2D-4D correspondence: Towers of kinks versus towers of monopoles in N=2 theories

We continue to study the BPS spectrum of the N=(2,2) CPN ^-1 model with the Z _N-symmetric twisted mass terms. We focus on analysis of the "extra" towers found previously in, and compare them to the states that can be identified in the quasiclassical domain. Exact analysis of the strong-coupling states shows that not all of them survive when passing to the weak-coupling domain. Some of the states decay on the curves of the marginal stability. Thus, not all strong-coupling states can be analytically continued to weak coupling to match the observable bound states. At weak coupling, we confirm the existence of bound states of topologically charged kinks and elementary quanta. Quantization of the U(1) kink modulus leads to formation of towers of such states. For the Z _N-symmetric twisted masses their number is by far less than N-1 as was conjectured previously. We investigate the quasiclassical limit and show that out of N possible towers only two survive in the spectrum for odd N, and a single tower for even N. In the case of CP2 theory the related curves of the marginal stability are discussed in detail. In these points we overlap and completely agree with the results of Dorey and Petunin. We also comment on 2D-4D correspondence.