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.

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