Tripled Structures of Algebroids in Gauged Double Field Theory

Abstract

Double field theory (DFT) is an effective theory of string theory. It has a manifest symmetry of T-duality. The gauge symmetry in DFT is related to some kind of algebroid structures, and they have a doubled structure. We focus on the gauge algebra of the O(D, D+n) gauged DFT and discuss an extension of the doubled structure. The gauge algebra of the O(D, D + n) gauged DFT has been described by the F-bracket. This bracket is related to some algebroid structures. We show that algebroids defined by the twisted C-bracket in the gauged DFT are built out of a direct sum of three Lie algebroids. They exhibit a “triple”, which we call the extended double, rather than a “double” structure. We also consider the geometrical realization of these structures in a (2D + n)-dimensional manifold.