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A note on the shifted Courant-Nijenhuis torsion

Marco Aldi, Sergio Da Silva, Daniele Grandini

TL;DR

The paper identifies the shifted Courant-Nijenhuis torsion $\mathcal{S}_J$ as the canonical tensorial integrability condition for skew-symmetric endomorphisms of the generalized tangent bundle $\mathbb{T}M$, and proves that the entire class of tensorial polynomial expressions in three variables is generated by the shifted Nijenhuis polynomial $S(x,y,z)=(x+y)(y+z)(z+x)$. Using Kosmann-Schwarzbach's polynomial framework and real algebraic geometry (Real Nullstellensatz), it shows $\mathcal{I}=\langle S\rangle$, i.e., any tensorial condition is a multiple of $S$ and $S\bullet_J\tau_C$ encapsulates integrability. The work further develops that the same tensoriality phenomenon has natural analogues for symmetric endomorphisms (with generator $S'=(x-y)(y-z)(x-z)$) and extends to broader algebroid settings (Proto-Courant algebroids), though the symmetric/Riemannian case can yield only trivial analogues in some instances. Overall, the results provide a precise, algebraic globalization of tensorial integrability for generalized geometric structures and clarify the limits of tensoriality-driven integrability conditions.

Abstract

We characterize the vanishing of the shifted Courant-Nijenhuis torsion as the strongest tensorial integrability condition that can be imposed on a skew-symmetric endomorphism of the generalized tangent bundle.

A note on the shifted Courant-Nijenhuis torsion

TL;DR

The paper identifies the shifted Courant-Nijenhuis torsion as the canonical tensorial integrability condition for skew-symmetric endomorphisms of the generalized tangent bundle , and proves that the entire class of tensorial polynomial expressions in three variables is generated by the shifted Nijenhuis polynomial . Using Kosmann-Schwarzbach's polynomial framework and real algebraic geometry (Real Nullstellensatz), it shows , i.e., any tensorial condition is a multiple of and encapsulates integrability. The work further develops that the same tensoriality phenomenon has natural analogues for symmetric endomorphisms (with generator ) and extends to broader algebroid settings (Proto-Courant algebroids), though the symmetric/Riemannian case can yield only trivial analogues in some instances. Overall, the results provide a precise, algebraic globalization of tensorial integrability for generalized geometric structures and clarify the limits of tensoriality-driven integrability conditions.

Abstract

We characterize the vanishing of the shifted Courant-Nijenhuis torsion as the strongest tensorial integrability condition that can be imposed on a skew-symmetric endomorphism of the generalized tangent bundle.
Paper Structure (6 sections, 8 theorems, 29 equations)

This paper contains 6 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Main Result
  4. Generalizations
  5. Symmetric endomorphisms
  6. More general algebroids

Key Result

Proposition 7

Let $P\in \mathbb{R}[x,y,z]$, and let $J:\mathbb{T} M\rightarrow \mathbb{T} M$ be skew. The pair $(P,J)$ is tensorial if and only if the following conditions are satisfied: for all $\mathbf{x},\mathbf{y},\mathbf{z}\in \Gamma(\mathbb{T} M)$ and $f\in C^\infty(M)$, where $P(x,y,z)=\sum_{i,j,k}a_{i,j,k}x^iy^jz^k$.

Theorems & Definitions (23)

  • Definition 1
  • Remark 2
  • Example 3
  • Definition 4
  • Example 5
  • Example 6
  • Proposition 7
  • proof
  • Theorem 8
  • proof
  • ...and 13 more