Higher derivative heterotic supergravity on a torus and supersymmetry

Ignoring ten-dimensional heterotic gauge fields, heterotic supergravity reduced on a $d$-dimensional torus gives rise to a half-maximal supergravity coupled to $d$ vector multiplets. The reduced theory has a continuous $O(d,d;\mathbb{R})\mathrm{/}O(d{)}_{-}\times O(d{)}_{+}$ symmetry that persists to all perturbative orders in the string ${\alpha }^{\mathrm{\prime }}$ expansion. We highlight this symmetry by explicitly reducing the bosonic sector of four-derivative heterotic supergravity as well as its fermionic supersymmetry variations. After appropriate field redefinitions, the resulting action and supersymmetry variations are manifestly $O(d{)}_{-}\times O(d{)}_{+}$ invariant. This reduction allows us to explore the interplay between the gravity and vector multiplets beyond leading order, where (in our conventions) $O(d{)}_{-}$ is the supergravity R-symmetry while $O(d{)}_{+}$ is a flavor symmetry of the $d$ vector multiplets.