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 $ R $ with another compactified around a distance of $ \frac{\alpha'}{R}=\frac{\ell_s^2}{R} $.

## T-Duality for Bosonic StringsEdit this section

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

$ x^9\sim x^9+2\pi R $

The ground Wavefunction is $ e^{ip_0^9x^9/\hbar} $. It is clear that this is single-valued only when

$ p_0^9=\hbar \frac nR $

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

$ \alpha_0^\mu=\tilde\alpha_0^\mu= \frac{\ell_s}{\hbar}\frac nR $

$ \alpha_0^\mu+\tilde\alpha_0^\mu= \frac{2\ell_s}{\hbar} \frac nR $ =

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 $ w $ Then,

$ x^9\sim x^9+2\pi R $

$ \alpha_0^\mu-\tilde\alpha_0^\mu= \frac{wR}{\ell_s} $

When $ w=0 $ (uncompactified), the RHS becomes 0.

If we consider the momentum, it is still $$ p=\frac nR $$, but the left- and right- moving momenta are:

$ p_-= \hbar\left(\frac nR - \frac1{\ell_s^2} wR \right) $

$ p_+= \hbar\left(\frac nR + \frac1{\ell_s^2} wR \right) $

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

$$ $ m=\frac{2\pi T\ell_s}{c_0^2} \sqrt{N+\tilde N-a-\tilde a + \ell_s^2 \frac{n^2}{R^2} \frac{1}{\ell_s^2}w^2R^2 } $ $$

If we talke the limit as $ R\to\infty $, $ w\to0 $ and if we take the limit as $ R\to0 $, then $ n\to0 $, 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.

$ w\leftrightarrow n $

$ R\leftrightarrow \frac{ell_s^2}{R} $

These transformations are called **T-Duality**. This is also equivalent to negating the right-moving mode of oscillation; and therefore, the field $ X^\mu $. 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 $ X^\mu $. By the manifest Worldsheet Supersymmetry of RNS String Theory, this also implies that the fermionic field $ \psi ^\mu $ is negated.

This immediately implies that the GSO Projection also flips sign:

$ \operatorname{T}: \mathcal P^-_\operatorname{GSO}\leftrightarrow \mathcal P^+_\operatorname{GSO} $

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 $ \frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}_2} $ (the gauge group of the Type HO String Theory is given by $ \Gamma^{16} $ while the weight lattice of $ E(8)\times E(8) $ is given by $ \Gamma^8\oplus\Gamma^8 $. Since $ \Gamma^{8}\oplus \Gamma^8\oplus\Gamma^{1,1}= \Gamma^{16} \oplus\Gamma^{1,1} $, it follows that Type HO String Theory is **T-Dual** to Type HE String Theory.