Geometry of Membrane Sigma Models

TL;DR

This paper explores the mathematical structures underlying membrane sigma models in string theory, focusing on generalized geometry and Leibniz algebroids to describe membrane backgrounds in curved spacetime.

Contribution

It provides a comprehensive introduction to Leibniz algebroids, generalized geometry, and Nambu-Poisson structures, emphasizing generalized metrics and orthogonal transformations relevant to membrane models.

Findings

Review of Leibniz algebroids and generalized geometry foundations

Introduction of generalized metrics and orthogonal transformations

Integration of well-known and new results in the mathematical framework

Abstract

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 out that string and membrane backgrounds can be conveniently described using objects defined on a direct sum of tangent and cotangent bundles of the spacetime manifold. Mathematical field studying such object is called generalized geometry. Its integral part is the theory of Leibniz algebroids, vector bundles with a Leibniz algebra bracket on its module of smooth sections. Special cases of Leibniz algebroids are better known Lie and Courant algebroids. This thesis is divided into two main parts (the second one is ommited in this preprint). In the first one, we review the foundations of the theory of Leibniz algebroids, generalized geometry, extended generalized geometry, and NambuPoisson structures. The main aim is to provide the reader with a consistent introduction to the mathematics used in the published papers. The text is a combination both of well known results and new ones. We emphasize the notion of a generalized metric and of corresponding orthogonal transformations, which laid the groundwork of our research.