Sigma Models on Riemann Surfaces

Question

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.

This post imported from StackExchange Physics at 2014-04-01 15:54 (UCT), posted by SE-user Matthew

Comments

Tip: You might get better/more focused/useful answers if you disclose what literature you are reading.

This post imported from StackExchange Physics at 2014-04-01 15:54 (UCT), posted by SE-user Qmechanic

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Maybe closed strings on spacetimes with compactified dimensions qualify. See e.g any introductory string theory text that makes mention of T-duality.

This post imported from StackExchange Physics at 2014-04-01 15:54 (UCT), posted by SE-user alexarvanitakis

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@Qmechanic thanks for the suggestion, I didn't have any specific literature in mind; I guess I just meant literature in the broader sense.

This post imported from StackExchange Physics at 2014-04-01 15:54 (UCT), posted by SE-user Matthew

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@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.

This post imported from StackExchange Physics at 2014-04-01 15:54 (UCT), posted by SE-user Matthew

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Oh right, the target space is supposed to be a Riemann surface. Sorry. This should be interesting then...

This post imported from StackExchange Physics at 2014-04-01 15:54 (UCT), posted by SE-user alexarvanitakis

Answer 1

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.

(This is not really an answer but an approach to the question.)

This post imported from StackExchange Physics at 2014-04-01 15:54 (UCT), posted by SE-user suresh

Answer 2

If C and C' are two Riemann surfaces, then a non-constant holomorphic function f : C' -> C is necessarely a ramified covering. This means that the study of two-dimensional sigma models of target a Riemann surface C is related to the study of (ramified) coverings of C.

Example of physical interest: If G is a connected Lie group, usual Chern-Simons theory is a QFT of gauge group G on a 3-dimensional M. If M is divided in two pieces by a Riemann surface C, then the classical phase space of the theory is given by the moduli space of flat G-bundles on C. In http://math.ucr.edu/home/baez/qg-winter2005/group.pdf, Dijkgraaf and Witten replace G by a finite group to obtain a "Chern-Simons theory with finite gauge group" on M. If G is finite group, a flat G-bundle on C is the same thing than a covering of C of group G. In this way, one is naturally driven to study some two-dimensional TQFTs which appears as sigma models of target C.

Something related: in http://arxiv.org/abs/math/0411037, Bryan and Pandharipande have solved some of these two-dimensional TQFT. This work shows the relevant of sigma-models of target a Riemann surface to a maybe more physical theory: a sigma-model of target a Calabi-Yau 3-fold X.If X contains a Riemann surface C which is isolated, that means which can not be deformed in X, then the holomorphic world-sheet instantons of the sigma-model of target X (whose counts are the Gromov-Witten invariants of X) localize around C to the holomorphic world-sheet instantons of the sigma model of target C. To understand this sigma-model is a first step in the study of the sigma model of target X.