Conformal blocks and positive cycles on the moduli space of curves

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

The moduli space of stable n-pointed curves of genus g, is endowed with vector bundles having interesting positivity properties and connections to many different areas of mathematics and mathematical physics. Over a point, the bundles are naturally described as conformal blocks, vector spaces that arise as basic objects in rational conformal field theory. The projects in this proposal are focused on three general themes, aimed at developing our understanding of vector bundles of conformal blocks bundles and their interplay with cycles on the moduli space. The first project aims to show the Chern classes of so-called "critical level bundles" are subject to ''strange identities'', given by interchanging roles of level and rank. The second project explores properties of conformal blocks divisors, including their corresponding morphisms, questions of finite generation, and applications. The third project is concerned with cones of positive cycles on varieties, illustrated for the moduli space of stable n-pointed rational curves via conformal blocks. Moduli spaces of stable curves with marked points occupy a unique and prominent position in the algebro-geometric universe. As moduli spaces, they give insight into the study of smooth curves and their degenerations, and as special varieties, they have played an important role in developing general theory. Recent work has revealed that many combinatorial aspects of the spaces are reflections of underlying geometric structures embodied by vector bundles of conformal blocks. The projects in this proposal aim both to use the features of these vector bundles to discover the nature of the moduli spaces, and also to use the architecture of the moduli spaces to reveal underlying relationships between the vector bundles.