At a conference in 1995, physicist Edward Witten proposed a bold resolution to the problem of five distinct string theories. In his theory, based on newly discovered dualities, each of the existing theories was a special case of one overarching string theory, which he enigmatically called M-theory. One of the key concepts required for M-theory was the introduction of branes (short for membranes) into string theory. Branes are fundamental objects in string theory that have more than one dimension.
Witten didn’t thoroughly explain the true meaning of the name M-theory, leaving it as something that each person can define for himself. There are several possibilities for what the “M” could stand for: membrane, magic, mother, mystery, or matrix. Witten probably took the “M” from membrane because those featured so prominently in the theory, but he didn’t want to commit himself to requiring them so early in the development of the new theory.
Although Witten didn’t propose a complete version of M-theory (in fact, we’re still waiting on one), he did outline certain defining traits that M-theory would have:
‘ 11 dimensions (10 space dimensions plus 1 time dimension)
‘ Dualities that result in the five existing string theories all being different explanations of the same physical reality
‘ Branes — like strings, but with more than one dimension
Translating one string theory into another: Duality
The core of M-theory is the idea that each of the five string theories introduced in topic 10 is actually a variation on one theory. This new theory — M-theory — is an 11-dimensional theory that allows for each of the existing theories (which are 10-dimensional) to be equivalent if you make certain assumptions about the geometry of the space involved.
The basis for this suggestion was the understanding of dualities that were being recognized among the various string theories. A duality occurs when you can look at the same phenomenon in two distinct ways, taking one theory and mapping it to another theory. In a sense, the two theories are equivalent. By the mid-1990s, growing evidence showed that at least two dualities existed between the various string theories; they were called T-duality and S-duality.
These dualities were based on earlier dualities conjectured in 1977 by Claus Montonen and David Olive. In the early 1990s, Indian physicist Ashoke Sen and Israeli-born physicist Nathan Seiberg did work that expanded on the notions of these dualities. Witten drew upon this work, as well as more recent work by Chris Hull, Paul Townsend, and Witten himself, to present M-theory.
Topology: The mathematics of folding space
The study of topology allows you to study mathematical spaces by eliminating all details from the space except for certain sets of properties that you care about. Two spaces are topologically equivalent if they share these properties, even if they differ in other details. Certain actions may be more easily performed on one of the spaces than the other. You then perform actions on that space and can work backward to find the resulting effect on the topologically equivalent space. It can be far easier than trying to perform these actions on the original space directly.
One of the key components of topology is the study of how different topological spaces relate to each other. Much of the time, these
different spaces involve some sort of manipulation of the space, which is what adds the complexity. If this manipulation can be performed without breaking or reconnecting the space in a new way, the two spaces are topologically equivalent.
To picture this, imagine a donut (or torus) of clay that you slowly and meticulously recraft into the shape of a coffee mug. The hole in the center of the donut never has to be broken in order to be turned into the handle of the coffee mug. On the other hand, if you start with a donut, there’s no way to turn it into a pretzel without introducing breaks into the space — a donut and a pretzel are topologically distinct.
Topological duality: T-duality
One of the dualities discovered at the time was called T-duality, which refers to either topological duality or toroidal duality, depending on whom you ask. (Toroidal is a reference to the simplest case, which is a torus, or donut shape. Topological is a precise way of defining the structure of that space, as explained in the nearby sidebar “Topology: The mathematics of folding space.” In some cases the T-duality has nothing to do with a torus, and in other cases, it’s not topological.) The T-duality related the Type II string theories to each other and the heterotic string theories to each other, indicating that they were different manifestations of the same fundamental theory.
In the T-duality, you have a dimension that is compactified into a circle (of radius R), so the space becomes something like a cylinder. It’s possible for a closed string to wind around the cylinder, like thread on a spindle. (This means that both the dimension and the string have radius R.) The number of times the closed string winds around the cylinder is called the winding number. You have a second number that represents the momentum of the closed string.
