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The textbook approach to explaining T-dualities is to show that a type of T-duality transformation "inverts the radius of the circle, that is, it maps $R\rightarrow\tilde{R} = \alpha'/R$ and it leaves the mass formula for the string invariant provided that the string winding number is exchanged with the Kaluza-Klein excitation number." 1

However, the broader classification of T-duality appears imply that theories are T-daul if there is any type of transformation that changes on theory into the other. So the question is whether there are other approaches to showing T-duality, or is there a "canonical" approach for T-duality.

String Theory and M-Theory: A Modern Introduction [Hardcover] Katrin Becker (Author), Melanie Becker (Author), John H. Schwarz (Author)

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"Any transformation that changes one theory into another" (or the same) theory is not called T-duality. It is just a "duality". A condition is that the two theories seemingly look different - otherwise the equivalence would be vacuous - but it must be true that their spectrum and the strength of interactions between their states must be totally isomorphic: physics has to be indistinguishable. A duality is therefore just a very fancy redefinition of coordinates that can't be quite done in the classical limit so you can't really write any "explicit" redefinition. But the impact is the same - the two theories behave in the same way. For each state/object on one side, you find a counterpart on the other side - whose geometrical interpretation may be very different but whose behavior is isomorphic.

T-duality in string theory

T-duality is a duality that changes the geometrical properties of the target spacetime but that holds order-by-order in the expansion in terms of the string coupling in perturbative string theory. In particular, a T-duality must preserve the notion of the "fundamental string" (or "world sheet") and its tension and it is a weak-weak duality when it comes to the interactions between strings. The world sheet of a given shape remains the world sheet of the same shape; however, the fields on it get transformed. That's T-duality in general.

Quite generally, T-dualities that we know may be interpreted as the duality you mentioned although it may also invert the radii of several circles (a torus) at the same moment, and the shape of these circles (and torus) may depend on other coordinates of spacetime (the so-called "Buscher procedure" is a standardized method to perform such a transformation). In particular, "mirror symmetry" relating two completely different Calabi-Yau shapes may be understood as a T-duality acting on a 3-dimensional toroidal fibers of the Calabi-Yau space (a special way to choose coordinates on the 6D shape so that 3 of them look like a torus whose shape depends on the remaining 3.) However, in recent years, people have also appreciated the importance of fermionic T-dualities that mostly act on fermionic fields on the world sheet but that are otherwise analogous.

Other dualities

There are several other important kinds of dualities. S-dualities invert the weak and strong coupling. In particular, for a very large value of $g$ much greater than one, they find another theory with a weak coupling $g'=1/g$ or $1/g^2$ - which is much smaller than one - that is equivalent to the previous one. So S-dualities usually invert a small and large couplings, much like T-dualities exchange small and large radii of circles. A fundamental string doesn't remain a fundamental string under an S-duality; it usually gets transformed into another string, e.g. a D-string (also known as a D1-brane), that used to be heavy and unimportant when the coupling was weak but becomes the lightest and most important object when the coupling is strong. Again, an S-duality exchanges fundamental strings and D-strings, and similar pairs of objects.

More generally, there are also U-dualities that combine the actions of S-dualities and T-dualities: they exchange a strong coupling with large radii with a weak coupling and small radii, among similar actions. In particular, M-theory (essentially 11-dimensional supergravity with the extra "stringy" stuff needed to make it consistent at all energies) compactified on a $k$-dimensional torus has a self-U-duality group which is something like $E_{k(k)}(Z)$, a discrete subgroup of a noncompact edition of the exceptional Lie groups (for $k\leq 8$: for $k\leq 5$, the exceptional group can be written in terms of non-exceptional groups). Such an exceptional U-duality group exchanges momenta in compact dimensions with wrapped membrane and fivebrane charges (and other discrete charges such as KK-monopole charges if there are at least 6 compact dimensions).

Several other equivalences between two seemingly different theories are also called "dualities" - and the AdS/CFT holographic correspondence itself is sometimes classified as a duality as well. It would triple this article if I wanted to cover all important kinds of dualities that appear in field theory and string theory.

Why the existence of dualities was surprising

Finally, it's useful to stress that dualities - a key notion of theoretical physics of the last 20 years or so - didn't have to exist. More precisely, people thought that a theory compactified on an extremely tiny circle, or a theory extrapolated to an insanely strong coupling, is a completely new mysterious beast filled with sea dragons that behaves like nothing that the people had previously encountered. However, it turned out that the behavior of those "mysterious limits" is almost always governed by another "mundane" theory that the people have essentially known, and that the two are equivalent. It's like sailing to India and America and expecting totally new creatures over there - but finding the same animals and plants we're used to.

Whenever some radius is taken extremely tiny, coupling is made very strong, or even the number of colors in a gauge theory is sent to infinity, so that the perturbative techniques of the original theory totally break down, there always seems to be a new description that is totally weakly coupled. That's one of the lessons of physics since the mid 1990s - something that we didn't previously know but nowadays, we do know it. The fact that we seem to understand the "opposite extreme" limits of the previously known theories places an upper bound on the amount of "new enigmas" that may still be hiding in field theory and string/M-theory. If there are totally new enigmas, they have to hide at the intermediate values - like Atlantis in the middle of the Atlantic Ocean, if you wish. ;-)

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+1 Sea dragons. LOL. Actually there are elephants and tigers in India and turkey and buffalo in America, neither of which are native to Europe. –  user346 Feb 4 '11 at 19:08

T-duality involves an interchange $x_9~\rightarrow~x_9~+~2\pi R$ with the compactification of a dimension. A particle running around this circle has a momentum $\sim~1/R$, and this does connect in part with the idea of a reciprocal lattice. The mass of the particle running around the circle is $M~=~n/R$. A closed string can also wrap around a circle. The winding of a string $w$ contributes energy $E~=~2\pi wRT$, $T~=~1/(2\pi\alpha’)$ the string. The mass is related to the winding number by $$ M^2~=~\Big(\frac{n}{R}\Big)^2~+~\Big(\frac{wR}{\alpha’}\Big)^2~+~\frac{1}{\alpha’}(2N~+~w~-~2) $$ This has the reciprocal structure as well. The connection between the lattice and compactification and string/brane wrappings will have to be brought in tighter with a connection to compactified spaces. T-duality relates type IIA superstring theory to type IIB superstring theory, and it relates heterotic $SO(32)$ superstring theory to heterotic $E_8\times E_8$ superstring theory. There is an interchange between mode and winding numbers with $R~\rightarrow~\alpha’/R$.

S-duality is a relationship between two similar field theories with reciprocal coupling strengths. This is based on the Montenen-Olive relationship which is a form of the Bohr-Sommerfeld quantization rule. Given charges $q,~g$ for a similar gauge field then $$ qg~=~n\hbar. $$ A low energy limit of a type IIB SUGRA with a complex field $\phi$ decomposed into an axion and diliton field as the real and imaginary parts $\phi~=~\psi~+~iexp(X)$ has an $SL(2,{\mathbb R})$ structure, where the discrete subset $SL(2,{\mathbb Z})$ has a modular (or M{\”o}bius) structure, where the field has a linear fractional transformation $$ \phi~\rightarrow~\frac{a\phi~+~b}{c\phi~+~d},~ad~-~bc~=~1 $$ where for $b~=~c~=~1$ and $a~=~d~=~0$ we have another reciprocating relationship.

The T and S dualities may be composed in various ways to define this less well understood U-duality.

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