Communications in Mathematical Physics

, Volume 246, Issue 3, pp 475–502 | Cite as

Gravity from Lie Algebroid Morphisms

  • T. StroblEmail author


Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from TΣ to any Lie algebroid E, where Σ is regarded as a d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S ij (X)A iA j , so as to permit an interpretation as a metric on Σ. This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the additional structure is identified with a Riemannian foliation of M, in the Poisson case E=T * M with a sub-Riemannian structure which is Poisson invariant with respect to its annihilator bundle. (For integrable image of S, this means that the induced Riemannian leaves should be invariant with respect to all Hamiltonian vector fields of functions which are locally constant on this foliation). This provides a huge class of new gravity models in d dimensions, embedding known 2d and 3d models as particular examples.


Manifold Sigma Model Gravity Model Hamiltonian Vector Integrable Image 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität JenaJenaGermany

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