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We generalize our analysis in [arXiv:1301.1977], and show that a 5d and 6dAGT correspondence for SU(N) -- which essentially relates the relevant 5d and6d Nekrasov instanton partition functions to the integrable representations ofa q-deformed and elliptic affine W_N-algebra -- can be derived, purelyphysically, from the principle that the spacetime BPS spectra of string-dualM-theory compactifications ought to be equivalent. Via an appropriate defect,we also derive a "fully-ramified" version of the 5d and 6d AGT correspondencewhere integrable representations of a quantum and elliptic affine SU(N)-algebraat the critical level appear on the 2d side, and argue that the relevant"fully-ramified" 5d and 6d Nekrasov instanton partition functions aresimultaneous eigenfunctions of commuting operators which define relativisticand elliptized integrable systems. As an offshoot, we also obtain variousmathematically novel and interesting relations involving the double loopalgebra of SU(N), elliptic Macdonald operators, equivariant elliptic genus ofinstanton moduli space, and more.

An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems