The derived moduli stack of shifted symplectic structures

Bach, Samuel; Melani, Valerio

Rendiconti del Seminario Matematico della Università di Padova, Tome 141 (2019), pp. 221-241.

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 in [8]. In particular, under reasonable assumptions on $X$, we prove that $\mathrm{𝐒𝐲𝐦𝐩}(X,n)$ 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].

Classification : 14, 53
Mots-clés : Derived algebraic geometry, moduli spaces, Lie algebroids, shifted symplectic structures