String Theory/TwoDimensional Conformal Field Theory
TwoDimensional Conformal Field Theory
editConformal Transformations
editThe Conformal Group
editThe story of string theory begins with twodimensional conformal invariance.
Conformal transformations on a manifold preserve angles at every point, an example of such a transformation being the Mercator projection of the Earth onto an infinite cylinder. They may be defined as transformations that leave the metric invariant up to a scale.
The set of invertable conformal transformations form a group. This is the conformal group.
Let us apply this rule to a two dimensional manifold.
For this transformation to be conformal the metrics must be proportional to one another, which means,
Writing out the components, the following conditions emerge:
These conditions turn out to be equivalent to the CauchyRiemann conditions for either holomorphic or antiholomorphic functions!
 and (holomorphic)
 and (antiholomorphic)
In two dimensions, therefore, the conformal group is the set of all invertable holomorphic maps, which is isomorphic to the set of all antiholomorphic maps. For this reason it is convenient to use complex coordinates when discussing twodimensional conformal fields.
The set of all reversible holomorphic functions is the set of fractional linear transformations

 where
It is easily verified by composing two such functions that their composition is equivalent to matrix multiplication for matrices of the form
It is clear that the conformal group in two dimensions is equivalent to the group of complex invertible matrices having a determinate of 1. This group is also known as .
The Virasoro Algebra
editModular Invariance
editSuperconformal Transformations
editClassical Strings
editThe Classical String
editLet us embed an action that is conformally invariant in two dimensions into a higher dimensional space. We will find that such an action generalizes the concept of the point particle.
Boundary Conditions
editThe Classical Superstring
editTwoDimensional Quantum Field Theory
edithere goes