Prerequisites: Superstring theory
See also: Mathematics and physics -- Geometry and topology -- Black holes -- Quantum gravity -- Noncommutative geometry -- Quantum geometry
In any given field or theory there will in general be more than one important problem -- usually many more than one. But only some of that number are "ripe" for attack. Those become the next targets.
In the case of superstring theory there are plenty of problems, of course. Out of them all, the one most ripe for attack (at least judging by the amount of effort in recent years) may be this: Why do there seem to be so many different "solutions" of the theory? Are they really all as different as they appear to be?
Surely, for any decent theory, it should be possible to determine which of many apparently different solutions is the "right" one which describes physics in our universe. Unfortunately, it hasn't yet been possible to do this for superstring theory.
To expand a bit, recall that, for various reasons, superstring theories work best in ten spacetime dimensions. Further, in ten dimensions there are only five different mathematically consistent types of solutions of superstring theory (named Type I, Type IIA, Type IIB, Heterotic-E, Heterotic-O).
But because these are 10-dimensional theories, while the universe we observe has only four spacetime dimensions, six of the dimensions must be tightly curled up ("compactified") to explain why they aren't observable. This introduces -- apparently -- a large number of different solutions for each case, because observable physics depends in a number of ways on the specific topology of the compact six dimensions.
If we could compute precisely enough (using perturbation theory) the physical consequences of any specific topology, we would hope to be able to eliminate most alternatives as unrealistic. But we don't know how to do that.
So how do we make any sense of this mess? Is it possible that the alternatives really aren't as different as they appear to be? It turns out that a lot of the recent progress that has been made has resulted from trying to show, in several different ways, that this is in fact the case.
One of the main outcomes of this effort has been the hypothesis (and that's all it is at this point) that there is really only one fundamental theory, of which the five consistent superstring theories are just special cases. This more fundamental theory seems to involve a spacetime of eleven, rather than ten, dimensions. And in addition, 11-dimensional supergravity seems to be a special case also.
M-theory is the name given to this more fundamental theory. But we know almost nothing about it -- certainly not its fundamental equations, let alone how to solve them. Even the name is (deliberately) left a bit vague. Whimsically, the "M" could stand for membrane, manifold, matrix, meta, magic, or mystery -- among other things.
If we know so little about this theory, why even suppose it exists? The answer is that there are several relationships, known as "dualities", which -- surprisingly -- connect the different superstring theories. Theories which are dual to each other should predict the same physics. And if that is so, the theories should in some way be limiting cases of a single theory that we simply can't identify yet.
For future reference, here's a table of the five superstring theories that you can refer to if you want to keep them straight:
|Name||String types||Gauge group||Chiral?|
|Type I||open, closed||SO(32)||yes|
In case you wonder about the precise difference between Type IIA and Type IIB theories, the explanation is fairly technical. Type II theories involve oriented closed strings. In group theoretical terms, both Type II theories have two "supersymmetry generators" (or "supercharges"). The supersymmetry generators correspond to the fermionic and bosonic vibrations. In the IIA case, the generators have different handedness, which is to say that the fermionic and bosonc vibrations move in opposite directions. In the IIB case, they have the same handedness, so both vibrations move in the same direction. This implies that there is, or is not, an invariance under reversal of the orientation of spacetime (as in a mirror image). Hence IIB is chiral, but IIA is not.
Although this seems like rather a technicality, we shall see that in some respects the Type IIA and Type IIB theories are affected differently by certain "dualities".
Human thinking is full of dualities, like yin/yang, male/female, rational/intuitive, etc. We are quite familiar with how the world tends to appear different from the male perspective as opposed to the female perspective, for instance. What is seen as "better" from one perspective can be seen as "worse" from the other. (Where "better" can be read as "more enjoyable", "less frightening", etc. depending on the context.)
Well-known dualities crop up in mathematics and physics. In mathematics, the discrete/continuous duality is a frequent theme. In physics -- quantum mechanics to be precise -- there is the particle/wave duality. This is the idea that the same phenomena can be understood in terms of either particles or waves. But the more you describe something as a particle, the less you may describe it as a wave. Nevertheless, the two points of view must ultimately be consistent with each other, since they pertain to the same reality. In other words, there is one underlying reality which is manifested differently in different situations. If you take the full theory and look at special limiting cases, you see two apparently different formulations of what is actually the same theory.
The particle/wave duality has little direct relevance to our present concern with string theory. But there are other dualities in physics which are relevant. And they turn out to relate special limiting cases of a single general theory.
The interesting thing about Maxwell's equations is that they have a lot of symmetry between the electric and the magnetic field. That is, you get almost the same equations if you switch the two field types. This is especially true for the equations in a region of space where there are only fields and no electric charges. When the equations are extended to allow for charges, only electric charges are considered, because magnetic charges (also known as magnetic monopoles) have never been observed in nature. Much of the symmetry is lost when the equations take account of electric charge.
However, there is no known inconsistency in the idea of magnetic charge. If the equations are further modified to describe what would happen with magnetic charge present, much of the symmetry is restored. The symmetry still isn't perfect, even then. There are some fundamental reasons why electric and magnetic fields are not completely interchangeable. But the near symmetry is intriguing.
In the traditional analysis of magnetic monopoles, they are regarded as "solitons". These are solutions of a wave equation that have finite spatial extent but do not disperse with time. Another term used for monopoles is that they are "collective excitations". This means that they arise out of the fields of the theory (in this case the electric field), for reasons that are essentially topological.
What is meant by a "topological" reason? For an example, visualize a sphere in ordinary 3-space. Imagine that the sphere is covered with hair. The hair corresponds to a vector field, that is, a 3-component vector attached to every point on the surface of the sphere. Now imagine trying to comb that hair, which amounts to trying to assign a direction to every vector such that vectors which are close together point in almost the same direction. If you think of combing your own hair, you should see that the "hair" on a sphere cannot be combed so that the direction of each hair (vector) changes smoothly at every point on the sphere. That is, there will be at least one spot around which the vectors point in nearly opposite directions. That spot corresponds to a soliton. It can move around on the surface, but it can't simply disappear. Mathematically speaking, this is how solitons may arise from vector fields on surfaces or higher dimensional manifolds.
