[Paper Review] Nijenhuis infinity and contractible dg manifolds

Sergei Merkulov|ArXiv.org|Mar 15, 2004

Homotopy and Cohomology in Algebraic Topology5 references4 citations

TL;DR

This paper introduces a minimal differential graded (dg) operad $N_{ u}$ whose generic representations in $\mathbb{R}^n$ classify formal germs of Nijenhuis tensors—endomorphisms of the tangent bundle satisfying the Nijenhuis integrability condition. The operad arises as the cobar construction of the quadratic operad of homologically trivial dg Lie algebras, and the paper establishes that minimal $N_{ u}$-algebras are homotopy equivalent to contractible dg manifolds, providing a strong homotopy generalization of Nijenhuis geometry.

ABSTRACT

We find a minimal differential graded (dg) operad whose generic representations in $R^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to $R^n$ which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar construction on the quadratic operad of homologically trivial dg Lie algebras. As a by product we obtain a strong homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifold.

Motivation & Objective

• To develop a strong homotopy generalization of Nijenhuis geometry using operadic methods.
• To identify the minimal dg operad governing formal germs of Nijenhuis tensors on $\mathbb{R}^n$.
• To establish a correspondence between $N_\infty$-algebras and contractible dg manifolds.
• To extend the framework of deformation theory to include geometric structures like complex and Nijenhuis structures via minimal resolutions of operads.

Proposed method

• Construct the minimal dg operad $N_{\infty}$ as the cobar construction on the quadratic operad of homologically trivial dg Lie algebras.
• Represent Nijenhuis structures as infinite collections of multilinear operations $J_k: \otimes^{k+1}V \to V$ satisfying quadratic relations.
• Define $N_{\infty}$-structures via a differential form $\Gamma$ on the formal neighborhood $\hat{V}$ satisfying $\mathcal{L}_{\eth}\Gamma + \frac{1}{2}[\Gamma \bullet \Gamma] = 0$.
• Use the cobar construction to derive the homological vector field $\hat{\eth} = \Psi(\eth + \Gamma)$ on $\mathcal{M} = \hat{V} \oplus \hat{V}[-1]$, which commutes with the standard differential $d$.
• Establish a one-to-one correspondence between $N_{\infty}$-algebras and homological vector fields on $\mathcal{M}$ commuting with $d$, leading to the contractibility result.
• Apply Kontsevich's classification of $L_\infty$-algebras to show that minimal $N_{\infty}$-algebras are $L_\infty$-isomorphic to linear contractible dg manifolds.

Experimental results

Research questions

• RQ1 What is the minimal dg operad whose generic representations classify formal germs of Nijenhuis tensors on $\mathbb{R}^n$?
• RQ2 How can the Nijenhuis integrability condition be generalized to a strong homotopy structure via operadic resolution?
• RQ3 What is the relationship between $N_{\infty}$-algebras and the homotopy theory of dg manifolds?
• RQ4 Is there a canonical correspondence between $N_{\infty}$-structures and contractible dg manifolds?
• RQ5 Can the deformation theory of Nijenhuis structures be fully captured by a minimal resolution of an operad?

Key findings

• The minimal dg operad $N_{\infty}$ is constructed as the cobar construction on the operad of homologically trivial dg Lie algebras.
• Generic representations of $N_{\infty}$ in $\mathbb{R}^n$ are in one-to-one correspondence with formal germs of Nijenhuis tensors satisfying $N_J = 0$.
• The $N_{\infty}$-structure is encoded by a differential form $\Gamma$ on $\hat{V}$ satisfying $\mathcal{L}_{\eth}\Gamma + \frac{1}{2}[\Gamma \bullet \Gamma] = 0$, generalizing the Nijenhuis condition.
• There is a one-to-one correspondence between $N_{\infty}$-algebras and homological vector fields on $\mathcal{M} = \hat{V} \oplus \hat{V}[-1]$ that commute with the standard differential $d$.
• Minimal $N_{\infty}$-algebras are $L_\infty$-isomorphic to linear contractible dg manifolds, establishing a strong homotopy equivalence.
• Any $N_{\infty}$-algebra is homotopy equivalent to a contractible dg manifold, showing that the homotopy theory of $N_{\infty}$-algebras is equivalent to that of contractible dg manifolds.