String Theory/Two-Dimensional Conformal Field Theory

Two-Dimensional Conformal Field Theory

Conformal Transformations

The Conformal Group

The story of string theory begins with two-dimensional 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.

${\underset{\_}{g}}_{\mu \nu }({\underset{\_}{x}}^{\xi })=\mathrm{\Lambda }(x^{\xi })g_{\mu \nu }(x^{\xi })$

The set of invertable conformal transformations form a group. This is the conformal group.

Let us apply this rule to a two dimensional manifold.

${\underset{\_}{g}}_{\underset{\_}{\mu }\underset{\_}{\nu }}({\underset{\_}{x}}^{\xi })=\left(\frac{\partial {\underset{\_}{x}}^{\underset{\_}{\mu }}}{\partial x^{\mu }}\right)\left(\frac{\partial {\underset{\_}{x}}^{\underset{\_}{\nu }}}{\partial x^{\nu }}\right)g_{\mu \nu }(x^{\xi })$

For this transformation to be conformal the metrics must be proportional to one another, which means,

$\left(\frac{\partial {\underset{\_}{x}}^{\underset{\_}{\mu }}}{\partial x^{\mu }}\right)\left(\frac{\partial {\underset{\_}{x}}^{\underset{\_}{\nu }}}{\partial x^{\nu }}\right)\propto {\delta }_{\mu }^{\underset{\_}{\mu }}{\delta }_{\nu }^{\underset{\_}{\nu }}$

Writing out the components, the following conditions emerge:

${\left(\frac{\partial z^0}{\partial {\underset{\_}{z}}^0}\right)}^2+{\left(\frac{\partial z^0}{\partial {\underset{\_}{z}}^1}\right)}^2={\left(\frac{\partial z^1}{\partial {\underset{\_}{z}}^0}\right)}^2+{\left(\frac{\partial z^1}{\partial {\underset{\_}{z}}^1}\right)}^2$

$\frac{\partial {\underset{\_}{x}}^0}{\partial x^0}\frac{\partial {\underset{\_}{x}}^1}{\partial x^0}+\frac{\partial {\underset{\_}{x}}^0}{\partial x^1}\frac{\partial {\underset{\_}{x}}^1}{\partial x^1}$

These conditions turn out to be equivalent to the Cauchy-Riemann conditions for either holomorphic or antiholomorphic functions!

$\frac{\partial {\underset{\_}{x}}^1}{\partial x^0}=\frac{\partial {\underset{\_}{x}}^0}{\partial x^1}$ and $\frac{\partial {\underset{\_}{x}}^0}{\partial x^0}=-\frac{\partial {\underset{\_}{x}}^1}{\partial x^1}$ (holomorphic)

$\frac{\partial {\underset{\_}{x}}^1}{\partial x^0}=-\frac{\partial {\underset{\_}{x}}^0}{\partial x^1}$ and $\frac{\partial {\underset{\_}{x}}^0}{\partial x^0}=\frac{\partial {\underset{\_}{x}}^1}{\partial x^1}$ (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 two-dimensional conformal fields.

The set of all reversible holomorphic functions is the set of fractional linear transformations

$f(z)=\frac{\mathrm{♡}z-\mathrm{♣}}{\mathrm{♠}z+\mathrm{♢}}$

where

$\mathrm{♡}\mathrm{♢}-\mathrm{♣}\mathrm{♠}=1$

It is easily verified by composing two such functions that their composition is equivalent to matrix multiplication for matrices of the form

$\left(\begin{array}{cc}\mathrm{♡}& \mathrm{♣}\\ \mathrm{♠}& \mathrm{♢}\end{array}\right)$

It is clear that the conformal group in two dimensions is equivalent to the group of complex invertible $2\times 2$ matrices having a determinate of 1. This group is also known as $SL(2,ℂ)$.

Classical Strings

The Classical String

Let 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.

Two-Dimensional Quantum Field Theory

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