Exploring $\beta$-symmetry in heterotic theory at order ${\alpha }^{\mathrm{\prime }}$ with boundary term

This paper investigates the $\beta$-symmetry of the heterotic string theory at order ${\alpha }^{\mathrm{\prime }}$ in the context of open spacetime manifolds. Our analysis reveals that the parity-odd component of the effective action at this order remains invariant under $\beta$-transformations. Furthermore, we demonstrate that the corresponding $\beta$-transformations leave the Gibbons-Hawking boundary term invariant.