The gauge algebra of double field theory and Courant brackets

TL;DR

The paper addresses the symmetry algebra of double field theory on a doubled torus, showing that the gauge algebra is governed by the C-bracket, an $O(D,D)$ covariant generalization of the Courant bracket. By restricting fields to a totally null subspace $N$, the authors recover the Courant bracket on $(T\oplus T^*)N$ and connect their construction to Siegel’s momentum-only gauge algebra, while maintaining duality covariance. The analysis reveals that the Jacobi identity fails for these brackets, but the Jacobiator reduces to a trivial gauge transformation, ensuring a consistent realization of the symmetry on fields. These results bridge double field theory with generalized geometry and Courant algebroids, and clarify how duality-covariant gauge algebras can be realized despite non-Lie-bracket structures.

Abstract

We investigate the symmetry algebra of the recently proposed field theory on a doubled torus that describes closed string modes on a torus with both momentum and winding. The gauge parameters are constrained fields on the doubled space and transform as vectors under T-duality. The gauge algebra defines a T-duality covariant bracket. For the case in which the parameters and fields are T-dual to ones that have momentum but no winding, we find the gauge transformations to all orders and show that the gauge algebra reduces to one obtained by Siegel. We show that the bracket for such restricted parameters is the Courant bracket. We explain how these algebras are realised as symmetries despite the failure of the Jacobi identity.