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

We show that the $D=10$ heterotic supergravity in a non-relativistic (NR) limit has a finite Lagrangian due to non-trivial cancellations of divergent parts arising from the Chern-Simons terms in the curvature of the $\hat B$-field and the Yang-Mills Lagrangian. This is similar to what happens in bosonic supergravity between the Ricci scalar, $\hat R$, and the $- \frac{1}{12} \hat H^2$ term, under the same limit. In this work we present the explicit form of the gauge transformations and curvatures after considering a NR limit and we compute the full finite Lagrangian in its covariant form. Particularly, we address both the ungauged part and the gauged part, and we write it using covariant derivatives and curvatures for the gauge fields. These results constitutes the first computation of a finite Lagrangian for heterotic supergravity under these kinds of limits. As an interesting feature of the expansion implemented, the Green-Schwarz mechanism for the two-form can be trivialized and terms equivalent to Chern-Simons contributions naturally arise from the previous property.