Singular Euclidean Structures and Riemann Surfaces

Abstract

As we have seen in Chap. 1, a Euclidean triangulated surface (T _l , M) characterizes a polyhedral metric with conical singularities associated with the vertices of the triangulation. In this chapter we show that around any such a vertex we can introduce complex coordinates in terms of which we can write down the conformal conical metric, locally parametrizing the singular structure of (T _l , M). This makes available a powerful dictionary between 2-dimensional triangulations and complex geometry. It must be noted that, both in the mathematical and in the physical applications of the theory, the connection between Riemann surfaces and triangulations typically emphasizes only the role of ribbon graphs and of the associated metric. The conical geometry of the polyhedral surface (T _l , M) is left aside and seems to play no significant a role. Whereas this attitude can be motivated by the observation (due to M. Troyanov, On the moduli space of singular Euclidean surfaces. In: Papadopoulos A (ed) Handbook of teichmuller theory, vol I. IRMA lectures in mathematics and theoretical physics, vol 11. European Mathematical Society (EMS), Zurich, pp 507–540, arXiv:math/0702666v2 [math.DG], 2007) that the conformal structure does not see the conical singularities (see below, for details), it gives a narrow perspective of the much wider role that the theory has to offer. Thus, it is more appropriate to connect a polyhedral surface (T _l , M) to a corresponding Riemann surface by taking fully into account the conical geometry of (T _l , M). This connection is many–faceted and exploits a vast repertoire of notion ranging from complex function theory to algebraic geometry. We start by defining the barycentrically dual polytope (P _T , M) associated with a polyhedral surface (T _l , M) and discuss the geometry of the corresponding ribbon graph. Then, by adapting to our case the elegant approach in Mulase and Penkava (Asian J Math 2(4):875–920, 1998. math-ph/9811024 v2), we explicitly construct the Riemann surface associated with (P _T , M). This directly bring us to the analysis of Troyanov’s singular Euclidean structures and to the construction of the bijective map between the moduli space $\mathfrak {M}_{g,N_0}$ of Riemann surfaces (M, N ₀) with N ₀ marked points, decorated with conical angles, and the space of polyhedral structures. In particular the first Chern class of the line bundles naturally defined over $\mathfrak {M}_{g,N_0}$ by the cotangent space at the i-th marked point is related with the corresponding Euler class of the circle bundles over the space of polyhedral surfaces defined by the conical cotangent spaces at the i-th vertex of (T _l , M). Whereas this is not an unexpected connection, the analogy with Witten–Kontsevich theory being obvious, we stress that the conical geometry adds to this property the possibility of a deep and explicit characterization of the Weil–Petersson form in terms of the edge–lengths of (T _l , M). This result (Carfora et al., JHEP 0612:017, 2006. arXiv:hep-th/0607146) is obtained by a subtle interplay between the geometry of (T _l , M) and 3-dimensional hyperbolic geometry, and it will be discussed in detail in Chap. 3 since it explicitly hints to the connection between polyhedral surfaces and quantum geometry in higher dimensions.