# Calabi-Yau Structures, Spherical Functors, and Shifted Symplectic Structures

## TL;DR

This paper develops a categorical framework connecting Calabi-Yau structures, spherical functors, and shifted symplectic structures, with applications to symplectic and algebraic geometry, and potential insights into derived hyperk"ahler geometry.

## Contribution

It introduces a formalism linking Calabi-Yau structures and spherical functors, along with a gluing principle for constructing shifted symplectic structures on derived moduli spaces.

## Key findings

- Comparison between relative Calabi-Yau and spherical functors
- A local-to-global gluing method for Calabi-Yau structures
- Construction of shifted symplectic and Lagrangian structures on moduli spaces

## Abstract

A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed by homological mirror symmetry, derived noncommutative geometry, and the theory of Fukaya categories with coefficients in a perverse Schober. The main technical results include (i) a comparison between the notion of relative Calabi-Yau structures and a certain refinement of the notion of a spherical functor, (ii) a local-to-global gluing principle for constructing Calabi-Yau structures, and (iii) the construction of shifted symplectic structures and Lagrangian structures on certain derived moduli spaces of branes. Potential applications to a theory of derived hyperk\"ahler geometry are sketched.