The net result is that magnetic monopoles, considered as solitons, are point-like knots in a quantum field. Now, the really interesting thing is that if one considers a magnetic field instead of an electric field, an "electric monopole" -- i. e. a particle with a unit of electric charge -- can arise as a collective excitation of the magnetic field. So in some sense, magnetic monopoles can be regarded as made up of knots in an electric field, and electric monopoles can be regarded as made up of knots in a magnetic field. In other words, there is a deep symmetry between electricity and magnetism.
This duality of electricity and magnetism was pointed out in 1977 by Claus Montonen and David Olive. More specifically, they noticed that it was under conditions of "weak coupling" that magnetic monopoles appeared to be collective excitations rather than elementary particles. "Weak coupling" means that the relevant coupling constant -- the so-called fine structure constant -- is small. But if one considered the strong coupling case where the coupling constant is large, then it is the "electric monopole" that appears as a collective excitation -- a knot in the field -- while the magnetic monopole would be an elementary particle.
The duality can be made precise by exchanging the fine structure constant with its inverse in formulas where it occurs. This exchange transforms particles into collective excitations, and vice versa. In Maxwell's equations, it exchanges the electric and magnetic fields. Naturally, this duality is referred to as electric-magnetic duality.
A similar phenomenon occurs in QCD, quantum chromodynamics, the theory of the strong force. There are different types of charge in that theory too, the color charges. Weak coupling is the situation where two quarks are close together. In fact, the closer they approach each other, the smaller the force between them. This is what is known as "asymptotic freedom", and it can be proven in QCD precisely because the coupling is weak. It is easy to do computations in the weak coupling case, because the perturbation expansions are power series in powers of the coupling constant. If that constant is small, the series converge rapidly.
On the other hand, when the coupling constant is large (especially if greater than 1) the perturbation expansions do not converge. The computations don't work. This is what has made it so difficult to study the behavior of quarks when they are seprated by increasing distance. Experimentally, the force remains constant, so that the potential energy due to separation increases without limit. This phenomenon is called "quark confinement". It is manifested in the fact that quarks seem never to occur by themselves, but always occur in groups of two, three, or possibly more. Although this is known experimentally, the computational difficulties have prevented giving a rigorous proof of confinement.
In string theory, gs has a simple physical interpretation. Namely, it is a measure of the likelihood that a single string will split into two separate strings on account of quantum fluctuations, yielding a virtual pair. The number is called a coupling constant because it measures the degree of coupling between the initial single string and the resulting pair. If it is small, strings split into pairs infrequently. If it is large, splitting is common, so that Feynman string diagrams will tend to have lots of loops. We will see presently that there is also a geometric interpretation of the string coupling constant.
If gs is small, then we say that is a weak coupling situation. If it is large, that is referred to as strong coupling. What string theorists found is that there is a duality between the two, analogous to the case with electromagnetism. Specifically, suppose you have two string theories A and B, chosen from among the Type I, Type IIA, etc. possibilities. Then A and B are said to be dual to each other if any physical quantity predicted by A for coupling constant gs will be identical to the quantity predicted by B when the coupling constant is the reciprocal, 1/gs.
Of course, if the coupling constant is large, its reciprocal is small, and vice versa. So to say that A and B are dual to each other is basically to say that they make the same physical predictions when one is under conditions of weak coupling and the other is under strong coupling. In symbols, if f denotes some physical quantity, then
fA(gs) = fB(1/gs)This duality is called "S-duality". (Which can be remembered if "S" stands for "strength" or "soliton".) It was formalized in 1990 as a generalization of the Montonen-Olive electric-magnetic duality.
The beauty of it is that one never needs to do computations in strong coupling conditions for a theory A if there is a dual theory B, since one can then do the computations in weak coupling for B. In order to do perturbative computations, it is necessary to work with weak coupling, since the series expansions contain powers of gs -- so it should be smaller than 1.
The story gets even better, since this is more than just an abstract possibility. In fact, string theorists found that there are several examples of S-duality among the five known consistent string theories. Specifically, the Type I theory is S-dual to the Heterotic-O theory. This makes some sense, as both theories have SO(32) as their gauge group. More surprisingly, perhaps, the Type IIB theory is S-dual to itself.
But even more surprising than that is what happens with the Type IIA and Heterotic-E theories under S-duality. Instead of being dual to each other, both theories have even more unexpected behavior under strong coupling as gs increases. Simply stated, both these types of strings grow an extra dimension. The Heterotic-E string grows from a 1-dimensional loop to a 2-dimensional closed ribbon, whose width increases with the coupling constant. The Type IIA string grows into a 2-dimensional torus, a hollow donut shape like an inner tube. The size of its extra dimension also increases with the coupling constant. In this sense, the coupling constant has a geometric significance, as mentioned earlier.
Hence, under S-duality, the Type IIA and Heterotic-E strings are not directly related to each other, but to an as-yet unknown 11-dimensional theory. Although we know little about this latter theory, it appears to be unique. This is the theory which is called M-theory. We'll discuss it more a bit later.
Let's recall one basic fact about closed strings. Their modes of vibration must have an integral number of wavelengths around the string. Say that the radius of the loop is R (in appropriate units), and the number or wavelengths is n. It isn't hard to figure out intuitively how the energy in the string varies with n and R. Just as with an electromagnetic wave, energy is inversely proportional to wavelength. The larger n is, for a fixed R, the smaller the length of each wave is, and hence the larger the energy. So energy varies directly with n. But for a fixed n, the wavelength is directly proportional to R, so the energy is inversely proportional to R. Therefore, in the appropriate units, the quantized energy of a particle that corresponds to the vibrating string is n/R.
As a result, we get a spectrum of energy states corresponding to positive integral wave numbers. As R increases, the states become closer together. If R is very large, we have almost a continuous spectrum, because the quantum differences are very small.
But there is another factor which affects the energy in any vibrating string, whether it's a guitar string or a string theory string, and that is the tension in the string. The higher the tension, the more energy there is for a given mode of vibration. So there is another way to generate a spectrum of energy states in a closed vibrating string. Simply imagine that the string is wrapped multiple times around a circle, like a rubber band you might wrap around your wrist. Like a rubber band, a closed string is of a fixed size to begin with. The tension in the string, and hence its energy, is directly proportional to how far it's stretched, which is in turn proportional to the number of times the string is wrapped around multiplied by the size of the circle it wraps around. Therefore, using the same energy units as before, the spectrum of energy states as a result of the string tension is given by mR/α', where m is the winding number.
