Metric-connection geometries on pre-Leibniz algebroids: A search for geometrical structure in string models

Abstract

Metric-affine and generalized geometries are arguably the appropriate mathematical frameworks for Einstein’s theory of gravity and low-energy effective string field theory, respectively. In fact, mathematical structures in a metric-affine geometry are constructed on the tangent bundle, which is itself a Lie algebroid, whereas those in generalized geometries, which form the basis of double field theories, are constructed on Courant algebroids. Lie, Courant, and higher Courant algebroids, which are used in exceptional field theories, are all known to be special cases of pre-Leibniz algebroids. As mathematical structures on these algebroids are essential in string models, it is natural to work on a more unifying geometrical framework. Provided with some additional ingredients, the construction of such geometries can all be carried over to regular pre-Leibniz algebroids. We define below the notions of locality structures and locality projectors, which are some necessary ingredients. In terms of these structures, E-metric-connection geometries are constructed with (possibly) a minimum number of assumptions. Certain small gaps in the literature are also filled as we go along. E-Koszul connections, as a generalization of Levi–Cività connections, are defined and shown to be helpful for some results including a simple generalization of the fundamental theorem of Riemannian geometry. The existence and non-existence of E-Levi–Cività connections are discussed for certain cases. We also show that metric-affine geometries can be constructed in a unique way as special cases of E-metric-connection geometries. Some aspects of Lie and Lie-type algebroids are studied, where the latter are defined here as a generalization of Lie algebroids. Moreover, generalized geometries are shown to follow as special cases, and various properties of linear generalized-connections are proven in the present framework. Similarly, uniqueness of the locality projector in the case of exact Courant algebroids is proven, a result that explains why the curvature operator, defined with a projector in the double field theory literature, is a necessity.