String Theory  

All Roads Lead to String Theory (Polchinski)  
Prior to the First Superstring Revolution
 
Early History  SMatrix 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
 
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  TDuality DBrane SDuality HoravaWitten String Theory MTheory Holographic Principle N=4 SuperYangMills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory FTheory String Field Theory Pure Spinor Formalism  
After the Revolutions
 
Phenomenology  String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology  
Bosonic String Theory is the original version of String Theory. It is completely determined by the Polyakov Action, and only includes bosons among it's mass spectrum. It also contains a tachyon in it's mass spectrum as the ground state.
NambuGoto ActionEdit this section
The NambuGoto Action is the action principle generally associated with Classical Bosonic String Theory. However, in principle, it is possible to use it for the fully quantised theory as well, but it is much harder. The NambuGoto Action can easily be obtained by looking at a Worldsheet suspended in spacetime. It can be stated as:
Where is the induced metric on the worldsheet when it is embedded in the background spacetime. This has an obvious geometrical meaning. However, due to the presence of the square root, this action is unsuitable for quantisation. For clarity, observe that under a suitable Diffeomorphism, i.e. reparameterisation, / \
Which is clearly hard to quantise.
Polyakov ActionEdit this section
The Polyakov Action is an easiertoquantise action for the Bosonic String Theory. It can be stated as:
This Polyakov Action is not so appealing geometrically, but it is easier to quantise. It can be shown that this is classically equivalent to the NambuGoto Action as follows (Please click here to view properly):
Proof that the Polyakov Action classically reduces to the NambuGoto Action 

We use the EulerLagrange Equation:
Where is the trace of the StressEnergyMomentum Tensor and is the induced metric on the worldsheet.
Also recall that:
From this we see that, if the intrinsic worldsheet metric satisfies the following field equation:
Then, the Polyakov Action becomes the NambuGoto Action:

However, as in the Polyakov Action, there is no square root over the ' s, it is much easier to quantise compared to the NambuGoto Action.
Virasoro AlgebraEdit this section
The Loperator in Bosonic String Theory is given by:
When , this results in ,.
The Loperators form an an algebra, known as the Virasoro Algebra. This is actually a quantisation of the Witt Algebra (also known as the Classical Virasoro Algebra, and is isomorphic to the Conformal Algebra) through a central extension.
The multiplication; of this algebra is given, by:
This algebra is actually useful in removing the negative norm states, as we will see.
This algebra also requires the Virasoro Constraints, which are stated simply as:
Imposing the Virasoro ConstrainstsEdit this section
Thus, /.
Therefore, /. 
Mass SpectrumEdit this section
In the Classical Bosonic String Theory, the mass spectrum can be given by ; however, in the quantised Bosonic String Theory, it is given by:
Since ,
.
The first equation is still true in superstring theories, though the second is not, since is different. Note that this is only for open bosonic strings. For closed bosonic strings, it becomes:
p
Where we took because both the left (Without the tilde) and right (with the tilde) moving sectors are from the Bosonic String Theory.