Comments on double field theory and diffeomorphisms

TL;DR

This paper discusses the nature of coordinates and diffeomorphisms in double field theory, proposing a gauge orbit perspective that ensures covariance and consistency of tensor transformations under finite reparametrizations.

Contribution

It introduces a gauge orbit-based interpretation of coordinates in double field theory and demonstrates the consistency of a tensorial transformation rule with full covariance.

Findings

Coordinate points correspond to gauge orbits, not physical points.

The tensorial transformation rule aligns with the exponential map.

Full covariance of derivatives and curvatures is maintained with proper projectors.

Abstract

As the theory is subject to a section condition, coordinates in double field theory do not represent physical points in an injective manner. We argue that a physical point should be rather one-to-one identified with a `gauge orbit' in the coordinate space. The diffeomorphism symmetry then implies an invariance under arbitrary reparametrizations of the gauge orbits. Within this generalized sense of diffeomorphism, we show that a recently proposed tensorial transformation rule for finite coordinate transformations is actually (i) consistent with the standard exponential map, and further (ii) compatible with the full covariance of the `semi-covariant' derivatives and curvatures after projectors are properly imposed.