# Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects

## TL;DR

This paper demonstrates how Calabi-Yau structures on dg categories induce symplectic forms on moduli spaces and how relative structures lead to Lagrangian structures, advancing the understanding of geometric structures in derived categories.

## Contribution

It establishes a connection between Calabi-Yau structures and symplectic and Lagrangian geometry on moduli spaces of objects in dg categories.

## Key findings

- Calabi-Yau structures induce symplectic forms of degree 2-d on moduli spaces.
- Relative Calabi-Yau structures induce Lagrangian structures on moduli space maps.
- The results extend the geometric understanding of dg categories and their moduli.

## Abstract

We show that a Calabi-Yau structure of dimension $d$ on a smooth dg category $C$ induces a symplectic form of degree $2-d$ on the moduli space of objects $M_{C}$. We show moreover that a relative Calabi-Yau structure on a dg functor $C \to D$ compatible with the absolute Calabi-Yau structure on $C$ induces a Lagrangian structure on the corresponding map of moduli $M_{D} \to M_{C}$.