# The derived moduli stack of shifted symplectic structures

## TL;DR

This paper studies the derived moduli stack of shifted symplectic structures on a derived stack, proving it carries a canonical shifted quadratic form, thus generalizing classical results to derived algebraic geometry.

## Contribution

It introduces the derived moduli stack of shifted symplectic structures and proves it has a canonical shifted quadratic form, confirming a conjecture in derived algebraic geometry.

## Key findings

- The derived moduli stack $ ext{Symp}(X,n)$ carries a canonical shifted quadratic form.
- Generalization of classical results from $C^{inity}$-setting to derived algebraic geometry.
- Confirmation of Vezzosi's conjecture regarding the structure of $ ext{Symp}(X,n)$.

## Abstract

We introduce and study the derived moduli stack $\mathrm{Symp}(X,n)$ of $n$-shifted symplectic structures on a given derived stack $X$, as introduced by [PTVV] (IHES Vol. 117, 2013). In particular, under reasonable assumptions on $X$, we prove that $\mathrm{Symp}(X, n)$ carries a canonical shifted quadratic form. This generalizes a classical result of Fricke and Habermann, which was established in the $C^{\infty}$-setting, to the broader context of derived algebraic geometry, thus proving a conjecture stated by Vezzosi.