Trees as the building blocks of quantum geometry


In the past decades two-dimensional quantum gravity (the path integral over metrics on surfaces) has been studied from many perspectives, including Liouville conformal field theory, matrix models, topological field theory, and lattice regularizations, and lots of intricate connections between these approaches have been drawn. In recent years mathematicians have uncovered a new common thread in the form of treelike structures embedded in surfaces. I will review the appearance of trees in both the lattice and Liouville theory approach and their importance in establishing fractal properties of 2d quantum geometry. Then I will demonstrate that similar trees are hidden in Witten’s topological gravity, providing an elementary route to evaluating the corresponding path integrals. Finally I will speculate on the relevance of trees to quantum gravity in higher dimensions.

Nikhef, Amsterdam