# A Glimpse of String Theory and Conformal Geometry

I am taking String Theory this quarter with Joe Polchinski, reading from his Little Book of String.  We have been covering bosonic string (no fermions) arising from the Polyakov string action .

$\displaystyle \frac{1}{4\pi \alpha} \int d\tau d\sigma\, \sqrt{-\det \gamma} \gamma^{ab} \partial_a X^\mu \partial_b X_\mu$

Technically, we should Wick rotate the time parameter $\tau \to i\tau$, the Lorentzian worldsheet metric becomes Riemannian $\gamma^{ab} \to g^{ab}$.  We can pick normal coordinates $g^{ab} \to \delta^{ab}$  and complexify   $w = \sigma^1 + i \sigma^2$ and $\overline{w} = \sigma^1 - i \sigma^2$ .  Then the action looks like the Nambu-Goto action again (measuring the area of the worldsheet).

$\displaystyle S_P = \frac{1}{2\pi \alpha} \int d^2 w \, \partial_w X^\mu \partial_{\overline{w}} X^\mu$

This action is invariant under conformal symmetries of the worldsheet
$z \to z + \sum_{n = -\infty}^\infty \epsilon_n z^{n+1}$, where $z$ parameterizes the worldsheet.  In 2 dimensions, the conformal group is especially large and there are infinitely many generators $\ell_n: z \to z + \epsilon_n z^{n+1}$ .

I couldn’t picture them so I drew them with mathematica.  Technically, these are infinitesimal transformations but I drew them globally.