The (inverse of the) constant of proportionality, α', is an important constant of string theory intimately associated with string tension. In fact, the string tension of the fundamental string is given by 1/2πα'. α' is sometimes referred to as the "Regge slope". It has dimensions of length squared, and its value in the appropriate system of units turns out to be the square of the characteristic string length scale -- about (10-32 cm)2.
In summary, we get two different energy spectra. They are sometimes referred to as the vibration and winding modes of a string. The energy levels are given, respectively, by
En = n/R and Em = mR/α'There are two geometric variables of importance here: the radius R of a circle that the string wraps around, and the number of times m that it wraps. Notice that if the radius R has the particular value (α')1/2, then the energy spectra are exactly the same. In other words, there is a special value of R where the spectra generated by wave number and winding number are the same. Does that ring any bells? Might it suggest some sort of duality, where there is a dual relationship between size and winding number? And isn't it interesting that this special value of R is the fundamental string length?
When theorists noticed this clue and investigated further, they found that there was indeed a duality here. Just observe that if you replace R in the formulas above for En and Em by a new variable, R' = α'/R, the formulas for the energy spectra are exchanged:
En = nR'/α' and Em = m/R'Is this just a fluke of algebra, or is this duality actually important? Well, at this point we have to pause and recall that the critical value of (α')1/2 is roughly 10-32 cm, only a factor of 10 from the Planck length. We do not expect physics at that scale to be much like the physics we are more familiar with. Space itself may be quantized at the Planck scale. Therefore, we can't feel too confident about working with length scales much smaller than (α')1/2. But fortunately, because of this duality, we don't have to! Any time we find ourselves working with very small values of R, we just replace every occurrence of R in our string theory formulas with α'/R. Doing this amounts to interchanging the vibration and winding modes of a string. We expect to get the same physical results.
Amazingly enough, this works. The name given to this duality is T-duality. (Which can be remembered if "T" stands for "tension".)
When T-duality was applied to our various seemingly different forms of string theory, it was found that the Type IIA theory is T-dual to the Type IIB theory, and the Heterotic-O theory is T-dual to the Heterotic-E theory. In other words, the theories in each pair are just different limiting cases of the same theory.
Let's summarize the results in a table:
|Type IIA||M-theory||Type IIB|
|Type IIB||Type IIB||Type IIA|
Note that this table is misleading in one respect. Namely, when we say a theory is dual to M-theory, we aren't being very precise, because it isn't known exactly what M-theory. All that's meant here is that the dual theory is 11-dimensional and behaves consistently with what little is known of the "real" M-theory. Also, the relation goes in one direction only, since not enough is known about how to apply a duality to it.
So, what is T-duality telling us? What it seems to be saying is that we do not need to even consider values of R smaller than the fundamental string length. For very small values of R (less than (α')1/2) we can just switch to another theory with R replaced by the larger quantity α'/R and go on our merry way. The limiting case for one theory with small R is just another theory with large R.
Nature seems to be saying that there's a smallest length scale beyond which we need not look.
One of the main reasons for the appeal of string theory early on was that a theory of extended objects (that is, having a dimensionality of 1 or more) makes it possible to avoid various problems associated with the point particle abstraction. We noted that strings, which are 1-dimensional, are the simplest alternative. But they are hardly the only alternative. Elementary particles could in principle be modeled as 2-dimensional objects (surfaces) or as something of an even higher dimension.
Strings were the first extended objects to be investigated, because their theory is the simplest. In particular, their trajectories through time are easily understood as 2-dimensional world-surfaces, and perturbative computations with these turned out to be especially easy (comparatively speaking, at least). With higher dimensional fundamental objects, the corresponding concept is a "world-volume". These are much harder to visualize, let alone deal with mathematically. It's also harder to have a consistent quantum theory (preserving unitarity) in this case.
But all these difficulties have to do with our mathematical tools. They don't mean that higher dimensional fundamental objects are not physically important. Indeed, detailed investigation of the properties of string theories has lead to an apparent necessity of considering higher dimensional objects. The simplest case of this is with S-duality of the Type IIA and Heterotic-E theories. We noted that as the strength of the coupling constant increases, the strings of those theories become 2-dimensional surfaces.
It therefore seems we cannot avoid dealing with fundamental objects of dimensionality higher than 1. Certainly we have to consider objects that are 2-dimensional surfaces, which might be called "membranes", or simply, "branes". More generally, for an object of dimension p, if one wants to be explicit about it, one can use the term "p-brane". It turns out that in M-theory we run into p-dimensional objects for any nonnegative integer p ≤ 9. Mathematically, a brane is just a certain type of manifold, that is, a geometric object which is "locally" (in a small region) like p-dimensional Euclidean space (Rp).
The example of 2-branes arising from S-duality is just one of various ways that branes come up in string theories. More generally, in the Type IIA and Type IIB theories, there are classical solutions that describe p-branes. These p-branes are topological solitons (spatial "defects" not unlike defects in a crystal lattice) carrying conserved charges that act as sources for various gauge fields.
Yet another important example is given by what physicists have called "BPS states". The term is from the names of the orginators of the idea: E. Bogomolnyi, Manoj Prasad, and Charles Sommerfeld. BPS states exist because of an important result from the theory of supersymmetry. This result states that there is a unique object that has a specified amount of some charge and a minimal mass. The charge might be electric, magnetic, color, or some other type of force charge. A BPS state is simply an object that satisfies this minimality condition.
The reason that BPS states are important is that their properties may be determined relatively easily without using perturbation theory, and (therefore) without regard for the size of any coupling constant. They therefore take advantage of supersymmetry to give us information about a theory without having to resort to perturbative methods, with all their attendant limitations and messiness. BPS states are just one of the miraculous "gifts" of supersymmetry. Since one of the problems in working with higher dimensional objects was the difficulty of using perturbation theory, it is fitting that -- free from this limitation -- BBS states are often higher dimensional objects. They can be 1-dimensional strings, but they can also be p-branes, for any p ≤ 9
Another special type of p-brane, D-branes, to be discussed next, also happen to be BPS states.
Theoretical work involving duality usually leads to branes. One of the earliest contributions along these lines was due to Andrew Strominger, who suggested that, in 10 total dimensions, a strongly interacting string could be dual to a weakly interacting 5-brane that is a soliton.
