Flops of any length, Gopakumar-Vafa invariants and 5d Higgs branches

The conifold is a basic example of noncompact Calabi-Yau threefold that admits simple flop, and in M-theory, gives rise to 5d hypermultiplet at low energies, realized by an M2-brane wrapped on the vanishing sphere. We develop novel gauge-theoretic method construct new classes examples generalize flop so-called length l=1,...,6. allows us naturally read off Gopakumar-Vafa invariants. Although they share similar properties beloved conifold, these threefolds are expected admit M2-bound states higher degree. demonstrate this through our computations GV Furthermore we fully characterize associated Higgs branches computing their dimensions flavor groups. With techniques extract more refined data such as charges hypers under group.