A Non-Relativistic Limit for Heterotic Supergravity and its Gauge Lagrangian

Motivated by the recent construction of non-relativistic (NR) heterotic Double Field Theory (HDFT), we analyze the $D=10$ bosonic sector of heterotic supergravity under a consistent NR limit. We show that the resulting theory admits a finite Lagrangian due to non-trivial cancellations of divergent contributions arising from the Chern-Simons terms in the curvature of the $\hat B$-field and from the Yang-Mills sector. This mechanism parallels the well-known cancellation between the Ricci scalar $\hat R$ and the $-\frac{1}{12}\hat H^2$ term in bosonic supergravity under the same limit. Extending previous analyses, we incorporate the full bosonic gauge sector emerging from the HDFT construction and derive the complete finite bosonic heterotic Lagrangian in manifestly gauge-covariant form, written in terms of gauge-covariant curvatures and derivatives. An interesting feature of the NR expansion is that the gauged Green-Schwarz transformation of the two-form trivializes (it can be eliminated by imposing field redefinitions), while terms equivalent to Chern-Simons contributions naturally re-emerge in the effective theory.