial dimensions of the spacetime. Instead, they're extended in some of them. It means that the metric does not depend on these dimensions parallel to the "worldvolume" of these branes.

OK, so we know what are the black 1-branes, black 2-branes, and so forth.


But you've also heard about D-branes, haven't you? How were they discovered? For closed string, a special symmetry called T-duality was known for the closed strings. If you study closed strings propagating on a spacetime whose one dimension has a shape of a circle, these closed strings are able to oscillate and wiggle but also do the following two simpler things:
  • they can have a momentum along the circle which is, much like for point-like particles, quantized, n/R
  • they can be wound on the circle w times
It turns out that both the momentum as well as winding adds a contribution to the energy of the string - more precisely, the squared energy (or mass) has contributions (n/R)^2 from the momentum as well as (w.2.pi.R.T)^2 from the winding. Here T is the tension of the string, the mass per unit length. Note that there is a symmetry between n,w: both contribute quadratically. If you exchange the momentum and the winding, and you change R to 1/(2.pi.T.R) - essentially the inverse value of the radius - you will obtain the same spectrum of possible energies because the two contributions will simply interchange.

This symmetry applies to all states of the closed string, including the oscillating ones, and it is preserved by the interactions. It's called the T-duality - a symmetry whose validity can be proved in perturbative string theory. It implies that a Universe with a dimension smaller than a critical size is exactly equivalent to another Universe with another dimension which is larger than this critical size.

Note that it was important that the string could wind. But the open strings cannot be wound around anything, but they can still have a momentum. How does T-duality act on them? What is the momentum exchanged with?

The answer was already found in 1989 by Petr Hořava and Joe Polchinski et al. The answer is that the action of T-duality on a theory with open strings actually creates a slightly different theory: a theory with a Dirichlet brane or a D-brane for short. The name "Dirichlet" refers to the Dirichlet boundary conditions satisfied by the coordinates transverse to the D-brane, which is a function defined on the string that must be equal to a specific value at the string end-points (this is called the Dirichlet boundary conditions).

T-duality then acts on the open strings as follows:
  • In the original theory, we have open strings ending anywhere in space, and the momentum can be anything. For example, the momentum along the z-direction is quantized if the z-direction is compactified. The fact that the open strings can end anywhere may also be rephrased by saying that the open strings must end on a D-brane which is filling the space.
  • If we T-dualize along the z-direction, for example, we create a D-brane that looks like a "horizontal" sheet of paper, given by the equation z=const. The original open strings, that have an arbitrary momentum and ended anywhere, are mapped to new open strings that must end on the D-branes. Their momentum in the z-direction must vanish, because their endpoints are glued to the brane, but their winding can now be effectively anything, because such an open string can be wound many times before the end-point terminates at the same brane.
These open strings therefore only have either momentum (for the open strings ending on the space-filling brane - which is a D25-brane for bosonic strings or a D9-brane for the ten-dimensional superstrings), or winding (for the open strings ending on the sheet, the codimension one D-brane), and these two integers are interchanged by T-duality. That was a nice observation back in 1989, but no one was interested in it because no one wanted to consider these ugly surfaces (D-branes) that were clearly violating various spacetime symmetries and were "unnatural".

In the early 1990s, people used to discuss various topics during the lunch. For example, in the first part of the lunch, they would discuss p-branes, and in the second part of the lunch, they would chat about the D-branes. No one noticed that these objects sounded similar. ;-)

Polchinski's revolution

Everything changed in 1995 when Polchinski revolutionized the whole field of string theory. He showed that these strange D-branes on which the open strings can end are actually not uninteresting, bizarre modifications of string theory. Instead, they're real physical objects that have several newly discovered properties:
  • D-branes are dynamical. They're very heavy, but they can still oscillate. They're sources for forces - such as gravity - and they're also affected by forces from others.
  • All features of these oscillations, as well as dynamics and interactions of various new fields "stuck" to the D-brane are described exactly by open strings and their interactions, much like gravity and other forces in spacetime are described by closed strings. The relevant technology is again two-dimensional conformal field theory, but the worldsheets are allowed to have boundaries ending on D-branes.
  • Most importantly, D-brane carry charges - you should imagine a generalization of the electric charge.
  • The previous property implies that D-branes are actually physically identical objects to the black p-branes. They carry the same charges and they have the same mass. Actually, I was oversimplifying when I described the black branes: they're more like the generalized extremal Reissner-Nordstrom black holes with extra dimensions.
How can a D-brane, an infinite sheet of "paper" inserted into spacetime, be equivalent to a black brane, a black-hole-like solution that curves spacetime so much?