Here’s where things get interesting. For certain types of string theory, if you wrap one string around a cylindrical space of radius R and the other around a cylindrical space of radius 1/R, then the winding number of one theory seems to match the momentum number (momentum, like about everything else, is quantized) of the other theory.
In other words, T-duality can relate a string theory with a large compactified radius to a different string theory with a small compactified radius (or, alternately, wide cylinders with narrow cylinders). Specifically, for closed strings, T-duality relates the following types of string theories:
Type IIA and Type IIB superstring theories ‘ Type HO and Type HE superstring theories
The case for open strings is a bit less clear. When a dimension of superstring space-time is compactified into a circle, an open string doesn’t wind around that dimension, so its winding number is 0. This means that it corresponds to a string with momentum 0 — a stationary string — in the dual superstring theory.
The end result of T-duality is an implication that Type IIA and IIB superstring theories are really two manifestations of the same theory, and Type HO and HE superstring theories are really two manifestations of the same theory.
Strong-Weak duality: S-duality
Another duality that was known in 1995 is called S-duality, which stands for strong-weak duality. The duality is connected to the concept of a coupling constant, which is the value that tells the interaction strength of the string by describing how probable it is that the string will break apart or join with other strings.
The coupling constant, g, in string theory describes the interaction strength due to a quantity known as the dilation field, (|. If you had a high positive dilation field (|, the coupling constant g = e( becomes very large (or the theory becomes strongly coupled). If you instead had a dilation field the coupling constant g = e~( becomes very small (or the theory becomes weakly coupled).
Because of the mathematical methods (see nearby sidebar “Perturbation theory: String theory’s method of approximation”) that string theorists have to use to approximate the solutions to string theory problems, it was very hard to determine what would happen to string theories that were strongly coupled.
In S-duality, a strong coupling in one theory relates to a weak coupling in another theory, in certain conditions. In one theory, the strings break apart and join other strings easily, while in the other theory they hardly ever do so. In the theory where the strings break and join easily, you end up with a chaotic sea of strings constantly interacting.
Trying to follow the behavior of individual strings is similar to trying to follow the behavior of individual water molecules in the ocean — you just can’t do it. So what do you do instead? You look at the big picture. Instead of looking at the smallest particles, you average them out and look at the unbroken surface of the ocean, which, in this analogy, is the same as looking at the strong strings that virtually never break.
S-duality introduces Type I string theory to the set of dual theories that T-duality started. Specifically, it shows that the following dualities are related to each other:
Type I and Type HO superstring theories Type IIB is S-dual to itself
If you have a Type I superstring theory with a very strong coupling constant, it’s theoretically identical to a Type HO superstring theory with a very weak coupling constant. So these two types of theories, under these conditions, yield the exact same predictions for masses and charges.
Perturbation theory: String theory’s method of approximation
The equations of string theory are incredibly complex, so they often can only be solved through a mathematical method of approximation called perturbation theory. This method is used in quantum mechanics and quantum field theory all the time and is a well-established mathematical process.
In this method, physicists arrive at a first-order approximation, which is then expanded with other terms that refine the approximation. The goal is that the subsequent terms will become so small so quickly that they’ll cease to matter. Adding even an infinite number of terms will result in converging onto a given value. In mathematical speak, converging means that you keep getting closer to the number without ever passing it.
Consider the following example of convergence: If you add a series of fractions, starting with 14 and doubling the denominator each time, and you added them all together (V2 + % % + . . . well, you get the idea), you’ll always get closer to a value of 1, but you’ll never quite reach 1. The reason for this is that the numbers in the series get small very quickly and stay so small that you’re always just a little bit short of reaching 1.
However, if you add numbers that double (2 + 4 + 8 + . . . well, you get the idea), the series doesn’t converge at all. The solution keeps getting bigger as you add more terms. In this situation, the solution is said to diverge or become infinite.
The dual resonance model that Veneziano originally proposed — and which sparked all of string theory — was found to be only a first-order approximation of what later came to be known as string theory. Work over the last 40 years has largely been focused on trying to find situations in which the theory built around this original first-order approximation can be absolutely proved to be finite (or convergent), and which also matches the physical details observed in our own universe.