The relevance of boundary values here is in connection with Type I strings, which are the only kind that may be open, rather than closed loops. Strings that are open have endpoints -- the boundaries of the string -- and hence the values of various quantities on these endpoints are of importance. D-branes are simply objects on which open strings may terminate. Going back to the origins of string theory in the theories of Veneziano, Nambu, and others related to the strong force, their prototypical strings were abstractions whose terminal points were quarks. D-branes, then, are a generalization which draws attention to the endpoints of strings.
Not all of the extended objects, the branes, which turn up in extensions of string theory are D-branes. But a rather large number are, and so this special case seems to be especially important.
Recall that T-duality as described above deals with closed strings of radius R and the symmetry which results when R is replaced with a new variable R' = α'/R. The duality is a consequence of the fact that the same energy spectra are obtained by varying the number of nodes in a standing wave on the string as are obtained when the string wraps a certain number of times around a circle. However, it's not clear how to apply this idea to open strings.
In 1989 Joseph Polchinski (with Jin Dai and Robert Leigh) figured out what needed to be done with open strings in order to extend T-duality to that case. Not surprisingly, the answer depends on what happens at string boundaries. Specifically, what they found is that after applying the duality the endpoints of a string are no longer completely free to move anywhere, but instead are required to lie on a fixed hyperplane. For instance, if one spatial dimension is dualized (i. e., subjected to the R ↔ R' transformation), then one degree of freedom of movement of the string endpoints is lost, so that the endpoints are constrained to move in a 9-dimensional hyperplane instead of full 10-dimensional spacetime. It is as though the endpoints had become stuck or attached to the hyperplane. The restriction on the endpoints amounts to a set of Dirichlet boundary conditions, and the hyperplane is a prototypical D-brane.
The resulting D-brane is not a fixed, static object, however. Due to the quantum effects of virtual strings appearing from and returning to the vacuum (just like virtual particles), D-branes can vibrate, move around in spacetime, and even interact with strings and other D-branes. But all of this dynamical behavior is a result of the strings that are "stuck" to the D-brane. In short, D-branes have all the characteristics of extended objects, even though from another point of view they are "topological defects" like solitons -- a sort of higher dimensional glitch in spacetime.
Later, in 1995, Polchinski showed that D-branes could carry charges analogous to electric charge. These charges represent the spectrum of states that would be expected to arise from the duality symmetry.
In addition to charge, there is also a concept of the "tension" in any p-brane, and in D-branes in particular. It is defined to the be the mass-energy per unit p-dimensional volume. (In particular, the tension in a string -- a 1-brane -- is mass per unit length.) The tension turns out to be simply proportional to the inverse of the string coupling constant (i. e., to 1/gs).
It is not clear what D-branes are "really". Although they are defined as a certain mathematical construct (a kind of soliton) built out of strings, they could be just as fundamental as strings. Or strings might be built out of D-branes. Or perhaps both strings and D-branes are composed of something else. This ambiguity arises from duality. From one point of view, D-branes can be defined in terms of weakly coupled open strings, where gs is small, gravitational effects are negligible, and spacetime is "flat". But in another point of view, where the coupling is large and gravity is significant, the D-branes may be regarded as p-brane solutions of a supergravity theory.
But whatever they "really" are, D-branes have already proved to be quite useful theoretically. They have, for instance, been used -- by Andrew Strominger and Cumrun Vafa -- to provide a string-theoretic account of black hole thermodynamics. Specifically, they enable a string-theoretic proof of the Beckenstein-Hawking entropy formula, as we shall discuss later.
|Source: The Official String Therory Web Site|
Referring to the diagram at right, M theory is like a large continent. We are far from the center of it, and at this time we can explore it only from the coast -- the fringes of the continent. Not only that, but most of the coast is shrouded in fog. All we can explore at this time amounts to just a few points of land that emerge from the fog.
Those points of land are the different types of consistent superstring theories. At first, each of those theories seemed isolated and separate from each other. For all we knew, we might be dealing with an archipelago of islands rather than a continent. But there were curious and intriguing similarities from place to place. The geology of what we could see seemed quite similar everywhere, suggesting common origins. The flora and fauna of each location, though of different species, nevertheless bore distinct family resemblances. All of this suggested hidden connections we simply couldn't see through the fog.
The points on the diagram labelled I, IIA, IIB, HO, and HE are the string theories we have been describing. 11d refers to 11-dimensional supergravity theory. You may recall that supergravity was a quantized theory of gravity, based on supersymmetry, which was thought, back in the late 1970s and early 80s, to hold great promise as the fundamental theory of matter and forces. Although it ultimately proved to be unsatisfactory in that role, for several reasons, it remained a theory with some very interesting properties. Among those properties -- and very germane to the present discussion -- were its relations to the various string theories which appeared as those latter theories revealed their properties under extensive investigation.
Now it happens that supergravity theories can be formulated in anywhere from four to eleven spacetime dimensions (but in no higher dimensions). The most promising versions were formulated in ten or eleven dimensions. Interestingly enough, it was found that all of the string theories worked best in ten dimensions. Is that a coincidence?
No, of course, it isn't a coincidence. As we have noted, string theories are all about physics near the Planck scales of energy and length. It is, in fact, fiendishly difficult to derive from them any predictions of what physics is like at energy scales 16 orders of magnitude smaller, where we can actually perform experiments. Nevertheless, there are some results relating string theories to low energy field theories that seem to fit our experimental knowledge. For example, the Heterotic-E theory seems to have a low-energy version that best resembles the Standard Model.
Now, in 10 dimensions there are actually four different supergravity theories, which differ in details about how supersymmetry is incorporated. When the different string theories are examined at low energies, where strings are very well approximated by point particles, three of the string theories (Type IIA, Type IIB, and Heterotic-E) have approximations corresponding to three of the 10-D supergravity theories.
Very interesting. But what about the other supergravity theory and Type I and Heterotic-O string theory? It turns out that the low-energy approximation of both of these is the same, remaining 10-D supergravity theory. Even more interesting. This result also suggests an unsuspected (at the time) kinship between Type I and Heterotic-O theory. Though not known at the time, we now know these string theories are S-duals of each other.
This is all well and good, except that it was 11-D supergravity rather than any of the 10-D versions which seemed to be most promising. Where does it fit in? Well... S-duality is the clue. It relates the Type I and Heterotic-O theories, and the Type IIB theory is S-dual to itself. So what about the Type IIA and Heterotic-E theories?
