## FANDOM

String Theory
Prior to the First Superstring Revolution
Early History S-Matrix Theory
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Type HO String Theory
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Second Superstring Revolution T-Duality
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(2,0) Theory
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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.