Non-formality of Voronov's Swiss-Cheese operads

Authors: N. I., Renato Vasconcellos Vieira.
Publication: In: Q. J. Math. 75.1 (2024), pp. 63-95.
Online on: Online on March 29, 2023.
Accepted on: Accepted on October 19, 2023.
Updated on: Updated on April 16, 2024.
Full text: Full text
DOI:10.1093/qmath/haad041: DOI:10.1093/qmath/haad041
arXiv:2303.16979: arXiv:2303.16979
hal-04053257: hal-04053257
MR4732946: MR4732946
Zbl 07936839: Zbl 07936839
Source: Source

Abstract

The Swiss-Cheese operads, which encode actions of algebras over the little $n$ -cubes operad on algebras over the little $(n−1)$ -cubes operad, comes in several variants. We prove that the variant in which open operations must have at least one open input is not formal in characteristic zero. This is slightly stronger than earlier results of Livernet and Willwacher. The obstruction to formality that we find lies in arity $(2,2^n)$ , rather than $(2,0)$ (Livernet) or $(4,0)$ (Willwacher).