We have already noted that as the string coupling constant gs increases (as prescribed by the recipe for S-duality), Type IIA and Heterotic-E strings become 11-dimensional. Bingo! More specifically, a Type IIA string acquires the topology of (the surface of) a torus in 11-dimensions, while a Heterotic-E string becomes, topologically, a ribbon. That is, as the coupling constant increases, an extra spatial dimension appears (giving a total of 10 spatial dimensions and 11 dimensions of spacetime).
And, sure enough, when one looks at the low-energy approximations of these strong-coupling theories (as best one can, given the limited theoretical tools available), one finds... something that looks a lot like 11-D supergravity. (The tools used for this must be non-perturbative, including the theory of BPS states.)
Now, the large-gs Heterotic-E string is topologically like a ribbon, and the extra dimension is a line segment, an interval on the real line. The large-gs Type IIA string is a torus, where the extra dimension is a circle. These "extra" dimensions are referred to as "compact" dimensions (when gs is small) -- an interval in the Heterotic-E case and a circle in the Type IIA case. Physicists speak of "compactifying" the respective theories on an interval or a circle. This explains the "interval" and "circle" labels in the M-theory diagram above. The other labels refer (obviously) to S and T dualities.
(Mathematicians note that a circle is just an interval with the endpoints identified. In mathematical notation, if S1 is a circle, then an interval is (topologically) S1/Z, the "quotient space" of S1 by Z, where Z denotes the integers.)
The bottom line here is that there are many relationships among the five string theories and 11-D supergravity. It's actually much more extensive than depicted in the diagram. There are other types of duality which have been discovered which relate the various theories to each other. Beyond that, there are many possibilities of combining dualities, as when S and T are both applied to study what happens when the coupling constant is varied at the same time as string vibration and winding modes are interchanged.
Type IIA string theory and 11-D supergravity play an especially important role. In technical terms, it happens that there are D0-brane solitons in the Type IIA theory. (That is, 0-dimensional D-branes -- pointlike objects.) These are extremely massive BPS states. One assumes there can be bound states with any number of such D0-branes. (I. e., they clump together.) It is found that in the strong coupling limit, there is a spectrum of low energy states that coincide with the spectrum of states of 11-D supergravity. Stated differently, at low energies 11-D supergravity provides a good description of the strong-coupling limit (the S-dual) of Type IIA theory. Much of what is known about M-theory is the result of exploiting this relationship, together with a few other facts, such as the existence of 2-branes and 5-branes in the 11 dimensional theory.
The general state of affairs is that there is actually an extensive web of relationships among the six theories. It is, surely, far too extensive to be merely a coincidence. The only reasonable conclusion is that something more is going on. There must be a theory we can't yet see clearly -- a whole continent -- that lies between the theories we (sort of) understand and which connects them.
To change the metaphor a bit, there should be a single theory with many knobs and dials and switches that can be varied and set independently. Mathematically, these are parameters, such as the string coupling constant and various topological size and shape quantities. For certain very specific choices of these parameters we get the five string theories and supergravity.
But most choices give a form of theory we don't really know how to work with presently. M-theory is simply the name applied to this presumed "master" theory we don't yet understand.
This is hardly an unprecedented circumstance in the history of physics. Physicists have been in this situation before. For example, there was the theory of black-body radiation just about 100 years ago.
At issue is the spectrum of electomagnetic radiation emitted by a body of matter heated to a given temperature. This system can be modelled by a study of the electromagnetic field inside a finite enclosure. Physicists in the 19th century developed theories for describing this mathematically in terms of the independent oscillation of many identical particles in the walls of the enclosure. Each particle was assumed to oscillate with a certain fixed frequency. Since each particle could oscillate with a different frequency, the overall result would be electromagnetic energy emitted in a whole spectrum of wavelengths. The problem was to come up, theoretically, with a formula for the spectrum -- that is, the emitted energy as a function of wavelength -- which would depend in turn upon the temperature.
There were problems with this theoretical framework, however. Notably, different physicists obtained different, and inconsistent results. In 1900 Lord Raleigh derived one formula -- "Raleigh's Law" -- for the amount of energy in the enclosure per unit volume for any given frequency (which is inversely related to wavelength) at a given temperature. His formula agreed well with experiment for low frequencies (i. e., long wavelengths) and high temperature. Unfortunately, it didn't work well when the ratio of frequency to temperature was large rather than small (high frequency, low temperature). On the other hand, a little earlier (1893) Wilhelm Wien had derived a different formula ("Wien's law") that did work when the frequency-temperature ratio was large. Reconciling these contradictory results was one of the principal open questions of physics 100 years ago. Was there a single theory which agreed with experiment over the full range of experimental parameters?
Max Planck (who gave us Planck's constant) resolved the quandry very soon after Raleigh published his formula. It required making the highly unintuitive (for the time) assumption that energy is quantized -- comes only in certain discrete values rather than in continuously varying amounts. But Planck's formula did agree with experiment for all accessible values of frequency and temperature.
Moreover -- and to the point -- Planck's single formula was a good approximation to Raleigh's when the frequency-temperature ratio was small, and to Wien's when the ratio was large. So, there is good precedent for subsuming disparate theories which each work for a certain range of parameters under a single theory which works over a larger range.
Just a little later, in 1905, Albert Einstein extended Planck's quantum ideas to explain the "photoelectric effect". Within little more than another 20 years, these theoretical efforts had coalesced into quantum theory, much as we know it today. This quantum theory not only gave a unified set of equations which covered electormagnetic and atomic phenomena, but it also provided underlying principles which -- however unintuitive they may have been (and still are) -- allowed the construction of a consistent and experimentally validated theory of all the phenomena within their purview.
Physics has continued to evolve far beyond that, but in essentially the same pattern. Which is why it seems at least a good bet that some underlying theory -- call it M-theory to give it a name -- can be found to unify and explain the partial theories ranging from the Standard Model on up to the various string theories that we have today.
So. We still don't know what M-theory is. But we can summarize our discussion (as well as a few points not yet covered) by listing a few things that we are pretty certain we know about it:
The existence of black holes is a relatively straightforward prediction of general relativity. So straightforward, in fact, that Karl Schwarzchild deduced it from Einstein's work in 1916, only a year after the theory was published. Yet up until fairly recently, many physicists doubted the physical reality of the black holes predicted by the "Schwarzchild solution", and other solutions, of Einstein's equations. In the last few years, observational evidence for the existence of black holes has become overwhelming.
