Multiple D2-brane action from M2-branes

We study the detailed derivation of the multiple D2-brane effective action from multiple M2-branes in the BaggerLambertGustavsson (BLG) theory and the AharonyBergmanJafferisMaldacena (ABJM) theory by employing the novel Higgs mechanism. We show explicitly that the high-order F³ and F ⁴ terms are commutator terms, and conjecture that all the high-order terms are commutator terms. Because the commutator terms can be treated as the covariant derivative terms, these high-order terms do not contribute to the multiple D2-brane effective action. Inspired by the derivation of a single D2-brane from a M2-brane, we consider the curved M2-branes and introduce an auxiliary field. Integrating out the auxiliary field, we indeed obtain the correct high-order F⁴ terms in the D2-brane effective action from the BLG theory and the ABJM theory with SU(2) × SU(2) gauge symmetry, but we cannot obtain the correct high-order F⁴ terms from the ABJM theory with U(N) × U(N) and SU(N) × SU(N) gauge symmetries for N > 2. We also briefly comment on the (gauged) BF membrane theory.