I have been given an action of the form:

$$S = \frac{1}{4\pi}\int d^2\sigma \ \sqrt{-g}\left(\frac{1}{2}\partial_\mu\phi \partial^\mu\phi + \frac{1}{\zeta}\phi R + \frac{\mu}{2\zeta^2}e^{\zeta\phi} \right).$$

I have been able to calculate its stress energy tensor in the coordinates $z = \sigma^0 + i\sigma^1$ and $\overline{z} = \sigma^0 - i\sigma^1$ as:

$$T_{zz} = \frac{1}{2}(\partial_z\phi)^2 - \frac{1}{\zeta}\partial^2_{z}\phi,$$ $$T_{\overline{z}\overline{z}} = \frac{1}{2}(\partial_{\overline{z}}\phi)^2 - \frac{1}{\zeta}\partial^2_{\overline{z}}\phi.$$

How do I calculate the Virasoro charges and its Poisson brackets?


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