In Theory of the cords and Supercorde S the duality T indicates a particular duality under which one or more Rayon of compactification is reversed.

A simple case

Initially let us consider the simplest case of duality T. If one compactifie the bosonic theory on a circle of radius $R\; \backslash ,$ then the states of Vide of the theory $|m,\; n>\; \backslash ,$ are doubly quantified in the following way:

• the quantum Number $m\; \backslash ,$ indicates that the associated cord (or more precisely its Center of mass) has one moment $m/R\; \backslash ,$ in the direction of compactification. If $n=0\; \backslash ,$ one speaks then about states of moment.
• the quantum number $n\; \backslash ,$ indicates that the associated cord is rolled up $n\; \backslash ,$ time around the direction of compactification. In this fundamental state such a cord has a length $nR\; \backslash ,$ in this direction. If $m=0\; \backslash ,$ one speaks then about states of rolling up.

The study of the Specter of the theory of the cords indicates that such states have a mass $m\; \backslash ,$ given by

$m^2=\; \{m^2\; \backslash \; over\; R^2\}\; +\; \{n^\; 2R\; ^2\; \backslash \; over\; \backslash \; alpha\text{'}^2\}\; \backslash ,$

The first term is strictly similar to the mass of a particle moving with one moment $m/R\; \backslash ,$ in the compact direction within the framework of the Théorie of Kaluza-Klein. The second term is natural since a cord is an object having a Tension $\backslash \; alpha\text{'}\; \backslash ,$, to impose a minimal length to him thus costs energy proportional to its longueur.

It is seen whereas the spectrum that one has just described is invariant under the transformation

$R\; \backslash \; leftrightarrow\; \{\backslash \; alpha\text{'}\; \backslash \; over\; R\}\; \backslash ,$

on the condition of carrying out the exchange simultaneously

$m\; \backslash \; leftrightarrow\; N\; \backslash ,$ I.e. in particular that the states of moments (which have a particulate interpretation by considering the center of mass of the cord) are exchanged with states of rolling up (which do not have particulate interpretation) during the operation of duality T.

The operation which one has just described corresponds precisely so that one understands by duality T in this particular case. From the point of view of the Space-target this operation is completely remarkable: with the subtlety of the constant $\backslash \; alpha\text{'}\; \backslash ,$ close which makes it possible to homogenize the relation of inversion of ray (it with the unit a length squared), one thus sees who compactifier the theory of the cords on a circle of very small radius (what should lead to a theory having a dimension of less if one followed the intuition resulting from the theory of Kaluza-Klein for physics from the particles) is strictly equivalent to a theory of the cords compactifiée on a circle of very large radius and who within the limit of very large ray reproduces the not compactifiée original theory.

In addition let us announce that contrary to the symmetry of reparametrisation which is a characteristic common to all the theories of the gravitation incorporating the General relativity and thus nonspecific to the only theory of the cords alone, the duality T is primarily cordist in measurement or to be realized from a quantum point of view it requires the taking into account of the states of rolling up of the cord around a direction of compactification gold of such states could not exist in a theory or the fundamental excitations are only particles (which would provide only states of moment but not of states of rolling up).

duality T and field B

References

, vol. 1, chapter 8.

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