Thursday, 25 August 2011

The Mathematics of Super String Theory



The single most important equation in (first quantisized bosonic) string theory is the N-point scattering amplitude. This treats the incoming and outgoing strings as points, which in string theory are tachyons, with momentum ki which connect to a string world surface at the surface points zi. It is given by the following functional integral which integrates (sums) over all possible embeddings of this 2D surface in 26 dimensions.

 A_N = \int{D\mu \int{D[X] exp \left( -\frac{1}{4\pi\alpha} \int{ \partial_z X_{\mu}(z,\overline{z}) \partial_{\overline{z}} X^{\mu}(z,\overline{z})}dz^2 + i \sum_{i=1}^{N}{k_{i \mu} X^{\mu}(z_i,\overline{z}_i) }  \right) }}

The functional integral can be done because it is a Gaussian to become:

This is integrated over the various points zi. Special care must be taken because two parts of this complex region may represent the same point on the 2D surface and you don't want to integrate over them twice. Also you need to make sure you are not integrating multiple times over different paramaterisations of the surface. When this is taken into account it can be used to calculate the 4-point scattering amplitude (the 3-point amplitude is simply a delta function):

 A_4 = \frac{ \Gamma (-1+\frac12(k_1+k_2)^2) \Gamma (-1+\frac12(k_2+k_3)^2)  } { \Gamma (-2+\frac12((k_1+k_2)^2+(k_2+k_3)^2)) }

Which is a beta function. It was this beta function which was apparently found before full string theory was developed. With superstrings the equations contain not only the 10D space-time coordinates X but also the grassman coordinates θ. Since there are various ways this can be done this leads to different string theories.

When integrating over surfaces such as the torus, we end up with equations in terms of theta functions and elliptic functions such as the Dedekind eta function. This is smooth everywhere, which it has to be to make physical sense, only when raised to the 24th power. This is the origin of needing 26 dimensions of space-time for bosonic string theory. The extra two dimensions arise as degrees of freedom of the string surface.




D-Branes are membrane-like objects in 10D string theory. They can be thought of as occurring as a result of a Kaluza-Klein compactification of 11D M-Theory which contains membranes. Because compactification of a geometric theory produces extra vector fields the D-branes can be included in the action by adding an extra U(1) vector field to the string action.

\partial_z \rightarrow \partial_z +iA_z(z,\overline{z})

In type I open string theory, the ends of open strings are always attached to D-brane surfaces. A string theory with more gauge fields such as SU(2) gauge fields would then correspond to the compactification of some higher dimensional theory above 11 dimensions which is not thought to be possible to date.

Why Five Superstring Theories?



For a 10 dimensional supersymmetric theory we are allowed a 32-component Majorana spinor. This can be decomposed into a pair of 16-component Majorana-Weyl (chiral) spinors. There are then various ways to construct an invariant depending on whether these two spinors have the same or opposite chiralities:

Superstring Model Invariant
Heterotic \partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}
IIA \partial_zX^\mu-i\overline{\theta_{L}}\Gamma^\mu\partial_z\theta_{L}-i\overline{\theta_{R}}\Gamma^\mu\partial_z\theta_{R}
IIB \partial_zX^\mu-i\overline{\theta^1_{L}}\Gamma^\mu\partial_z\theta^1_{L}-i\overline{\theta^2_{L}}\Gamma^\mu\partial_z\theta^2_{L}

The heterotic superstrings come in two types SO(32) and E8xE8 as indicated above and the type I superstrings include open strings.