Non-formality of Voronov's Swiss-Cheese operads

TL;DR

The paper proves that Voronov’s sub-operad SC^{\mathrm{vor}}_{n+1} is not formal over any field with characteristic not equal to 2, for all $n\ge 1$. It adapts Livernet’s Massey-product approach to a cubical operad framework, building a hierarchy of obstructions starting from dimension 2 and inductively constructing higher arity obstructions in $(2,2^n)$. The authors define explicit chains $\beta_n$, $\eta_n$, and $\gamma_n$, along with $\mu_n$, $\alpha_n$, and $\ell_n$, to realize nontrivial homology classes that cannot be killed by any quasi-isomorphism, thus establishing non-formality. The work uses a relative two-colored operad formalism and a cubical $\omega$-groupoid perspective (with a Schwede–Shipley bridge) to produce a small, combinatorial obstruction theory that applies in all dimensions, impacting deformation-quantization models and rational homotopy models for Swiss-Cheese-type operads.

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).

Figures (7)

• Figure 1: The path used in Livernet2015 for $n=2$. The hatched square is an open input.
• Figure 2: Examples of thin squares obtained from a one-dimensional segment.
• Figure 3: Operadic composition in $\mathcal{C}_2$.
• Figure 4: The three inclusions $\mathcal{C}_2 \to \mathcal{C}_3$.
• Figure 5: The 1-chain ${ \eta ^{} _{1} } \in C_1 \mathcal{SC}^{\mathrm{vor}}_2(2, \pm)$.

Theorems & Definitions (72)

• Theorem A
• Remark
• Remark
• remark 1.1
• definition 1.2
• remark 1.6
• definition 1.7
• definition 1.8: BrownHigginsSivera2011
• example 1.9
• definition 1.12: BrownHigginsSivera2011