Higher Haantjes Brackets and Integrability

Integrability of Lie Brackets

In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-mode...

Integrability of Poisson Brackets

We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas. As an application ...

Lectures on Integrability of Lie Brackets

Compatibility, multi-brackets and integrability of systems of PDEs

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher JacobiMayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer δ-cohomology of generalized complete intersections and eval...

Brackets, Sigma Models and Integrability of Generalized Complex Structures

It is shown how derived brackets naturally arise in sigma-models via Poissonor antibracket, generalizing a recent observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression for the derived bracket is obtained. The generalized Nijenhuis tensor of generalized complex geometry is shown to coincide up to a de-Rham closed term with the...