Cubic Action in Double Field Theory

TL;DR

This paper develops a gauge-invariant cubic action in Double Field Theory for massive states on a torus, revealing that momentum and winding modes cannot coexist in a single compact doubled space.

Contribution

It provides the first derivation of a gauge-invariant cubic action for massive states in Double Field Theory, including the unique constraints for $N_L eq N_R$ and the analysis of mode truncation.

Findings

Massive states require a gauge-invariant cubic action with specific couplings.

Momentum and winding modes cannot both appear in a single compact doubled space.

Naive truncation of modes does not yield a consistent theory at cubic order.

Abstract

We study target space theory on a torus for the states with $N_L+N_R=2$ through Double Field Theory. The spin-two Fierz-Pauli fields are not allowed when all spatial dimensions are non-compact. The massive states provide both non-vanishing momentum and winding numbers in the target space theory. To derive the cubic action, we provide the unique constraint for $N_L\neq N_R$ compatible with the integration by part. We first make a correspondence of massive and massless fields. The quadratic action is gauge invariant by introducing the mass term. We then proceed to the cubic order. The cubic action is also gauge invariant by introducing the coupling between the one-form field and other fields. The massive states do not follow the consistent truncation. One should expect the self-consistent theory by summing over infinite modes. Hence the naive expectation is wrong up to the cubic order. In the end, we show that the momentum and winding modes cannot both appear for only one compact doubled space.