# Is the restriction of the Cartan 3-form on conjugacy classes exact?

Let $$G$$ be a complex semisimple group and $$\mathcal{O} \subset G$$ a conjugacy class, i.e. $$\mathcal{O} = \{gag^{-1} : g \in G\}$$ for some $$a \in G$$. Let $$\Omega$$ be the Cartan 3-form on $$G$$ defined by $$\Omega(x^L, y^L, z^L) = \langle x, [y, z] \rangle,\quad (x,y,z \in \mathrm{Lie}(G))$$ where $$\langle,\rangle$$ is the Killing form, and $$x^L, y^L, z^L$$ are the left-invariant vector fields generated by $$x,y,z$$.

Question: Is the pullback of $$\Omega$$ to $$\mathcal{O}$$ trivial in cohomology?

• What do you mean by $x^L$?
– YCor
Jul 29 at 20:25
• The left-invariant vector field generated by $x$. Jul 29 at 20:33
• Thanks. One particular case: if $G=\mathrm{SL}_2(\mathbf{R})$ then conjugacy classes are manifolds of dimension $\le 2$ so their cohomology vanish in degree 3.
– YCor
Jul 29 at 20:53

Yes, it is exact, and there is in fact a canonical 2-form on each conjugacy class whose derivative is your $$\Omega$$. This was an important observation when studying D-branes in WZW models, see, e.g. https://arxiv.org/abs/hep-th/0008038 or https://arxiv.org/abs/hep-th/0205233.