In Euclidean <conformal gauge>, complex coordinates reduce the <worldsheet ghost action> to
$$
S_{\rm Gh}=\frac1{2\pi}\int d^2z\,
\left(b_{zz}\bar\partial c^z+
b_{\bar z\bar z}\partial c^{\bar z}\right)
$$
up to a convention-dependent common normalization absorbed into the fields. Inverting $\bar\partial$ and $\partial$ gives
$$
\boxed{\displaystyle
\langle b_{zz}(z)c^z(w)\rangle=\frac1{z-w}},
\qquad
\boxed{\displaystyle
\langle b_{\bar z\bar z}(\bar z)c^{\bar z}(\bar w)\rangle
=\frac1{\bar z-\bar w}},
$$
with mixed holomorphic-antiholomorphic correlators equal to zero. These are the standard local-plane propagators; compact worldsheets require the zero-mode qualifications described in part (a).

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 4 / b / Solution

For the traceless diffeomorphism operator
$$
(P_1c)_{\alpha\beta}=\nabla_\alpha c_\beta+\nabla_\beta c_\alpha-g_{\alpha\beta}\nabla_\gamma c^\gamma,
$$
the ghost action is
$$
S_{\rm gh}=\frac1{2\pi}\int d^2\sigma\,\sqrt g\,
b^{\alpha\beta}(P_1c)_{\alpha\beta}.
$$


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