$E$.

Each choice of a representation $E$ and of an invertible multiplicative ${\bf U}$-equivariant characteristic class $c$ determines an $(E,c)$-twisted $\ke$-quasimap CohFT, see §6.2 of [@CKM]. (We restrict here to bundles arising from representations for simplicity, but general ${{\bf G}}$-equivariant bundles on $W$ may be considered.) These generalize to arbitray stability parameter $\ke$ the twisted Gromov-Witten CohFT’s studied by Coates and Givental, [@CG].

It is shown in [@CKM], and explained again in §7.2.1 of [@CK0], that when $W$ is smooth, the representation $E$ is convex, and the class $c$ is the (equivariant) Euler class, the resulting [*genus zero*]{} twisted $\ke$-quasimap theory recovers (most of[^1]) the $\ke$-quasimap theory of the zero locus $Z/\!\!/{{\bf G}}\subset {W/\!\!/{{\bf G}}}$ of a regular section of $\underline{E}$ (after specializing the invariants at $\lambda =0$, where $\lambda$ is the equivariant parameter for ${\bf U}$). For example, all Calabi-Yau complete intersections in toric varieties are covered by this construction, but there are many more cases with indecomposable bundle $\underline{E}$ when the group ${{\bf G}}$ is non-abelian.

When twisting by the [*inverse*]{} Euler class of a concave representation $E$, the resulting theory coincides with the (untwisted) ${\bf U}$-equivariant theory of the total space of the bundle $\underline{E}$ over ${W/\!\!/{{\bf 