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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 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

Compactication on a circle

    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

Energy of compact momentum state

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 is wrapped around the circle.

Energy of winding state

where

The string length

tells us the length scale Ls of string theory.
    The total mass squared for each mode of the closed string is

String mode mass with winding

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

T duality

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.

Next: S-Duality >>

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