Equivariant Derived Categories Associated to a Sum of Two Potentials

Suppose $f,g$ are homogeneous polynomials of degree $d$ defining smooth hypersurfaces $X_f = V(f)\subset \mathbb{P}^{m-1}$ and $X_g = V(g)\subset\mathbb{P}^{n-1}$. Then the sum $f(x)+g(y)$ defines a smooth hypersurface $X=V(f(x)+g(y))\subset\mathbb{P}^{m+n-1}$ with an action of $\mu_d$ scaling the $g$ variables. Motivated by the work of Orlov, we construct a semi-orthogonal decomposition of the derived category of coherent sheaves on $[X/\mu_d]$ provided $d\geq \mathrm{max}\{m,n\}$.