# 2D CFT from sigma models

$$X$$ 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.)

• For (1), it’s a complicated differential equation written as a power series in $\alpha’$. The leading term is the vacuum EFE. Whether the power series converges, and whether there’s a nonperturbative solution that isn’t just the flat space, I don’t know. Maybe someone more knowledgeable in string theory can comment on this Commented Apr 8, 2023 at 7:45