Global Aspects of Doubled Geometry and Pre-rackoid

TL;DR

This paper explores the global geometric structures underlying double field theory, introducing pre-rackoids as a new way to integrate algebroids, with applications to topological sigma models.

Contribution

It introduces the concept of pre-rackoids as global objects for DFT algebroids and proposes realizations related to cotangent paths and formal exponential maps.

Findings

Pre-rackoids generalize rackoids in DFT.

Reduction to Courant algebroid when constraints are applied.

Application to a 3D topological sigma model.

Abstract

The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid which are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose that several realizations of pre-rackoid structures. One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we exhibit an implementation of the (pre-)rackoid in a three-dimensional topological sigma model.