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 E_{8}XE_{8}
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 midnineties 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 E_{8}XE_{8}
 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 Tduality,
and comes about from the compactification of extra space dimensions
in a ten dimensional superstring theory. Let's take the X^{9}
direction in flat tendimensional 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 n^{th} 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
E_{w} to the string energy is
equal to the string tension T_{string} 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.
where
tells us the length scale L_{s} 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 rightmoving and leftmoving 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 Tduality. Notice that if the
compactification radius R is much smaller than
the string scale L_{s}, then the compactification radius
after the winding and momentum modes are exchanged is much
larger than the string scale L_{s}. So Tduality obscures
the difference between compactified dimensions that are much bigger
than the string scale, and those that are much smaller than the string
scale.
Tduality relates type IIA
superstring theory to type IIB superstring theory, and it relates
heterotic SO(32) superstring theory to heterotic
E_{8}XE_{8} 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 nonchiral theory, whereas type IIB theory is a chiral theory and
has massless fermions with only a single chirality.
Tduality 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:
SDuality >>
