Topics in Quantum Geometry of Riemann Surfaces: Two-Dimensional Quantum Gravity

In these lectures, we present geometric approach to the two-dimensional quantum gravity. It became popular since Polyakov’s discovery that first-quantized bosonic string propagating in IR can be described as theory of d free bosons coupled with the two-dimensional quantum gravity [1]. In critical dimension d = 26, the gravity decouples and Polyakov’s approach reproduces results obtained earlier by different methods (see, e.g., [2] for detailed discussion and references). Classically, the two-dimensional gravity is a theory formulated on a smooth oriented twodimensional surface X, endowned with a Riemannian metric ds, whose dynamical variables are metrics in the conformal class of ds. Classical equation of motion is the two-dimensional Einstein equation with a cosmological term and it describes the metric with constant Gaussian curvature. Since in two dimensions, conformal structure uniquely determines a complex structure, a surface X with the conformal class of ds has a structure of a one-dimensional complex manifold—a Riemann surface. We will consider the case when X is either compact (i.e. an algebraic curve), or it is non-compact, having finitely many branch points of infinite order. Except for few cases, the virtual Euler characteristic χ(X) of the Riemann