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How arestring theories related?

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 the two heterotic 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. These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation.
    These dualities link quantities that were also thought to be separate. Large and small distance scales, strong and weak coupling strengths -- these quantities have always marked very distinct limits of behavior of a physical system, in both classical field theory and quantum particle physics. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related.

Large and small distance

    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.
    Suppose we're in ten spacetime dimensions, which means we have nine space and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L=2pR takes you around the circle and brings you back to where you started.
    A particle traveling around this circle will have a quantized momentum around the circle, and this will contribute to the total energy of the particle. But a string is very different, because in addition to traveling around the circle, the string can wrap around the circle. The number of times the string winds around the circle is called the winding number, and that is also quantized.
    Now the weird thing about string theory is that these momentum modes and the winding modes can be interchanged, as long as we also interchange the radius R of the circle with the quantity Lst2/R, where Lst is the string length.
    If R is very much smaller than the string length, then the quantity Lst2/R is going to be very large. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.
    This type of duality is called T-duality. T-duality relates Type IIA superstring theory to Type IIB superstring theory. That means if we take Type IIA and Type IIB theory and compactify them both on a circle, then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other! The same is also true for the two heterotic theories.
    So T-duality obscures the difference between large and small distances. What looks like a very large distance to a momentum mode of a string looks, looks to a winding mode of a string like a very small distance. This is very counter to how physics has always worked since the days of Kepler and Newton.

Strong and weak coupling

    What is a coupling constant? This is some number that tells us how strong an interaction is. Newton's constant is the coupling constant for the gravitational force, for example. If Newton's constant were twice the size it is measured to be now, then we would feel twice as much gravitational force from the Earth, and the Earth would feel twice as much from the Moon and the Sun, and so on. A larger coupling constant means a stronger force, and a smaller coupling constant means a weaker force.
    Every force has a coupling constant. For electromagnetism, the coupling constant is proportional to the square of the electric charge. When physicists study the quantum behavior of electromagnetism, they can't solve the whole theory exactly, so they break it down to little pieces, and each little piece that they can solve has a different power of the coupling constant in front of it. At normal energies in electromagnetism, the coupling constant is small, and so the first few little pieces make a good approximation to the real answer. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.
    This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the oscillation modes of the string, called the dilaton. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one.
    This symmetry 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. Notice that the theory with strong coupling cannot be understood by means of expanding in a series, but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory. For a physicist, that is the proverbial two-for-one deal!
    Superstring theories related by S-duality are: Type I superstring theory with heterotic SO(32) superstring theory, and Type IIB theory with itself.

What does it mean?

    T-duality is something unique to string physics. It's something particles cannot do, because a particle cannot get wrapped around a circle like a string. If string theory is a correct theory of Nature, then this implies that one 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.
    The same goes for S-duality, which teaches us that the strong coupling limit of one string theory can describe the weak coupling limit of a different string theory.
    This sounds like it goes against all traditional physics, but this is indeed a reasonable outcome for a quantum theory of gravity, because Einstein's theory of gravity tells us that gravity is about how the sizes of objects and magnitudes of interactions are measured in curved spacetime.

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