The derived moduli stack of shifted symplectic structures

Abstract

We introduce and study the derived moduli stack $\mathbf{Symp}(X,n)$ of $n$-shifted symplectic structures on a given derived stack $X$, as introduced in [8]. In particular, under reasonable assumptions on $X$, we prove that $\mathbf{Symp}(X,n)$ carries a canonical quadratic form, in the sense of [14]. This generalizes a classical result of Fricke and Habermann (see [13]), which was established in the $C^{\infty }$-setting, to the broader context of derived algebraic geometry, thus proving a conjecture stated in [14].