A map is a 2-cell embedding of a graph in a Riemann surface. The generating
series for a class of maps is called the map series for the class. I shall discuss
two questions, one from mathematical physics and the other from algebraic geometry, where
map theory reveals the presence of deeper structure and connexions between the two.
I): The φ4-model and log(1-φ)-1-model
(due to Penner) are early
models of topological quantum field theory. The relationship between the partition
functions for these two models may be explained as a consequence of a functional
relationship between two classes of maps in orientable surfaces, one in which all vertices
have degree 4 and the other in which there is no restriction on vertex degrees.
Moreover, comparable relationships hold for other classes of maps and there is evidence
of a natural bijection accounting for these relationships.
II): The generating series for the virtual Euler characteristics for the moduli
spaces of complex and for real algebraic curves, respectively, may be shown to be
specialisations of the map series for all surfaces through an algebraic parameter
associated with Jack symmetric functions. This parameter is conjectured to have an
interpretation as an invariant of maps, which then opens the possibility of passing it
through the Strebel derivative construction used by Harer and Zagier, to the level of the
moduli spaces.
In this talk I shall show how these conjectures arose in the first place.
|