Calabi-Yau categories are abundant in physics, geometry, and algebra. In some sense, they can be interpreted as non-commutative spaces with shifted symplectic structure. I will discuss a « counting » (or « prequantization ») functor from a category of spans of Calabi-Yau categories of fixed odd dimension to the category of vector spaces. The key ingredient is a replacement of the notion of homotopy cardinality, well-suited for the setting of even-dimensional Calabi–Yau categories and their relative generalizations. This includes cases where the usual definition does not apply, such as Z/2-graded dg categories. As a first application, this allows us to define a version of Hall algebras for odd-dimensional Calabi-Yau categories. I will explain its relation to some previously known constructions and old problems. If time permits, I will also briefly present an application of the 2-categorical version of the main construction in the context of invariants of smooth and graded Legendrian links, where we prove a conjecture of  Ng-Rutherford-Shende-Sivek relating ruling polynomials with augmentation categories. The talk is based on joint work with Fabian Haiden, arxiv:2409.10154.

https://indico.math.cnrs.fr/event/15718/