The Decoupling of $\bar{\mathrm{\Omega }}$ in String Theory

In this note, we study the deformation of the topological string by $\bar{\mathrm{\Omega }}$. Namely, adopting the perturbative string amplitudes approach, we identify the $\bar{\mathrm{\Omega }}$-deformation in terms of a physical state in the sting spectrum. We calculate the topological amplitudes $F_g$ in heterotic string theory in the presence of the latter. In particular, we show that it is crucial to include quadratic terms in the effective action in order for $\bar{\mathrm{\Omega }}$ to decouple. It turns out that this decoupling happens at the full string level, suggesting that this holds non-perturbatively.