13.3. Polyakov–Wiegman identity Consider the action of the sigma model with a Wess-Zumino topological term S(g) = k 16π

d2xTr(∂μg−1∂μg)+k.

Prove the identity S(gh−1) = S(g)+S(h)+ k 2π

d2xTr(g−1∂¯ zgh−1∂z h).

Show that this identity gives rise to the invariance of the action under the transformation g(z, ¯ z)→G(z)g(z, ¯ z)G−1( ¯ z).