|All Roads Lead to String Theory (Polchinski)|
Prior to the First Superstring Revolution
|Early History|| S-Matrix Theory|
|Bosonic String Theory|| Worldsheet|
Bosonic String Theory
String Perturbation Theory
|Supersymmetric Revolution|| Supersymmetry|
|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
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
Twistor String Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
|Phenomenology|| String Theory Landscape|
Minimal Supersymmetric Standard Model
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
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:
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.