Differential geometry with a projection: Application to double field theory

TL;DR

This paper develops a new differential geometry framework based on an O(D,D) symmetric projection, providing a mathematical foundation for double field theory and its gauge symmetries.

Contribution

It introduces a differential operator compatible with the O(D,D) projection, replacing ordinary derivatives in double field theory's gauge transformations.

Findings

Constructed gauge covariant tensors including scalars and tensors with O(D,D) indices.

Provided a covariant differential operator compatible with the O(D,D) projection.

Established a geometric structure underlying double field theory.

Abstract

In recent development of double field theory, as for the description of the massless sector of closed strings, the spacetime dimension is formally doubled, i.e. from D to D+D, and the T-duality is realized manifestly as a global O(D,D) rotation. In this paper, we conceive a differential geometry characterized by a O(D,D) symmetric projection, as the underlying mathematical structure of double field theory. We introduce a differential operator compatible with the projection, which, contracted with the projection, can be covariantized and may replace the ordinary derivatives in the generalized Lie derivative that generates the gauge symmetry of double field theory. We construct various gauge covariant tensors which include a scalar and a tensor carrying two O(D,D) vector indices.