As has been stated before, there are five types of string theory that are very similar, but different in certain fundamental aspects. They are Type I theory, Type IIA theory, Type IIB theory, Heterotic type O(32) theory (read "oh-thirty-two"; Heterotic-O for short), and Heterotic type E8 × E8 theory (read "e-eight times e-eight"; Heterotic-E for short). The differences between them are subtle but distinct. In Type IIA, vibrations traveling clockwise around a string are identical to those traveling counterclockwise; the result of this is that particles can spin in both directions (have both chiralities). In Type IIB, clockwise and counterclockwise vibrations are exactly opposite, resulting in its particles spinning in only one direction (having one chirality). In the two heterotic theories, clockwise vibrations resemble those of a Type II string (clockwise vibrations in IIA and IIB are the same), but counterclockwise vibrations are those of the original bosonic theory. Interestingly, the bosonic string requires 26 dimensions, but the Type II string requires 10, resulting in an odd duality in which clockwise vibrations occur in 10 dimensions and counterclockwise vibrations occur in 26. The extra 16 dimensions must be contained in one of two torus-like shapes; these give rise to the two separate heterotic theories. The Type I theory is similar to Type IIB except that it includes open strings, or strings whose ends are loose and unconnected, as well as the usual closed strings.
Before the second superstring revolution, the five string theories seemed to be completely distinct and unrelated except in their formulation. There seemed to be a vast rift between all of them, separating them from each other and forcing physicists to accept that their candidate theory of everything apparently failed the inevitability test. However, the second superstring revolution showed that, in a complex series of dualities to be explained later in this page, the five theories are actually different parts of one unifying theory called M-theory. Essentially, all the work is still being done on the exact form of M-theory, but it has two fundamental features already known. First, as mentioned earlier, the ten dimensions of the basic five string theories arises from the use of approximate equations. Just as Kaluza found an exciting union of relativity and electromagnetism in the addition of a dimension, modern theorists find a union of the five distinct theories in the addition of another dimension. In reality, the theory's exact equations yield eleven dimensions. Second, also mentioned earlier in the series, M-theory is not limited to one-dimensional vibrating strings; two-dimensional membranes, three-dimensional nuggets or blobs called three-branes, and so on up the dimensional ladder are also included.
Perturbation theory is in essence an attempt to give a rough, ballpark answer, then gradually refine it by adding more and more precise details to the calculation. When applied within certain limits, it is a powerful and useful tool whose original estimate is reasonably close to the final answer, but is successively refined by addition of minor details. However, when the premise of the initial approximation was flawed, the final answer can differ significantly from the original estimate. This is called the failure of perturbation theory. The incompleteness of our understanding of string theory derives mainly from the previous use of this theory outside its range of applicability. Much of contemporary research is focused on the discovery and use of nonperturbative techniques for calculations in string theory.
Calculations of the Earth's motion through the solar system provide an excellent if often-used example of perturbation theory. Since everything exerts an gravitational force on everything else, we are quickly led to an impossibly complex calculation involving all bodies in the solar system and, in theory, all cosmological objects in the universe. However, the influences of most of these factors are immeasurably slight, so they can safely be ignored. The sun's mass and location near the Earth make it the dominant factor in this calculation. Therefore, performing this calculation with only the sun as a factor will give a ballpark estimate; adding the gravitational influences of the next-closest bodies, such as planets passing close to the Earth at the time of calculation, will refine the result still further. This approach works because there is a dominant body in the calculation; the following is an example of failure of perturbation theory. If we wanted to calculate the gravitational relationship between the three stars of a tertiary system, there is no dominant influence. Therefore, if one attempted to use a perturbative approach by first modeling the interactions of two stars, then refining this estimate with the addition of the third, one would find that the estimate differed significantly from the actual figure. This large refinement is also not the final, actual figure; the third star's motion has an unaccounted-for effect on the motion of the other two, which have a similar effect on the motion of the third, etc.
Within the theoretical framework of string theory, particle interactions are based on the splitting and joining of strings, as in the diagram at right. Within string theory is a precise mathematical formula determining the influence of each string in an interaction on the motion of the others. (This formula actually differs slightly in each individual string theory, but the differences are not important here.) However, this formula is not the absolute determinant of string interactions: by the uncertainty principle, particle/antiparticle pairs (or in stringy terms, two strings in opposite vibrational patterns) can temporarily borrow energy and leap into existence for a short period of time before annihilating themselves. These pairs, called virtual string pairs, must quickly recombine into a single loop of string. This is called a one-loop process and is illustrated at right. Obviously, this process can continue indefinitely, extending with two, three, sixteen, or any infinite number of loops. Like there was for the original interaction, there is a formula to describe each of these increasingly complex interactions. String theorists could exactly understand the interactions between strings by adding together the expressions for each of these interactions, but this implies an infinite number of expressions since there is an infinite number of interactions, each with more loops than the preceding one. This is clearly impossible, but string theorists thus far have coped by using the perturbative methods discussed above, first calculating the value for the simple interaction, then refining it by adding successive numbers of loops until desired accuracy is obtained.
A coupling constant is a number in string theory that determines the probability of one string splitting into two, thus forming a virtual pair. The mathematics show that coupling constants larger than 1 yield an increasing likelihood of virtual string pairs; constants less than 1 yield an increasing unliklihood of virtual string pairs. (To be more specific, every virtual string pair contributes a factor to be multiplied into the constant; if it is less than 1, the multiplications will yield increasingly small contributions; if it is greater than 1, the multiplications yield equal or greater contributions.) If the constant is less than 1, most or all of the loops can be safely ignored and the perturbative approach works perfectly. However, if the constant is greater than 1, the estimate given by perturbation theory is widely off the mark because the contributions of increasing numbers of loops are so great.
