The quantum geometry of polyhedral surfaces: Variations on strings and all that

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 MgIN0 , and allows one to prove that
the partition function of 2D quantum gravity is directly related to computation of
theWeil–Petersson volume ofMgIN0 . 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 MgIN0 . 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.