Leibniz algebroids, generalized Bismut connections and Einstein-Hilbert actions

'Anti-Commutable' Pre-Leibniz Algebroids and Admissible Connections

The concept of algebroid is convenient for constructions of geometrical frameworks. For example, metric-affine and generalized geometries can be written on Lie and Courant algebroids, respectively. Furthermore, string theories might make use of many other algebroids such as metric algebroids, higher-Courant algebroids, or conformal Courant algebroids. Working on the possibly most general algebroid structure is fruitful as it creates a chance to study all of them at once. Local pre-Leibniz algebroids are such general ones in which metric-connection geometries are possible to construct. Yet, the…

Metric-Connection Geometries on Pre-Leibniz Algebroids: A Search for Geometrical Structure in String Models

The metric-affine and generalized geometries, respectively, are arguably the appropriate mathematical frameworks for Einstein's theory of gravity and the low-energy effective massless oriented closed bosonic string field theory. In fact, mathematical structures in a metric-affine geometry are written on the tangent bundle, which is itself a Lie algebroid; whereas those in generalized geometries introduced as the basis of double field theories, are written on Courant algebroids. The Lie, Courant and the higher Courant algebroids used in exceptional field theories, are all special cases of pre-L…

Geometry of Membrane Sigma Models

String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or p-branes) in a curved spacetime. On the classical field theory level, they are described by an action functional extremalising the volume of a manifold swept by a propagating membrane. This and related field theories are collectively called membrane sigma models. Differential geometry is an important mathematical tool in the study of string theory. It turns…

Exceptional algebroids and type IIB superstrings

In this note we study exceptional algebroids, focusing on their relation to type IIB superstring theory. We show that a IIB-exact exceptional algebroid (corresponding to the group $E_{n(n)}\times \mathbb{R}^+$, for $n\le 6$) locally has a standard form given by the exceptional tangent bundle. We derive possible twists, given by a flat $\mathfrak{gl}(2,\mathbb{R})$-connection, a covariantly closed pair of 3-forms, and a 5-form, and comment on their physical interpretation. Using this analysis we reduce the search for Leibniz parallelisable spaces, and hence maximally supersymmetric consistent t…

例外代数胚与IIA型超弦

Exceptional algebroids and type IIA superstrings

本文研究IIA型弦论翘曲紧化至n≤6维时的例外代数胚，发现其非恰当且局部模空间含5个点，关联两种标量形变，还可通过多形式扭曲括号，并将Leibniz平行化空间分类转化为代数问题，讨论了相关对偶性及例子。