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.