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

We show that the $D=10$ heterotic supergravity under 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 gauge curvatures after considering a NR limit. We compute the full finite Lagrangian in its gauge covariant form, using gauge covariant curvatures and gauge covariant derivatives. These results constitute 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. Terms equivalent to Chern-Simons contributions naturally arise from the previous property.