The answer is that string theory contains a coupling constant g - actually it is a dynamical scalar field in string theory called "the exponential of the dilaton", not an arbitrary parameter. Newton's constant is proportional to g^2. If the coupling constant is small, the best description of the brane is in terms of a D-brane that has almost no effect on the surrounding geometry. If the coupling constant becomes large, gravity grows stronger and it is better to describe the object as a black p-brane. More generally, the object both curves the geometry around, but it is also affected by excited open strings connected to it, and one needs the full string theory to describe what happens.

At the popular level, this equivalence between the black branes and D-branes sounds almost like Maldacena's AdS/CFT correspondence, even though it took two more years for people to appreciate it. One can focus on the "near-horizon" geometry of the black 3-branes, for example and it looks like five-dimensional anti de Sitter space (5 is 3+1+1, where 3 spatial dimensions come from the 3-branes, 1 is time, and the last 1 is the radial dimension) multiplied by a five-dimensional sphere, and type IIB superstring theory lives there. Equivalently, one can see that physics near the horizon is equivalent to the low-energy limit of the D3-branes.

Consequently, the gravitational type IIB theory (string theory that can be approximated by type IIB supergravity at low energies) is equivalent to a simple, four-dimensional gauge theory defined in flat space - the low-energy limit of the D-brane description. The two descriptions are more useful and "natural" in different regions of the parameter space and the AdS/CFT correspondence is the key quantitative example of the holographic principle in quantum gravity.

Wrapped branes and charges

When people finally understood that p-branes were really D-branes, they started to apply them in many ways. Andy Strominger was the first person who calculated that "wrapped" D3-branes actually regularize some divergences of the "conifold", a singular generalized manifold with a cone-like singularity in it. The low-energy theory for strings propagating on a certain singular space looks singular, but Strominger showed that there is a perfectly non-singular theory that contains the fundamental strings but also wrapped D3-branes. The singularity in the theory with strings only is reproduced if these D3-branes are "integrated out".

People studied various ways to "wrap" the branes on various "cycles" - submanifolds of the manifold of hidden dimensions of string theory. Each wrapped brane behaves like a charged object - each kind of wrapping a brane around a different homology cycle gives you a different type of an electric charge. These possible charges were first classified by homology, but homology was then seen to be superseded by K-theory - that used to be a purely mathematical subject, but was realized to be a more precise way to describe the possible ways how to wrap branes. I recommend you e.g. the recent article of Greg Moore for the mathematicians if you're interested.

Calculating with branes

But what I really want to fix in this article are the comments about our ability to calculate facts about the branes of different types. As indicated above, the D-branes are completely calculable at the weak coupling, and their internal dynamics is described by string theory - all the fields that propagate on their worldvolume (such as the scalar fields that remember the position in the transverse coordinates) are associated with open string states - open strings attached to this D-brane.

Besides string theory in 10 dimensions, there is also M-theory, a sibbling of string theory that exists in 11 dimensions. M-theory allows two types of branes to exist: M2-branes and M5-branes. M-theory in 11 dimensions has no strings, so there can't be any D-branes in 11 dimensions because D-branes are the branes on which the strings can end, and it's not possible without the strings.

Instead, M-theory has membranes which are kind of "fundamental" objects, but not quite. When we compactify M-theory on a circle, we obtain a 10-dimensional theory - namely type IIA string theory - and the M2-branes with one dimension wrapped on the circular dimension become the fundamental type IIA string.

As explained above, we can't really create a field theory based on M2-branes because they have worldvolume short-distance divergences. But that does not mean that M2-branes are unpredictable, as Peter and Warren kind of wanted to indicate. The dynamics of M2-branes is completely unique and fixed - for example, we can calculate it from Matrix theory that may be understood as "the correct" regularization of the worldvolume theories of M2-branes.