While the doubters persisted in their skepticism, other physicists set about to take black holes seriously and to investigate theoretically what could be said about their properties. Stephen Hawking was one of that number. Beginning in the late 1960s he made several fundamental theoretical discoveries about black holes. Notably, in 1970, he found that the area of a black hole's event horizon always increases as a result of any physical interaction.
To Jacob Bekenstein, who was at that time only a graduate student, this was suspiciously similar to the fact about the property of thermodynamic entropy that it, too, always increases in physical interactions -- the famous "second law of thermodynamics". This similarity reinforced other ideas Bekenstein had about black holes and entropy. In particular, since all matter which fell into a black hole would be "lost" to the universe outside the hole, the entropy of the matter would also be lost to the outside universe. Therefore, in order not to violate the second law, the black hole would have to possess entropy, and this entropy should increase by an amount at least as large as what belonged to any matter which fell into the hole. Thus, the black hole accumulates not only the mass but also the entropy of any matter it acquires. Motivated in part by Hawking's discovery, Bekenstein went on to conjecture that the entropy of a black hole was proportional to the area of its event horizon.
At first Hawking himself doubted this suggestion. It just didn't seem right for a black hole to possess entropy. After all, the only distinguishing characteristics of a black hole are its mass, its angular momentum, and several types of force charge. That doesn't seem to leave room for a lot of "disorder". And further, entropy is ultimately a quantum mechanical concept, based on the quantum mechanical states of all the constituent particles of a system. But quantum mechanics seemed to have little relevance to a construct of general relativity like a black hole. As we have noted a number of times, the two theories do not play well together. And worst of all, if a black hole had entropy, it would also have to have a well-defined temperature, which means that the black hole would have to radiate energy. This seemed contradictory, almost, to the definition of a black hole.
Nevertheless, in 1974 Hawking proved the now well-known result that black holes can radiate energy and therefore have a well-defined temperature. Of course, it's not much: the temperature of a black hole of several solar masses is only about 10-8 degrees. (Microscopic black holes radiate much more strongly and therefore have much higher temperatures.) But from the temperature Hawking could calculate the entropy, and when he did, he found that Bekenstein was right. Indeed, Hawking obtained a simple formula for the entropy of a black hole:
S = A / 4Gℏwhere S is the entropy, A is the area of the event horizon, G is Newton's gravitational constant, and ℏ is Planck's constant divided by 2π. For a typical 3-solar-mass black hole, this is a rather large number, about 1078.
So, black holes have entropy, and there is even a simple formula for it. But entropy means disorder. That implies there is something to be disordered. What is it that a black hole's entropy is a disordering of? In a classical thermodynamic system, such as a volume of gas, entropy is a measure of disordering of the gas molecules, and can be calculated from the microscopic details of the system using quantum statistical mechanics. More precisely, entropy is defined to be proportional to the number of quantum microstates of the system. Now, Hawking's calculation did use quantum mechanics but made no assumptions about microstates of the black hole. Consequently, it was hard to understand the origin of black hole entropy.
In 1996, Andrew Strominger and Cumrun Vafa were able to come up with an answer to this puzzle. Basically, what they did was to show how to build up (theoretically) a certain type of black hole out of D-branes. Then, using standard techniques of string theory and supersymmetry they were able to count the quantum states of the system of D-branes. Out of this computation emerged Hawking's exact formula for the entropy.
At first, the computation was done for a rather special case. One of the special conditions was for the black hole to be in a "ground state" from which it cannot radiate. This required a black hole that represents a different solution of Einstein's equation than the Schwarzchild solution. The appropriate solution is called a "Reissner-Nordström" black hole, which (unlike the Schwarzchild case) has a certain amount of electric charge. For any given amount of charge the black hole must have more than a specific amount of mass. When the hole has the minimal amount of mass possible for the charge, it is called an "extremal" black hole. In that circumstance it is unable to radiate (since that would entail loss of mass), and hence the hole is in a ground state.
Since the presence of charge was necessary for this approach to work, another key ingredient was a demonstration that D-branes could carry charge. This piece of the puzzle was supplied by Joseph Polchinski in 1995.
Recall that BPS states in string theory also have the property of having a minimal amount of mass for a given amount of charge. The D-branes of which the black hole is constructed are BPS states. One of the convenient facts about BPS states is that many of their properties are preserved when theoretical parameters like coupling constants are changed. This makes it possible to do computations under the assumption of weak coupling (using perturbation theory) and still infer the corresponding results under strong coupling conditions -- which is what one has in an extremal black hole.
That was the strategy pursued by Strominger and Vafa. Interestingly enough, they were at first unable to do the computations for a 4-dimensional black hole (3 space, 1 time dimension), but they succeeded in deriving a forumla for the entropy in the 5-dimensional case. At that point, they had to go back and derive the entropy formula for a 5-dimensional black hole using Hawking's methods -- since no one had considered that problem before. The formulas derived by the two different methods matched perfectly. In other words, string calculations in fact made correct predictions in cases that had not previously been done using the old methods.
The calculations have since been done successfully in the 4-dimensional ("real world") case, and the resulting entropy formula is just what Hawking originally derived. Further computations have also been done using other assumptions, such as slightly nonextremal black holes. In all cases, the string theory computations agree with the Bekenstein-Hawking formula computed by "classical" means.
This is perhaps the best piece of evidence to date that string theory (and concepts like duality and D-branes, in particular) are on the right track. And by implication, M-theory (whatever it is), whose foundations are in those other concepts, is on firmer ground.
As far as black holes themselves are concerned, these developments may bring us a little closer to solving some of their remaining theoretical problems. Chief among those is the "black hole information problem".
Quantum theory and relativity -- quantum gravity.
The holographic principle.
Ideas of Susskind -- counting string states, holographic principle, matrix theory.
What M(atrix) theory (or more simply, the matrix model) is, rather, is a conjecture for what M-theory itself is. The model was first proposed by Tom Banks, Willy Fischer, Stephen Shenker, and Leonard Susskind in a 1996 paper. (The authors are sometimes referred to simply as "BFSS".) Susskind had been involved in string theory since the earliest days, made suggestions about counting string states which eventually led to the calculation of black hole entropy described above, and also contributed significantly to the holographic principle, to be discussed later. Another formulation of the matrix model was offered a little later by N. Ishibashi, H. Kawai, I. Kitazawa, and A. Tsuchiya (referred to as IKKT), in which connections with noncommutative geometry are more evident.
