The Quantum Geometry of Polyhedral Surfaces: Variations on Strings and All That

This chapter is an introduction to the basic ideas of 2-dimensional quantum field theory and non-critical strings. This is classic material which nevertheless proves useful for illustrating the interplay between quantum field theory, the moduli space of Riemann surfaces, and the properties of polyhedral surfaces which are the leitmotiv of this LNP. At the root of this interplay lies 2D quantum gravity. It is well known that such a theory allows for two complementary descriptions: on the one hand we have a conformal field theory (CFT) living on a 2D world-sheet, a description that emphasizes the geometrical aspects of the Riemann surface associated with the world-sheet; while on the other, the theory can be formulated as a statistical critical field theory over the space of polyhedral surfaces (dynamical triangulations). We show that many properties of such 2D quantum gravity models are related to a geometrical mechanism which allows one to describe a polyhedral surface with N0 vertices as a Riemann surface with N0 punctures dressed with a field whose charges describe discretized curvatures (related to the deficit angles of the triangulation). Such a picture calls into play the (compactified) moduli space of genus g Riemann surfaces with N0 punctures \(\mathfrak {M}_{g;N_0}\), and allows one to prove that the partition function of 2D quantum gravity is directly related to computation of the Weil–Petersson volume of \(\mathfrak {M}_{g;N_0}\). By exploiting the large N0 asymptotics of such Weil–Petersson volumes, characterized by Manin and Zograf, it is then easy to relate the anomalous scaling properties of pure 2D quantum gravity, the KPZ exponent, to the Weil–Petersson volume of \(\mathfrak {M}_{g;N_0}\). We also show that polyhedral surfaces provide a natural kinematical framework within which we can discuss open/closed string duality. A basic problem in such a setting is to provide an explanation of how open/closed duality is generated dynamically, and in particular how a closed surface is related to a corresponding open surface, with gauge-decorated boundaries, in such a way that the quantization of this correspondence leads to an open/closed duality. In particular, we show that from a closed polyhedral surface we naturally get an open hyperbolic surface with geodesic boundaries. This gives a geometrical mechanism describing the transition between closed and open surfaces. Such a correspondence is promoted to the corresponding moduli spaces: \(\mathfrak {M}_{g;N_0}\times \mathbb {R}_{+}^{N}\), the moduli spaces of N0-pointed closed Riemann surfaces of genus g whose marked points are decorated with the given set of conical angles, and \(\mathfrak {M}_{g;N_0}(L)\times \mathbb {R}_{+}^{N_0}\), the moduli spaces of open Riemann surfaces of genus g with N0 geodesic boundaries decorated by the corresponding lengths. Such a correspondence provides a nice kinematical setup for establishing an open/closed string duality, by exploiting the results by M. Mirzakhani on the relation between intersection theory over \(\mathfrak {M}(g;N_0)\) and the geometry of hyperbolic surfaces with geodesic boundaries. The results in this chapter connect directly with many deep issues in 3D geometry, ultimately relating to the volume conjecture in hyperbolic geometry and to the role of knot invariants.