Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology
Sarah Scherotzke, Nicolò Sibilla, Paolo Tomasini
TL;DR
This work bridges two parallel approaches to elliptic Hochschild homology by introducing a Fourier--Mukai duality for one-dimensional formal groups, enabling a precise comparison between MRT's $\mathbb{G}$-Hochschild homology and Sibilla--Tomasini's elliptic Hochschild homology. It constructs a universal, $R$-linear framework $\mathrm{HH}^{\mathbb{G}}_*$ for any one-dimensional formal group $\mathbb{G}$ and proves that, when $\mathbb{G}=\widehat{E}$ for an elliptic curve $E$, the $\widehat{E}$-Hochschild homology coincides with elliptic Hochschild homology defined via mapping stacks. Using Lurie’s FM theory for dual abelian varieties, the authors relate the moduli-based elliptic data to a global picture, producing three global TMF-family Hochschild theories $\mathcal{HH}^{\mathrm{TMF}}_*$, $\mathcal{HH}^{\mathrm{Tmf}}_*$, and $\mathcal{HH}^{\mathrm{tmf}}_*$ that interpolate between elliptic, nodal, and cuspidal degenerations and recover ordinary and Hodge Hochschild homology as appropriate limits. This framework offers a universal, modular perspective on TMF-Hochschild homology and connects to the filtered and equivariant variants studied in related works, with future directions including extensions to non-affine and equivariant settings and a deeper exploration of cubics. The results establish a coherent bridge between algebraic geometry, operadic dualities, and topological modular forms in the context of Hochschild-type invariants.
Abstract
This paper establishes a unifying framework for various forms of twisted Hochschild homology by comparing two definitions of elliptic Hochschild homology: one introduced by Moulinos--Robalo--Toën and the other by Sibilla--Tomasini. Central to our approach is a new Fourier--Mukai duality for formal groups. We prove that when $\widehat{E}$ is the formal group associated to an elliptic curve $E$, the resulting $\widehat{E}$-Hochschild homology coincides with the mapping stack construction of Sibilla--Tomasini. This identification also recovers ordinary and Hodge Hochschild homology as degenerate limits corresponding to nodal and cuspidal cubics, respectively. Building on this, we introduce global versions of elliptic Hochschild homology over the moduli stacks of elliptic and cubic curves, which interpolate between these theories and suggest a universal form of TMF-Hochschild homology.