Conformal symmetry, its motivations, its applications

Conformal invariance

Conformal transformations

On a given space or spacetime $M$ with coordinates $x^{\mu }$, distances are defined using a metric $g_{\mu \nu }$. In particular, the length of an infinitesimal vector $v^{\mu }$ is $\|v\|={\sqrt {g_{\mu \nu }v^{\mu }v^{\nu }}}$. If we know distances, we can also compute angles. The angle $\theta$ between two infinitesimal vectors $v^{\mu },w^{\mu }$ obeys

$\cos \theta ={\frac {g_{\mu \nu }v^{\mu }w^{\nu }}{{\sqrt {g_{\mu \nu }v^{\mu }v^{\nu }}}{\sqrt {g_{\mu \nu }w^{\mu }w^{\nu }}}}}$

A map $f:M\to M$ is called an isometry if it preserves distances, equivalently if it preserves the metric:

$f{\text{ isometry }}\iff g_{\mu \nu }(f(x))df^{\mu }df^{\nu }=g_{\mu \nu }(x)dx^{\mu }dx^{\nu }$

A map $f:M\to M$ is called a conformal transformation if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function $\lambda :M\to \mathbb {R}$, the metrics $g_{\mu \nu }$ and $\lambda g_{\mu \nu }$ define the same angles. To preserve angles, a map therefore only needs to preserve the metric up to a scalar factor:

$f{\text{ conformal transformation }}\iff \exists \lambda ,\ g_{\mu \nu }(f(x))df^{\mu }df^{\nu }=\lambda (x)g_{\mu \nu }(x)dx^{\mu }dx^{\nu }$

case of flat metric, dilations etc.

not conformal example. $z^{2}$? Not conformal at a point.

conformal symmetry: inv. under conf. transfo.

Conformal symmetry and gravitation

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)