At this point, we actually need to be a little more precise about the term "M-theory". The term may be used in a broader and a narrower sense. In the broader sense, M-theory refers to the whole unknown theory which (presumably) connects each of the consistent string theories and 11-D supergravity. Each of these different theories is a particular limiting case of the general theory. More specifically, all the different relevant parameters (such as coupling constants and topological parameters) form what physicists call "moduli space". The consistent string theories and supergravity correspond to small regions of the moduli space -- quite limited values of the parameters. M-theory (in the broader sense) corresponds to the whole moduli space.
But sometimes "M-theory" is used in a narrower sense to refer only to that part of the whole theory whose low-energy limit is 11-D supergravity. This is also the part which is related to the Type IIA and Heterotic-E theories by S-duality (i. e., as their strong coupling limit). The matrix model, then, is a conjecture about what M-theory in this narrower sense might be. Of course, if the conjecture is correct, some means may eventually be found to extend to model to cover M-theory in the broader sense too.
The actual description of the matrix model is rather technical. For one thing, it involves a very exotic reference frame, known variously as the "infinite momemtum frame", the "light cone frame", or the "light front gauge". This basically means that the observer is moving at the speed of light.
In this terminology, the matrix model conjecture says that a simple supersymmetric Yang-Mills (SYM) theory of 0-branes is equivalent to 11-D supergravity in the infinite momentum reference frame. This is sometimes expressed by saying that the SYM theory is the matrix model.
0-branes are the fundamental ingredient. They are, of course, 0-dimensional, and so are, in some sense, like point particles. They behave like point particles at large distances, but rather differently at small distances. The matrix model gets its name because the spacetime coordinates of the 0-branes are N×N matrices instead of ordinary numbers. Thus it is the matrices which are actually the "degrees of freedom" of the theory.
To complete the model, it's necessary to take the limit as N → ∞. The reason for this is that the coupling constant is proportional to 1/N, so for large N we have a weak coupling situation. It is actually the theory in this limit which corresponds to supergravity in the light cone frame.
The value of the matrix model lies in the fact that it is known how to do computations with the theory, even though matrices replace ordinary numbers as coordinates. According to the conjecture, then, these computations tell us things about 11-D supergravity beyond the low energy limit -- and hence about M-theory -- which could not be computed perturbatively.
So these computed results are, in effect, predictions of the conjecture. They are necessary (but not sufficient) conditions for the conjecture to be valid. In a number of cases where the correct answers are known, in the low-energy limit, these predictions have been successfully verified. Here are some of the things which need to be checked:
To summarize, matrix theory is a quantum mechanical theory in which computations are possible. It can, if the conjecture is correct, be used to give a non-perturbative formulation of M-theory (in the narrow sense). In many cases that have been examined, involving a variety of string theories and compactifications (i. e., choices of topology of the compact dimensions of spacetime), matrix theory agrees with the string-theory perturbation expansions in different limiting situations.
In addition to further checking of special cases, matrix theory is involved in other research efforts. It appears to play a role in the "holographic principle" (discussed below). And it seems to offer a fertile field for the application of a powerful new branch of mathematics known as "noncommutative geometry".
Juan Maldacena's 1996 PhD thesis dealt with string theory and black holes. Some of the ideas from this work led to his conjectures. One thing they have in common is the importance of the boundary of a topological object for the physics of the entire object.
Typically, these conjectures relate a string theory (or M-theory) on some space (called the "background") to a quantum field theory. Since quantum field theories have intrinsic non-perturbative definitions, the conjectures provides such a definition for string theories on specific backgrounds. In the general conjecture a background space has the form K×AdS5, where K is a 5-dimensional manifold with positive curvature and AdS5 is the 5-dimensional manifold known as Anti-deSitter space. The most obvious choice of K is the 5-sphere, S5, so the simplest case of the conjecture states that Type IIB string theory on S5×AdS5 is equivalent to a quantum field theory called N=4 supersymmetric SU(N) gauge theory in (3+1) dimensions.
Let's look at the string theory part first. The space of interest is a 10-dimensional space that is the "direct product" of two 5-dimensional spaces. Recall that string theory works best in 10 dimensions. Half of the space is S5, the ordinary 5-sphere of radius R, which may be defined as a manifold embedded in 6-dimensional Euclidean space (R6) by means of the equation relating coordinates of any point on the sphere:
x12 + x22 + x32 + x42 + x52 + x62 = R2That is, every point on the sphere can be represented as a 6-tuple of coordinates (x1, x2, x3, x4, x5, x6) which must satisfy the equation above.
The other half, AdS5, is the anti-deSitter space of 5 dimensions. It is likewise just another 5-dimensional manifold sitting in R6, and it is defined very similarly to S5 by an equation relating the coordinates:
x12 + x22 - x32 - x42 - x52 - x62 = R2As with the sphere, this relation among coordinates has the effect of reducing the dimensionality of the object from 6 to 5. AdS5 is of interest because it has constant negative curvature (whereas a sphere has constant positive curvature).
The other theory dealt with by the conjecture is a version of gauge theory in (3+1) dimensions, with SU(N) as the gauge group. Recall that SU(N) is the special unitary groups of N×N matrices with complex entries. SU(N) enters here, since (as was also the case with the matrix model) the limit as N → ∞ is of interest. "(3+1)" means 3 space and 1 time dimensions, so this is an ordinary 4-dimensional theory. Supersymmetry can be added to such a theory in different ways. Here it is in the form "N=4", which means, technically, that there are 4 "Majorana fermions". (N and N are not the same.)
What the conjecture says is that the physics of these two theories is equivalent. The result is that computations using the field theory, which are reasonably well understood, can be used to determine properties of the string theory, which is not well understood. But the reasoning can go the other way too. It is possible to start with known results in string theory and obtain results in gauge theory. For instance, this can provide results in QCD about quark confinement and the existence of a "mass gap".
One interesting point about the correspondence in the example is that it relates a theory on a 10-dimensional space to one defined in four dimensions. This happens, as it turns out, because behavior of the string theory on the 4-dimensional boundary of AdS5 is crucial. The "holographic principle", to be discussed a little later, also deals with boundary behavior. An important application of the conjectured correspondence is to provide a proof of the holographic principle for the AdS case.
