Solution
= Solution
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>.