$X$ is a closed manifold with a positive-definite metric $g$. $M_2$ is a 2D oriented closed manifold with a positive-definite metric $G$ and a compatible volume form $\omega$. We can then consider the following Euclidean path integral for a $X$-valued scalar field $\phi$ on spacetime $M_2$: $$\int\mathcal{D}\phi\,\mathrm{e}^{-\mathcal{S}[\phi]}\,,\qquad \mathcal{S}[\phi]=\frac{1}{2e^2}\int_{M_2}\omega\ \langle G, \phi^*g \rangle\,,$$ or written in components: $$\mathcal{S}[\phi]=\frac{1}{2e^2}\int_{M_2}\sqrt{|G|}\,d^2x\ \ G^{\mu\nu}g^{ab}\partial_{\mu}\phi_a\,\partial_{\nu}\phi_b\,.$$ Since the coupling $e^2$ is dimensionless, the theory has a chance to become conformal. At the one-loop level, this requires $X$'s Ricci curvature to vanish.
Question 1: What's the sufficient and necessary condition on $X$ (and $g$) for having a conformal field theory?
Question 2: What if we add a $\theta$-angle $\in H^2(X,U(1))$ or a Wess-Zumino-Witten term $\in H^3(X,\mathbb{Z})$?
(I know these are tough questions even nowadays. So I'm actually asking about the best result we knew so far.)
