# T-Duality

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String Theory
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All Roads Lead to String Theory (Polchinski)
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Prior to the First Superstring Revolution
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Early History
| S-Matrix Theory Regge Trajectory | |

Bosonic String Theory
| Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation | |

Supersymmetric Revolution
| Supersymmetry RNS Formalism GS Formalism BPS | |

Superstring Revolutions
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First Superstring Revolution
| GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory | |

Second Superstring Revolution
| T-DualityD-Brane S-Duality Horava-Witten String Theory M-Theory Holographic Principle N=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism | |

After the Revolutions
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Phenomenology
| String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology | |

**T-Duality** is a duality, or equivalence between two String Theoryies. In contrast to S-Duality, **T-Duality** is a purely stringy concept. **T-Duality** equates a String Theory compactified around a distance with another compactified around a distance of .

## T-Duality for Bosonic StringsEdit this section

Beginning with a simple toy model, Bosonic String Theory, we compactify a spatial dimension, say (9 is randomly chosen as Bosonic String Theory is 26-dimensional, not 10-dimensional), such that: .

The ground Wavefunction is . It is clear that this is single-valued only when

There fore, the momenta is quantised by the above equation. Then,

=

However, a Closed String can actually wrap as many times around a circle as it wishes, not necessarily once. This "as many times" is called the winding number Then,

When (uncompactified), the RHS becomes 0.

If we consider the momentum, it is still $$, but the left- and right- moving momenta are:

The mass spectrum is also intuitively modified as (which is clear from the relation between the mass and the momenta); :

$$ $$

If we talke the limit as , and if we take the limit as , then , and the compactified dimension reappers. This is explained by the following transformations between the winding number and the momentum quantisation number; and between the compactification radii.

These transformations are called **T-Duality**. This is also equivalent to negating the right-moving mode of oscillation; and therefore, the field . Considering **T-Duality** for Open Strings, we see that a normal derivative becomes a tangential derivative, therefore exchanging Dirchilet and Newmann Boundary conditions.

This immediately makes the existence of D-Branes necessary.

## T-Duality for Type II StringsEdit this section

Most of the ideas of the previous section carry over to Type II (Type IIA and Type IIB) Strings; however there is an additional result. In the previous section, we learnt that T-Duality negates the bosonic field . By the manifest Worldsheet Supersymmetry of RNS String Theory, this also implies that the fermionic field is negated.

This immediately implies that the GSO Projection also flips sign:

As the Type IIA String Theory and Type IIB String Theory differ only by the GSO Projection, this means that **T-Duality** exchanges Type IIA String Theory and Type IIB String Theory.

## T-Duality for Type H StringsEdit this section

Type H String Theory, or Heterotic String Theory, is also affected by **T-Duality**. The weight lattice of (the gauge group of the Type HO String Theory is given by while the weight lattice of is given by . Since , it follows that Type HO String Theory is **T-Dual** to Type HE String Theory.