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arXiv · 2408.11157

Higher holonomy for curved L${}_\infty$-algebras 1: simplicial methods

Ezra Getzler
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Abstract

We construct a natural morphism $ρ$ from the nerve $\text{MC}_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L)$ of a pronilpotent curved L${}_\infty$-algebra $L$ to the simplicial subset $γ_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L,s_\bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $γ_\bullet(L) \hookrightarrow \text{MC}_\bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_\infty$-algebras to curved L${}_\infty$-algebras. The morphism $ρ$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $ρ^\square$ of $ρ$ to identify $ρ$ with higher holonomy for semiabelian curved \Linf-algebras.

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Ezra Getzler. 2025-07-15. Higher holonomy for curved L${}_\infty$-algebras 1: simplicial methods. https://arxiv.org/abs/2408.11157

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