Short Courses and Seminars - Summer 2025

Abstract: The chiral de Rham complex, introduced by Malikov, Schechtman, and Vaintrob in 1999, is a canonical sheaf of vertex algebras defined on any smooth algebraic variety. This construction refines the classical de Rham complex by embedding it as the weight-zero subcomplex, and provides a rich algebro-geometric model for two-dimensional conformal field theories.
One of the remarkable features of the chiral de Rham complex is its deep connection with mirror symmetry—it is widely believed to realize the A-model on a given variety. Moreover, subsequent developments have revealed that it encodes significant geometric invariants, including a particularly beautiful link with the elliptic genus, which emerges as the partition function of the associated conformal field theory.
In this series of talks, I will introduce a generalization of the chiral de Rham complex in the setting of noncommutative geometry, which we call the chiral cyclic complex. This framework unifies the classical theory with its orbifold extension, and opens the door to new examples arising in singularity theory.
In particular, we will explore how this formalism sheds light on the known relationship between the elliptic genus and the Steenbrink spectrum of isolated hypersurface singularities, offering a novel perspective on both invariants through the lens of vertex algebra geometry.