Let $(W,{{\bf G}},\theta)$ be semi-positive. Assume that $W$ admits an action by a torus ${{\bf {T}}}$, commuting with the action of ${{\bf G}}$ and such that the fixed points of the induced ${{\bf {T}}}$-action on ${W/\!\!/{{\bf G}}}$ are isolated. Then the $g=0$ cases of Conjectures \[main conj intro\] and \[equivalent\] hold. Moreover, if $E$ is a convex ${{\bf G}}$-representation such that for all $\theta$-effective $\beta$ we have $\beta(\det(T_W))-\beta(W\times \det(E))\geq 0$, then the conjectures also hold at $g=0$ for the $E$-twisted $\ke$-quasimap theories of ${W/\!\!/{{\bf G}}}$.

The inclusion of $E$-twisting means that Theorem \[genus zero intro\] covers the compact Calabi-Yau targets which are realised as zero loci of sections of homogeneous vector bundles on toric manifolds, on flag manifolds of classical types, or on products of such. The proof of this result given in §3 below can be summarized as follows: by first extending Dubrovin’s genus zero reconstruction [@Dubrovin] to all $\ke$-theories, we reduce to the case of big $J$-functions, which was already established in [@CK0].

The second evidence is the main result of the paper.

\[Main Thm\] Let $X$ be a nonsingular quasi-projective toric variety of dimension $n$, obtained as the GIT quotient of a semi-positive triple $(\CC^{n+r}, (\CC^*)^r, \theta)$. 