# Sigma Models on Riemann Surfaces

I'm interested in knowing whether sigma models with an $n$-sheeted Riemann surface as the target space have been considered in the literature. To be explicit, these would have the action \begin{align*} S=\frac{1}{2}\int d^2x\, \left(\partial_a R\partial^a R+R^2\partial_a \theta {\partial}^a\theta\right), \end{align*} where $R$ and $\theta$ represent radial and angular coordinates on the target space respectively. Also, $\theta\sim \theta+2\pi n$ for an $n$-sheeted Riemann surface.

Has anyone seen anything like this? One thing that I would be particularly happy to see is a computation of the partition function.

-
Tip: You might get better/more focused/useful answers if you disclose what literature you are reading. –  Qmechanic Mar 7 at 14:37
Maybe closed strings on spacetimes with compactified dimensions qualify. See e.g any introductory string theory text that makes mention of T-duality. –  alexarvanitakis Mar 7 at 16:20
@Qmechanic thanks for the suggestion, I didn't have any specific literature in mind; I guess I just meant literature in the broader sense. –  Matthew Mar 7 at 19:23
@alexarvanitakis I don't think that works. In that case the target space is $S^1\times \mathbf{R}^{25}$, which is a quotient of $\mathbf{R}^{26}$. The target space that I have in mind isn't a quotient of the plane. –  Matthew Mar 7 at 19:26
Oh right, the target space is supposed to be a Riemann surface. Sorry. This should be interesting then... –  alexarvanitakis Mar 7 at 19:30