In Euclidean worldsheet signature, the <string nonlinear sigma model> action can be written
$$
\boxed{
S=\frac1{4\pi\alpha'}\int_\Sigma d^2\sigma\sqrt h\,
\left[h^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu
+i\epsilon^{ab}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu
+\alpha'\Phi(X)R^{(2)}(h)\right]}.
$$
The factor of $i$ in the B-field term is absent in Lorentzian signature. Here $G$ is the target metric, $B$ the <Kalb–Ramond field>, and $\Phi$ the <dilaton>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 4 / i / Solution

The string nonlinear sigma model couples the worldsheet embedding fields to target-space backgrounds such as a metric, a <Kalb–Ramond field> and a <dilaton>. Quantum Weyl invariance requires its beta functions to vanish.

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