Log Geometric Models for Little Disks Operads in Even Dimensions
Oliver Lindström
TL;DR
This work extends log-geometric modeling of framed configuration spaces from dimension 2 to arbitrary even dimensions by constructing a pseudo-operad $\textnormal{CGK}^{\log}_d$ in Deligne–Faltings log schemes whose KN analytification is $\textnormal{FM}_{2d} \rtimes S^1$, with underlying moduli spaces $T_{d,n}$ generalizing $\overline{\mathcal{M}}_{0,n+1}$. It further introduces a log-geometric operad $\textnormal{CGK}^{\text{V-log}}_d$ whose KN analytification recovers $\textnormal{FM}_{2d}$, and shows how the constructions relate to the algebraic Fulton–MacPherson spaces, rooted-tree moduli, and their functorial descriptions. The paper also discusses cohomological and Hodge-theoretic implications, including Galois actions on étale cooperads and mixed Hodge structures, while noting purity constraints that complicate formality in the framed case for $d\ge 2$, contrasted with the unframed little disks which are formal in all dimensions. Collectively, these results suggest a motivic and algebro-geometric underpinning for framed configuration spaces in even dimensions and pave the way for further formality and computational investigations.
Abstract
We construct a model for the (non-unital) S^1-framed little 2d-dimensional disks operad for any positive integer d using logarithmic geometry. We also show that the unframed little 2d-dimensional disks operad has a model which can be constructed using log schemes with virtual morphisms.