Hilbert series for contractads and modular compactifications

TL;DR

This paper studies the Hilbert series of contractads related to configuration spaces, providing explicit formulas, exploring their algebraic properties, and connecting them to modular compactifications and known operads.

Contribution

It introduces functional equations for Hilbert series of Koszul dual contractads and describes their structure and homology in relation to modular compactifications.

Findings

Explicit Hilbert series for fundamental contractads.

Demonstration that certain compactifications coincide with Smyth's modular compactifications.

Homology of the contractads is quadratic and Koszul.

Abstract

Contractads are operadic-type algebraic structures well-suited for describing configuration spaces indexed by a simple connected graph $\Gamma$. Specifically, these configuration spaces are defined as $\mathrm{Conf}_{\Gamma}(X):=X^{|V(\Gamma)|}\setminus \cup_{(ij)\in E(\Gamma)} \{x_i=x_j\}$. In this paper, we explore functional equations for the Hilbert series of Koszul dual contractads and provide explicit Hilbert series for fundamental contractads such as the commutative, Lie, associative and the little discs contractads. Additionally, we focus on a particular contractad derived from the wonderful compactifications of $\mathrm{Conf}_{\Gamma}(\mathbb{k})$, for $\mathbb{k}=\mathbb{R},\mathbb{C}$. First, we demonstrate that for complete multipartite graphs, the associated wonderful compactifications coincide with the modular compactifications introduced by Smyth. Second, we establish that the homology of the complex points and the homology of the real locus of the wonderful contractad are both quadratic and Koszul contractads. We offer a detailed description of generators and relations, extending the concepts of the Hypercommutative operad and cacti operads, respectively. Furthermore, using the functional equations for the Hilbert series, we describe the corresponding Hilbert series for the homology of modular compactifications.