We show that the higher rank triplet W-algebra is simple when p bigger than or equal to h-1. Furthermore, we Feigin-Tipunin's module over introduced in arXiv:1002.5047 if parameter closure of fundamental alcove, and give decomposition as a direct sum modules affine W-algebras at level p-h.

We exploit the critical locus structure on Quot scheme $\mathrm{Quot}_{\mathbb A^3}(\mathscr O^{\oplus r},n)$, in particular associated symmetric obstruction theory, order to define rank $r$ K-theoretic Donaldson-Thomas invariants of Calabi-Yau $3$-fold $\mathbb A^3$. compute partition function as a plethystic exponential, proving conjecture proposed string theory by Awata-Kanno and Benini-Bone...