There exist questions about branes in sufficiently complicated situations that we're not able to calculate yet - for some of them, not even in principle - but there exists absolutely no indication that a question about brane dynamics could be ill-defined or ambiguous. There have been a plenty of questions and we could have answered a large portion of them. All of these answers turned out to be unique, beautiful, and illuminating, and there is simply no rational reason to think that something about the branes could have ambiguous answers or make string theory "unpredictable", as Peter Woit tried to suggest. The exact dynamics of branes is governed by the full structure of string/M-theory which is not just some naive local theory in spacetime. It is not a naive local theory in the worldvolume either - the fundamental strings are perhaps the only exception because their dynamics is a local two-dimensional field theory.

We have learned quite a lot about string theory and its accurate calculations in lots of different contexts - some of these calculations are dual or equivalent to each other. We have not learned everything about the question "what string theory is", but it may be just a temporary state of affairs. Warren's statement that "string theory is just a toy model" cannot be substantiated by anything material today - toy model of WHAT? - and I tend to guess that eventually we will prove that Warren's statement is simply wrong.

M-theory: broad and narrow meaning of the word

Also, Warren's viewpoint that M-theory is more fundamental than string theory is kind of obsolete, I think. 11-dimensional M-theory is just another limit of the "big string/M-theory" - it differs from the five perturbative string theory limit by its having no "fundamental strings". It also differs by a larger spacetime dimension - in this sense, M-theory is more "geometrical" than 10-dimensional string theory. But otherwise it is just another description of physics that has many other descriptions - each of them being very useful in a different limit of the parameter space.

Before people knew much about M-theory in 11-dimensions, they thought that it would answer all questions about string theory. The BFSS Matrix theory was found and it became clear that it contains all answers about M-theory in 11 dimensions, but in order to address the questions of various compactifications, new matrix models must be found. These insights made it clear that there is a "narrow" meaning of M-theory - the description in terms of an eleven-dimensional spacetime that looks like eleven-dimensional supergravity at low energies - and the "broad" definition of M-theory that includes everything good that the string theorists have ever studied.

One must distinguish these two definitions carefully. It is not correct to say that 11-dimensional M-theory is more fundamental than type IIB string theory in 10 infinite dimensions, for example. In fact, the latter can't be obtained by any finite compactification of M-theory. Because Imperialist Capitalist Pig found this statement strange, let me say a couple of words about their duality.

M-theory / type IIB duality

Eleven-dimensional M-theory is a description of string/M-theory in an 11-dimensional spacetime. By compactifying M-theory on a circle (i.e. taking the spacetime to be a product of R10 times S1), we obtain type IIA string theory in 10 dimensions. Its coupling constant is determined by the radius of the circle.

Now, type IIA is T-dual to type IIB. T-duality is explained at the beginning of the text. It means that type IIA on a short circle is equivalent to type IIB on a long circle, and vice versa. Both of them are theories with 9 large dimensions.

If you combine two previous paragraphs, you will see that M-theory compactified on a two-torus (i.e. the product of 2 circles) is equivalent to type IIB string theory compactified on a circle. Let me emphasize that this equivalence between M-theory and type IIB is absolutely exact. The relation between the parameters is a bit surprising:
  • The coupling constant of type IIB string theory is given simply by the ratio of the radii of the two circles that form the torus in M-theory
  • The radius of the circle in type IIB is inversely proportional to (a positive power of) the area of the torus of M-theory
The first point, incidentally, implies S-duality of type IIB. It's the statement that the theory with a large value of a coupling constant g is equivalent to the same theory with a small value of the coupling, g'=1/g. Using the relation between type IIB and M-theory, S-duality simply follows from the symmetry of the torus interchanging its two "sides".

The second relation between the two parameters is even more interesting: if we want to get the "decompactified" 10-dimensional type IIB string theory - i.e. the limit in which the circle in type IIB goes to infinity - we must take M-theory on a torus whose area goes to zero!

