3 mar 2016 -- 15:00
Sala riunioni, DipMat, Università di Salerno
Abstract.
Lie algebroids are vector bundles with a little extra structure that allows us to think of them as a generalisation of the tangent bundle of a manifold. A key example of a Lie algebroid is the cotangent bundle of a Poisson manifold. The mantra is 'whatever you can to on a tangent bundle you can do on a Lie algebroid'. Interestingly, this mantra includes Riemannian geometry. The notion of a metric on a vector bundle is standard, but it is the structure of a Lie algebroid that allows the theory to develop parallel to the classical case. This opens up the possibility of sigma models with Riemannian Lie algebroids as their target spaces (1). These ideas have already been used to describe novel gravity theories on Poisson manifolds (2). Although this talk will primarily be about mathematical ideas, I will also suggest possible physical applications along the way.
(1) Andrew James Bruce, Killing sections and sigma models with Lie algebroid targets, to appear in Reports on Mathematical Physics.
(2) Tsuguhiko Asakawa, Hisayoshi Muraki, Satoshi Watamura, Gravity theory on Poisson manifold with R-flux, Fortschritte der Physik, vol. 63, issue 11-12, pp. (2015) 683-704.