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We build the bosonic $\eta$-deformed $AdS_4\times\mathbb{CP}^3$ backgroundgenerated by an $r$-matrix that satisfies the modified classical Yang-Baxterequation. In a special limit we find that it is the gravity dual of thenoncommutative ABJM theory.

Bosonic $\eta$-deformed $AdS_4\times\mathbb{CP}^3$ Background