You know, in field theory you would expect that if you take an 11-dimensional theory and define it in a spacetime whose 2 dimensions are very tiny, you will obtain a 9-dimensional theory. The two dimensions simply disappear and you subtract them. However, string theory is not just some garden variety field theory: in string theory, you will obtain a ten-dimensional theory! Why is it so? It's because M-theory also admits M2-branes (membranes), as discussed previously, and these M2-branes may wrap on the small torus. Because the torus is small, these wrapped M2-branes will be very light (the mass is their tension times the area, in the simplest configuration), and by wrapping a large number of them, we can change the energy almost continuously. In fact, the winding number of these M2-branes is the same as the winding number w of the fundamental type IIA strings, and as we explained previously, using T-duality it is interpreted as the momentum n along the circle in the T-dual type IIB string theory.

For a small two-torus in M-theory, the wrapped membranes form a continuum; it is the continuum of momentum modes in type IIB, and if momentum becomes continuous, it means that the dimension becomes non-compact. It's not just this qualitative argument: we can prove the equivalence more exactly.

The left hand is not the right hand

Another interesting feature of type IIB string theory is that it is left-right asymmetric. For example, all of its gravitinos (supersymmetric partners of the graviton) are left-handed, if you define the word properly. In field theory, you could never obtain a left-right asymmetric theory by compactifying another theory on a circle: there is always a symmetry under the left-right flip in the "large" spacetime combined with the reversal of the circle (it was a rotation by pi before you compactified) - and this combined operation can always be named "parity" or "left-right symmetry". In string/M-theory, you often obtain "chiral" theories i.e. theories that distinguish between the left and the right. It's important in the Standard Model - for example, neutrinos are always left-handed. Heterotic string theory is also chiral, even if you compactify it on the Calabi-Yau manifold (which was the most classical "unique" way how superstring theory was believed to contain the real world).

One also obtains left-right asymmetric, chiral theories if she includes boundaries into M-theory. The boundaries of spacetime in 11-dimensional M-theory are called the "Hořava-Witten domain walls". They can be proved - by anomaly cancellation and other means - to carry a gauge field with a gaugino whose gauge group is the exceptional group E_8, and these domain walls also distinguish between the left and the right.

String/M-theory implies not only the existence of gravity: it also implies the existence of all other types of fields that we know and need - the fermions arranged in families, gauge fields, the Higgs fields - and the stringy vacua also naturally predict all the qualitative features we require - gauge interactions, Yukawa interactions, Higgs mechanism, confinement, left-right asymmetry, CP-violation, and so forth. The detailed portfolio of the particles and forces depends on the choice of the stringy vacuum - and we can get the right ones, although the right vacua don't seems sufficiently unique for us to make new unambiguous predictions beyond the Standard Model (at least so far). But the fact that we get all the qualitative phenomena and features right is highly non-trivial. Be sure that in loop quantum gravity or any other non-quantum-field-theoretical descriptions, you never get things like chiral interactions. All competitors are, in this sense, incompatible with the experimental discoveries in physics from the 1950s when parity violation was discovered. And it's not just parity violation.


  1. Thank you Prof.Lubos Motl for the clear explanation. it is very much appreciated.

  2. Nice post, but I have a dumb question. You say It is not correct to say that 11-dimensional M-theory is more fundamental than type IIB string theory in 10 infinite dimensions, for example. In fact, the latter can't be obtained by any finite compactification of M-theory. Does this imply that one or the other must be false or only approximate? Or am I missing the point?

  3. Hi Plato! Thanks, and you may call me Lubos - especially if you're one of the most famous philosophers in history - or is it you? :-) The same comment applies to others who call me an idiot.

    Capitalist: thanks, and I'm gonna add a sentence about the M/IIB duality to the article.

  4. Very nice, professor, and thank you. If you succeed, like here, to bridle your scorn of lqg you can educate us all.

    Also: congratulations! It seems with your previous
    entry you convinced Gerhard Schroeder to raise the German flood help to 500 MegaEuros.

  5. especially if you're one of the most famous philosophers in history - or is it you? :-) The same comment applies to others who call me an idiot.-------------------------------

    The gangs all here, and we do not think you a idiot. Right Aristotle? That's right Plato.:)"PLato saids, I Look to the perfection of the heavens for truth," while Aristotle saids "look around you at what is, if you would know the truth"

    When you see Plato's finger up like that, and Aristotle with his hand the way it is, the struggle by whose hand reveals what in today's world, is not about rock, scissors, or paper:)