Operads of moduli spaces of points in \(\mathbb C^d\) (Q2845452)

Let \(\mathrm{TH}_{d,n}\) be the variety of configurations of \(n\) points in affine \(d\)-space modulo the action of the affine group, and \(\mathrm T_{d,n}\) its compactification. In dimension \(d=1\), these varieties return the moduli spaces of points in \(\mathbb P^1\): \(\mathrm{TH}_{1,n}=\mathcal M_{0,n+1}\) and \(\mathrm T_{1,n}\) is the Deligne-Mumford compactification \(\overline{\mathcal M}_{0,n+1}\). Write \(H_*(\mathrm{TH}_d)\) for the operad whose \(n^{\mathrm{th}}\) term is \(\Sigma H_*(\mathrm{TH}_{d,n})\). Let \(\text{Grav}_d\) be the operad of graded \(\mathbb Z\)-modules generated by \(k\)-ary operations. If \(d=1\), this operad is the gravity operad introduced by \textit{E. Getzler} [Commun. Math. Phys. 163, No. 3, 473--489 (1994; Zbl 0806.53073)], where it was shown to be isomorphic to \(\Sigma H_*(\mathcal M_{0,n+1})\). In the paper under review, the author extends this result to the higher-dimensional setting: There is an isomorphism of operads \(H_*(\mathrm{TH}_{d})\cong \text{Grav}_d\) in arity \(n>1\). As an application, the author shows that if \(X=\Omega^{2d}Y\) for an \(SO(2d)\)-space \(Y\), then the shifted equivariant homology \(\Sigma H^{S^1}_*(X)\) is an algebra over the suboperad \((\text{Grav}_d)_{>1}\) of arity \(>1\).