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Gauged Gromov-Witten theory for small spheres

For smooth projective G-varieties, we equate the gauged Gromov-Witten invariants for sufficiently small area and genus zero with the invariant part of equivariant Gromov-Witten invariants. As an application we deduce a gauged version of abelianization for Gromov-Witten invariants. In the symplectic setting, we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map.