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We continue our investigation on the Nambu-Poisson description of M5-brane ina large constant C-field background (NP M5-brane theory) constructed inRefs.[1, 2]. In this paper, the low energy limit where the NP M5-brane theoryis applicable is clarified. The background independence of the NP M5-branetheory is made manifest using the variables in the BLG model of multipleM2-branes. An all order solution to the Seiberg-Witten map is also constructed.

More on the Nambu-Poisson M5-brane Theory: Scaling limit, background independence and an all order solution to the Seiberg-Witten map