Changing the target metric by $\delta G_{\mu\nu}=\epsilon_{\mu\nu}$ changes the action by
$$
\delta S=-\frac1{4\pi\alpha'}\int_\Sigma d^2\sigma\sqrt{|h|}\,
h^{ab}\epsilon_{\mu\nu}\partial_aX^\mu\partial_bX^\nu.
$$
Since the path-integral weight is $e^{iS}$, the leading change is the insertion
$$
\delta\langle\phi_1\cdots\phi_n\rangle
=\int_\Sigma d^2\sigma\,
\langle\mathcal O(\sigma,\tau)\phi_1\cdots\phi_n\rangle_{\rm conn},
$$
where
$$
\boxed{\mathcal O=-\frac{i}{4\pi\alpha'}\sqrt{|h|}\,
h^{ab}\epsilon_{\mu\nu}\partial_aX^\mu\partial_bX^\nu}.
$$
A change of worldsheet metric within a gauge orbit changes only the redundant description and cannot alter gauge-invariant observables. The target-space metric is instead a physical coupling of the <string nonlinear sigma model>, so changing it changes the theory and its correlation functions.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 1 / 4 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-306/1/4/solution