String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History S-Matrix Theory
Regge Trajectory
Bosonic String Theory Worldsheet
Bosonic String Theory
String Perturbation Theory
Tachyon Condensation
Supersymmetric Revolution Supersymmetry
RNS Formalism
GS Formalism
Superstring Revolutions
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-Duality
Horava-Witten String Theory
Holographic Principle
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
(2,0) Theory
Twistor String Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
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.