We study the moduli space of twisted quasimaps from a fixed smooth projective curve to a Nakajima's quiver variety and the moduli space of $\delta$-stable framed twisted quiver bundles with moment map relations. We show that they carry symmetric obstruction theories and when $\delta$ is large enough, they exactly coincide. These results generalize works by D.E. Diaconescu  about the ADHM quiver, in the framework of the quasimap theory of I. Ciocan-Fontanine, D. Maulik and the author [8, 9].
Digital Object Identifier: 10.2969/aspm/07110139