# 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.

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Tip: You might get better/more focused/useful answers if you disclose what literature you are reading. –  Qmechanic Mar 7 '13 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 '13 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 '13 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 '13 at 19:26
Oh right, the target space is supposed to be a Riemann surface. Sorry. This should be interesting then... –  alexarvanitakis Mar 7 '13 at 19:30

One usually considers Ricci-Flat target spaces for string propagation. If one doesn't care about the conformal anomaly, then given that one can think of an n-sheeted Riemann surface as given by the quotient of the Poincare upper half plane by some Fuchsian group. A first step to your question then is to compute the partition function for the sigma model with the Poincare upper half plane as the target space. The upper half plane can be be modeled by the coset space $PSL(2,\mathbb{R})/SO(2)$. Thus it might suffice to look at the WZW model for this coset space. Such cosets were considered in the context of CFT's for two-dimensional blackholes in the early 90's (the action there contains additional fields to cancel the conformal anomaly). The paper by Witten titled "On string theory and black holes" (http://inspirehep.net/record/314576) might be a good starting point. The next step would be consider orbifolding by the Fuchsian group but that is another story.