Polynomial Structures in Generalized Geometry

TL;DR

This paper investigates polynomial-structured skew-symmetric endomorphisms on generalized tangent bundles, exploring their compatibility with geometric operators and conditions for integrability, thus advancing understanding of generalized geometric structures.

Contribution

It introduces a framework for analyzing polynomial structures in generalized geometry and links algebraic conditions to geometric integrability criteria.

Findings

Identifies conditions for compatibility with the de Rham operator.

Establishes criteria for integrability of generalized almost complex structures.

Provides a unified approach to polynomial structures in generalized geometry.

Abstract

On the generalized tangent bundle of a smooth manifold, we study skew-symmetric endomorphism satisfying an arbitrary polynomial equation with real constant coefficients. We study the compatibility of these structures with the de Rham operator and the Courant-Dorfman bracket. In particular we isolate several conditions that when restricted to the motivating example of generalized almost complex structure are equivalent to the notion of integrability.