At present, no one knows the coupling constants for any of the five string theories. We do know that all conclusions thus far that are based on perturbation are not valid if the constant is greater than 1. String theory includes an equation to find the constant, but it is currently only an approximate version solvable only to yield the answer that the constant times zero is zero (thus implying that it could be any number according to the approximate equations). However, these equations themselves were formed based on perturbation, which assumes the answer will be less than 1. Further non-perturbative work will likely reveal the constants with greater accuracy.
As revealed by Edward Witten, building on the previous work of many others, the five known string theories appear completely distinct but in actuality are dual to one another. When the theories are weakly coupled - that is, when they have coupling constants less that 1 - they appear distinct. However, when analyzed using new nonperturbative methods to be discussed later, with the coupling constant greater than 1, they prove quite interrelated. Specifically, the strongly coupled behavior of one theory has an alternate description in the weakly coupled behavior of another. We will return to this topic shortly after a discussion of nonperturbative methods.
BPS states, named after three physicists recognized for their work in the area - E. Bogomoln'yi, Manoj Prasad, and Charles Sommerfeld - are methods invoking supersymmetry properties to determine certain aspects of physics without using a perturbative approach. To give an overused example, suppose you are told that a closed box is hiding some unknown object with, say, a total electric charge of +2. This could be two protons, three protons and an electron, six up-quarks, any of these accompanied by any number of neutrons, or any other combination of particles with electric charges totaling +2. However, you are also told that the contents are supersymmetric and have the minimum mass allowed given the +2 charge. Thus, the contents are two protons. Based on the conclusions of the physicists mentioned above, a structure such as supersymmetry and a "minimality constant" (Brian Greene), or the stipulation that the contents be the minimum mass possible, can allow the contents of any such theoretical "box" to be analyzed quickly and exactly. The minimum-mass components consistent with the charge value given are known specifically as BPS states.
If you begin a thought experiment involving a Type I string with a coupling constant much less than 1, perturbation theory is valid and the properties of the theory can be accurately described. If the constant's value is increased but still kept below 1, the accuracy of calculations will decrease, but they will still provide good approximations of reality. As the constant's value increases beyond 1, perturbation theory breaks down and BPS states must be used. Based on analysis with BPS, the behavior of the Type I strongly coupled string is exactly the same as the weakly coupled behavior of the Heterotic-O string. Studies suggest that the reverse is also true: the strongly coupled Heterotic-O string is equivalent to the weakly coupled Type I string. This property is known as strong-weak duality. Type IIB has another remarkable property in the same vein: it is self-dual. This means that the IIB coupling constant and its reciprocal describe the same physics.
Before the major appearance of strings, the most successful theories were those that incorporated supersymmetry and gravity into quantum field theory using ten or eleven dimensions. These theories were called supergravity theories. In the end, four ten-dimensional supergravity theories were developed - three of them turn out to be low-energy, point-particle approximations of the Type IIA, IIB, and Heterotic-E strings. The fourth gives an analog to both the Type I and Heterotic-O strings. However, eleven-dimensional supergravity was also found to be a major player in this web of dualities. It is actually a low-energy, point-particle approximation to the strongly coupled Type IIA string.
How can eleven-dimensional supergravity correspond to ten-dimensional string theory? Researchers found that strongly coupled strings, particularly Type IIA and Heterotic-E strings, actually have two dimensions. This extra dimension is a tenth spatial dimension, yielding a total of eleven. In Type IIA, the string becomes less like a string and more like the two-dimensional surface of a bicycle tire or inner tube. In Heterotic-E, the string becomes a membrane (or two-brane) whose size is governed by the coupling constant.
There is an additional, remarkable set of dualities between the string theories. Recall the earlier discussion of wound strings encircling dimensions and the conclusion reached: the dimension of radius R is equivalent to that of radius 1/R. The specific statement of this radius duality is this: the physics of a Type IIA string in a universe with a dimension of radius R is the same as the physics of a Type IIB string in a universe with a dimension of radius 1/R. The same applies to the Heterotic-O and Heterotic-E strings.
The Type IIA and Heterotic-E strings are in reality two-dimensional figures existing in a universe of eleven dimensions. However, the theory of this eleven-dimensional universe is virtually unknown to theorists. At low energies, it is approximated by eleven-dimensional supergravity, a point-particle theory. This theory, the strongest string candidate yet for the theory of everything, has been dubbed "M-theory" by Edward Witten; however, there is no consensus on what the mysterious name means (in fact, Mystery Theory was a suggestion). Despite our lack of knowledge of its detailed properties, we do know that it unifies the five string theories in one all-encompassing framework through a web of dualities. This implies that a problem too complicated to solve in one theory may be rephrased in another theory, possibly with a much greater likelihood of finding a successful solution. For example, strongly coupled strings are difficult to work with, but weakly coupled strings are much easier. Therefore, the weakly coupled Type I string might replace the strongly coupled Heterotic-O string in a physicist's calculation, but they both describe the same underlying physical properties.
M-theory, and therefore the other theories as well, contain more than just one-dimensional strings. As mentioned before, the theories include everything from one-branes to nine-branes. More generally, an object with p spatial dimensions is called a p-brane. Interestingly, physicists have proven that the mass of all such objects except strings is inversely proportional to the string constant of the theory within which a calculation is made. With weak coupling, all but strings will be much heavier than even the Planck mass. Since a brane therefore requires a great deal of energy to be created, branes are unlikely to have much of an effect on physics. However, with strong coupling, the masses of these extended objects get lighter and thus have a greater effect.