For M-theory, however, it appears that not only is the required mathematics as yet undeveloped, but in fact it's not even entirely clear what directions need to be taken. Nevertheless, there are some promising ideas. In this situation, the needs of physics are stimulating the development of new mathematics in a number of directions to an extent which perhaps hasn't occurred since Isaac Newton developed calculus.
The key insight is that associated naturally with any topological space X, such as a manifold, there is an algebraic structure (which is called rather unimaginatively an "algebra"). The most common example is the algebra C(X) consisting of "continuous" functions on X. (A "continuous" function is one whose value does not jump too abruptly between points that are very close to each other in X.) It turns out that all of the important information about the space X is contained in the algebra C(X), in the sense that any topological question about X can be answered by considering algebraic properties of C(X). In effect, X can be precisely reconstructed from a knowledge of C(X). One may, therefore, study X -- and usually more easily -- by studying C(X). This is a lot like the dualities of string theory which enable answering questions about one theory by working with another.
The next step is to dispense with the topological space X as a starting point, and simply consider the mathematical structures called algebras. It turns out that for certain types of algebras it is possible to construct another mathematical object that has many of the same characteristics as a topological space. When this is done for the algebra C(X) (which is commutative) of a space X, one gets the original space back as the result. But the really interesting thing is that this construction works for some types of noncommutative algebras. What falls out of the construction, then, is a noncommutative geometry.
This is more than just an idle exercise in abstraction, because many of the notions of quantum field theory and Yang-Mills gauge theory are developed in terms of topological spaces (manifolds). These geometric notions have specific analogs in noncommutative geometry. Many theorists hope that it may actually be easier to study these more general noncommutative analogs in order to better understand, eventually, the traditional commutative case. And, in addition, the noncommutative analogs may be precisely what is needed to formulate the mysterious M-theory.
This is more than just a wistful hope. Alain Connes, who is the primary inventor of noncommutative geometry, is also keenly interested in applications to string theory, M-theory, and the like. He has worked with string theorists such as John Schwarz to apply noncommutative geometry to the matrix model of M-theory.
More about noncommutative geometry
More precisely, we recall that the compact dimensions assume the form of a 6-dimensional Calabi-Yau space, a particular sort of manifold. Unfortunately, there are a large number of inequivalent possibilities for such a space -- on the order of ten thousand at least. However, the exact topological form of this space, in each instance, determines to a large extent the physics we actually observe, so for consistency with observed physics, there are substantially fewer possibilities. For instance, the number of particle family generations that exist (which is three, as far as we can tell), must be one half of the absolute value of the Euler number of the Calabi-Yau space. The Euler number, in turn, is a topological invariant which is depends on the number of holes of various dimensions which exist in the space. Since we know that the number of generations is three, the Euler number must be ±6. This eliminates most spaces from consideration.
Still, there remain a large number of Calabi-Yau spaces with Euler number ±6. Around 1988 several physicists (including Brian Greene, Ronen Plesser, and others) who were studying these spaces looked into a technique (called "orbifolding") for transforming one space into another. This process has the property that odd-dimensional holes in the space become even-dimensional, and vice versa. Since (as it happens), the Euler number doesn't change under this sort of exchange, the number of particles generations doesn't either.
The investigators wondered whether other physical quantities, such as particle charges, would also be preserved. What they eventually found was as good as could have possibly been hoped for -- all of the physics could be preserved under the right sort of transformations which exchanged holes of even and odd dimensions. And this is true even though the spaces involved are not topologically equivalent. Pairs of spaces which had this relationship were said to be "mirror symmetric", and the relationship was called "mirror symmetry". (It has nothing to do with the symmetry of reflection in a mirror. The name refers to a symmetry of abstract mathematical diagrams.)
Mirror symmetry is, then, a kind of duality in which two apparently different theories result in the same physics. The importance of this, as usual, is that when computations in one theory become difficult, a translation can be made into the other theory where (one hopes) the computation is easier. This trick often succeeds, and it has helped illuminate many properties of Calabi-Yau manifolds and the resulting string theory.
We mention this phenomenon here, because it is an example of how string theory has contributed directly to advances in mathematics as well as physics. What mirror symmetry is saying is that pairs of Calabi-Yau spaces which are quite different mathematically nevertheless are related by virtue of giving rise to the same string theory. It's easy to understand how this can be useful. A very common trick in mathematical reasoning is to calculate the same quantity in two different ways. Since it is known in advance that the answer must be the same, the results of the two different calculations can be equated, thus giving an identity which might otherwise not be at all obvious.
As is often the case, the type of reasoning used by physicists is often not sufficiently formalized and rigorous to satisfy mathematicians. But in this case, is was eventually possible to put all the steps involved on a mathematically rigorous foundation. The pay-off is that mirror symmetry has been of great use to mathematicians in solving long-standing problems, such as problems of enumerative geometry within the general field of algebraic geometry. A concrete example of such a problem is that of producing a formula for the number of spheres that can be packed in a particular Calabi-Yau space.
And yet, there is so much theoretical evidence that something of significance must lie behind the astonishing coincidence that a complex network of dualities connects the five consistent string theories and supergravity. It's a situation with little precedent in the history of physics to have so much reinforcing evidence in the form of consistencies for a body of theory which is, as yet, so inaccessible to experimental testing.
We may expect to see much more of this sort of circumstance in related areas of physics, such as cosmology, where our ability to model "extreme" conditions has outstripped our ability to investigate them experimentally. Some people are concerned that this is treading outside the bounds of "proper" science. Others are less timorous about pressing the envelope. After all, there have been cases in the past when theories have been developed largely by mathematical deduction from postulated principles, and still have proven to yield remarkably accurate predictions about the real world. General relativity is a primary example. In that case, the first verifications of its predictions were quite soon in coming. (Yet other predictions are still awaiting experimental confirmation -- gravitational waves, for example.)
It could be that experimental tests of superstring and/or M-theory will remain out of reach for a rather longer period of time. It would be a surprising coincidence if our technology for the appropriate experiments were so well matched to our theoretical achievements as was the case with general relativity. At all events, the future is a long time, and patience is a virtue.
In the interim, theoretical activity in this area -- "work in progress" -- is intense. It is quite possible -- even quite likely -- that out of this will emerge predictions which are within our experimental or observational capabilities. The computation of the Beckenstein-Hawking entropy of a black hole is one significant early example.
Here is a quick survey of other work in progress.
String theory and its extension, M-theory, have made two very significant contributions to a resolution of the problem:
Copyright © 2002 by Charles Daney, All Rights Reserved