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A new picture of string theory At one time, string theorists believed there were five distinct superstring theories: type I, types IIA and IIB, and heterotic SO(32) and E8XE8 string theories. The thinking was that out of these five candidate theories, only one was the actual correct Theory of Everything, and that theory was the theory whose low energy limit, with ten dimensions spacetime compactified down to four, matched the physics observed in our world today. The other theories would be nothing more than rejected string theories, mathematical constructs not blessed by Nature with existence. But now it is known that this naive picture was wrong, and that the the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory, of which there is only one. In the mid-nineties it was learned that superstring theories are related by duality transformations known as T duality and S duality. These dualities link the quantities of large and small distance, and strong and weak coupling, limits that have always been identified as distinct limits of a physical system in both classical and quantum physics. These duality relationships between string theories have sparked a radical shift in our understanding of string theory, and have led to the reasonable expectation that all five superstring theories -- type I, types IIA and IIB, and heterotic SO(32) and E8XE8 -- are special limits of a more fundamental theory. T duality The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality, and comes about from the compactification of extra space dimensions in a ten dimensional superstring theory. Let's take the X9 direction in flat ten-dimensional spacetime, and compactify it into a circle of radius R, so that A particle traveling around this circle will have its momentum quantized in integer multiples of 1/R, and a particle in the nth quantized momentum state will contribute to the total mass squared of the particle as A string can travel around the circle, too, and the contribution to the string mass squared is the same as above. But a closed string can also wrap around the circle, something a particle cannot do. The number of times the string winds around the circle is called the winding number, denoted as w below, and w is also quantized in integer units. Tension is energy per unit length, and the the wrapped string has energy from being stretched around the circular dimension. The winding contribution Ew to the string energy is equal to the string tension Tstring times the total length of the wrapped string, which is the circumference of the circle multiplied by the number of times w that the string sometimes is wrapped around the circle. where tells us the length scale Ls of string theory. The total mass squared for each mode of the closed string is The integers N and Ñ are the number of oscillation modes excited on a closed string in the right-moving and left-moving directions around the string. The above formula is invariant under the exchange In other words, we can exchange compactification radius R with radius a'/R if we exchange the winding modes with the quantized momentum modes. This mode exchange is the basis of the duality known as T-duality. Notice that if the compactification radius R is much smaller than the string scale Ls, then the compactification radius after the winding and momentum modes are exchanged is much larger than the string scale Ls. So T-duality obscures the difference between compactified dimensions that are much bigger than the string scale, and those that are much smaller than the string scale. T-duality relates type IIA superstring theory to type IIB superstring theory, and it relates heterotic SO(32) superstring theory to heterotic E8XE8 superstring theory. Notice that a duality relationship between IIA and IIB theory is very unexpected, because type IIA theory has massless fermions of both chiralities, making it a non-chiral theory, whereas type IIB theory is a chiral theory and has massless fermions with only a single chirality. T-duality is something unique to string physics. It's something point particles cannot do, because they don't have winding modes. If string theory is a correct theory of Nature, then this implies that on some deep level, the separation between large vs. small distance scales in physics is not a fixed separation but a fluid one, dependent upon the type of probe we use to measure distance, and how we count the states of the probe. This sounds like it goes against all traditional physics, but this is indeed a reasonable outcome for a quantum theory of gravity, because gravity comes from the metric tensor field that tells us the distances between events in spacetime. What is a coupling constant? This is some number that tells us how strong an interaction is. Newton's constant GN, which appears in both Newton's law of gravity and the Einstein equation, is the coupling constant for gravitational interactions. For electromagnetism, the coupling constant is related to the electric charge through the fine structure constant In both particle physics and string theory, usually the scattering amplitudes and other quantities have to be computed as an expansion in powers of the coupling constant or loop expansion parameter, which we've called g2 below: At low energies in electromagnetism, the pretend dimensionless coupling constant a is very small compared to unity, and the higher powers in a become too small to matter. The first few terms in the series make a good approximation to the real answer, which often can't be calculated at all because the mathematical technology doesn't exist to solve the whole theory at once. If the coupling constant gets very large compared to unity, perturbation theory becomes useless, because higher powers of the expansion parameter are bigger, not smaller, than lower powers. This is called a strongly coupled theory. Coupling constants in quantum field theory end up depending on energy because of quantum vacuum effects. A quantum field theory can be weakly coupled at low energies and strongly coupled at high energies, as is true with the fine structure constant a in QED, or strongly coupled at low energies and weakly coupled at high energies, as is true with the coupling constant for quark and gluon interactions in QCD. Some quantities in a theory cannot be caluclated at all using perturbation theory, espcially not for weak coupling. For example, the amplitude below cannot be expanded around the value g2=0 because the amplitude is singular there. This is typical of a tunneling transition, which is forbidden by energy conservation in classical physics and hence has no expansion around a classical limit. String theories feature two kinds of perturbative expansions: an expansion in powers of the string parameter a' in the conformal field theory on the two-dimensional string worldsheet, and a quantum loop expansion for string scattering amplitudes in d-dimensional spacetime. But unlike in particle theories, the string quantum loop expansion parameter is not just a number, but depends on one of the dynamic modes of the string, called the dilaton field f(x) This relationship between the dilaton and the string loop expansion parameter is important in understanding the duality relation known as S-duality. S-duality can be examined most easily in type IIB string theory, because this theory happens to be S-dual to itself. The low energy limit of type IIB theory (meaning the lowest nontrivial order in the string parameter a') is a type IIB supergavity field theory, which features a complex scalar field r(x) whose real part is the axion field c(x) and whose imaginary part is the exponential of the dilaton field f(x): This field theory is invariant under a global transformation by the group SL(2,R) (broken by quantum effects down to SL(2,Z)), with the field r(x) transforming as If there is no contribution from the axion field, then the expectation value of the field r(x) is given by the dilaton alone. Because the dilaton is identified with gst, the SL(2,Z) transformation with b=-1,c=1 tells us that the theory at coupling gst is the same as the theory at coupling 1/gst! This transformation is called S-duality. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. Type IIB superstring theory is S-dual to itself, so the strong and weak coupling limits are the same. Therefore this duality allows an understanding of the strong coupling limit of the theory that would not be possible by any other means. Something more surprising is that type I superstring theory is S-dual to heterotic SO(32) superstring theory. This is surprising because type I theories contain open and closed strings, where as heterotic theories contain only open strings. What's the explanation? At very strong coupling, heterotic SO(32) string theory has excitations that are open strings, but these open strings are highly unstable in the weakly coupled limit of the theory, which is the limit in which heterotic string theory is commonly understood.