Using two dualities to unite five superstring theories
Both T-duality and S-duality relate different string theories together. Here’s a review of the existing string theory relationships:
Type I and Type HO superstring theories are related by S-duality. Type HO and Type HE superstring theories are related by T-duality. Type IIA and Type IIB superstring theories are related by T-duality.
With these dualities (and other, more subtle ones, which relate IIA and IIB together with the heterotic string theories), relationships exist to transform one version of string theory into another one — at least for certain specially selected string theory conditions.
To solve these equations of duality, certain assumptions have to be made, and not all of them are necessarily valid in a string theory that would describe our own universe. For example, the theories can only be proved in cases of perfect supersymmetry, while our own universe exhibits (at best) broken supersymmetry.
String theory skeptics aren’t convinced that these dualities in some specific states of the theories relate to a more fundamental duality of the theories at all levels. Physicist (and string theory skeptic) Lee Smolin calls this the pessimistic view, while calling the string theory belief in the fundamental nature of these dualities the optimistic view.
Still, in 1995 it was hard not to be in the optimistic camp (and, in fact, many had never stopped being optimistic about string theory). The very fact that these dualities existed at all was startling to string theorists. It wasn’t planned, but came out of the mathematical analysis of the theory. This was seen as powerful evidence that string theory was on the right track. Instead of falling apart into a bunch of different theories, superstring theory was actually pulling back together into one single theory — Edward Witten’s M-theory — which manifested itself in a variety of ways.
The second superstring revolution begins: Connecting to the 11-dimensional theory
The period immediately following the proposal of M-theory has been called the “second superstring revolution,” because it once again inspired a flurry of research into superstring theory. The research this time focused on understanding the connections between the existing superstring theories and between the 11-dimensional theory that Witten had proposed.
Witten wasn’t the first one to propose this sort of a connection. The idea of uniting the different string theories into one by adding an 11th dimension had been proposed by Mike Duff of Texas A&M University, but it never caught on among string theorists. Witten’s work on the subject, however, resulted in a picture where the extra dimension could emerge from the unifications inherent in M-Theory — one that prompted the string theory community to look at it more seriously.
In 1994, Witten and colleague Paul Townsend had discovered a duality between the 10-dimensional superstring theory and an 11-dimensional theory, which had been proposed back in the 1970s: supergravity.
Supergravity resulted when you took the equations of general relativity and applied supersymmetry to them. In other words, you introduced a particle called the gravitino — the superpartner to the graviton — to the theory. In the 1970s this was pretty much the dominant approach to trying to get a theory of quantum gravity.
What Witten and Townsend did in 1994 was take the 11-dimensional super-gravity theory from the 1970s and curl up one of the dimensions. They then showed that a membrane in 11 dimensions that has one dimension curled up behaves like a string in 10 dimensions.
Again, this is a recurrence of the old Kaluza-Klein idea, which comes up again and again in the history of string theory. By taking Kaluza’s idea of adding an extra dimension (and Klein’s idea of rolling it up very small), Witten showed that it was possible — assuming certain symmetry conditions — to show that dualities existed between the existing string theories.
There were still issues with an 11-dimensional universe. Physicists had shown supergravity didn’t work because it allowed infinities. In fact, every theory except string theory allowed infinities. Witten, however, wasn’t concerned about this because supergravity was only an approximation of M-theory, and M-theory would, by necessity, have to be finite.
It’s important to realize that neither Witten nor anyone else proved that all five string theories could be transformed into each other in our universe. In fact, Witten didn’t even propose what M-theory actually was.
What Witten did in 1995 was provide a theoretical argument to support the idea that there could be a theory — which he called M-theory — that united the existing string theories. Each known string theory was just an approximation of this hypothetical M-theory, which was not yet known. At low energy levels, he also believed that M-theory was approximated by the 11-dimensional supergravity theory.
