G}}}$, viewed as the GIT quotient $(W\times E)/\!\!/{{\bf G}}$, in [*all genera*]{}, see Example 2.8.5 and §7.2.2 in [@CK0]. If the base ${W/\!\!/{{\bf G}}}$ is projective, one can specialize the ($\beta\neq 0$) invariants at $\lambda =0$. The typical examples we have in mind here occur as follows: take a projective Fano triple $(W,{{\bf G}},\theta)$, with $W$ a vector space, and take $E=\det (W)^\vee$. These are [*local Calabi-Yau targets*]{}, i.e., the total space of the canonical bundle of a Fano GIT quotient.

Unless specified otherwise, whenever we talk about twisting by $E$ in this paper, it should be understood as twisting by the Euler class.

Genus zero theory of semi-positive targets
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Recall from [@CK0] that a triple $(W,{{\bf G}},\theta)$ is called [*semi-positive*]{} if $$\beta(\det(T_W))\geq 0$$ for every $\theta$-effective class $\beta\in {\mathrm {Eff}}(W,{{\bf G}},\theta)$.

The $\ke$-wall-crossing for genus zero invariants with descendant insertions at one point and any number of primary insertions is treated in detail in [@CK0] . In this section we recall first what those results say for semi-positive targets, then we discuss the extension to the full genus zero descendant theories in that case.

From now on, we will assume that $W$ has an action by a torus 