Cubic Action in Double Field Theory

We study Double Field Theory on a 2d-dimensional doubled torus with $$N_L+N_R=2$$ N L + N R = 2 , where $$N_L$$ N L and $$N_R$$ N R are the numbers of left- and right-moving oscillators. The massive states, $$N_L\ne N_R$$ N L ≠ N R , provide the momentum and winding numbers simultaneously. The fields of Double Field Theory need to satisfy a constraint imposed on the string states. In the target space, we provide a unique constraint up to the cubic order compatible with the integration by part. We make a correspondence of fields between the massless and massive cases. We then use the gauge symmetry to build the action. For the quadratic order, the mass term at the order of $$1/\alpha ^{\prime }$$ 1 / α ′ appears when $$N_L\ne N_R$$ N L ≠ N R . We can also introduce the additional interacting term to construct the gauge-invariant cubic action. Since the massive states do not follow a consistent truncation, a consistent theory possibly cannot appear from the states of $$N_L+N_R=2$$ N L + N R = 2 . We show that the expectation is wrong up to the cubic order.