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Localization of supersymmetric $\mathcal{N}=2$ Chern-Simons-Matter theory ona squashed $S^3$ with $SU(2)\times U(1)$ isometry has been studied by differentgroups of authors. In this paper, we localize the theory on a squashed $S^3$with $SU(2)\times U(1)$ isometry and a class of complex background. We see thatcertain kinds of shifts of the background gauge fields are crucial in obtainingnontrivial results, and the previously found results on this manifold can beincorporated in our results as special limits.

Localization of Supersymmetric Chern-Simons-Matter Theory on a Squashed $S^3$ with $SU(2)\times U(1)$ Isometry