  B. Kim and the first author proved a result comparing the virtual fundamental
classes of the moduli spaces of stable quasimaps and stable LG-quasimaps by
studying localized Chern characters for 2-periodic complexes. In this paper, we
study a K-theoretic analogue of the localized Chern character map and show that
for a Koszul 2-periodic complex it coincides with the cosection localized Gysin
map by Y.-H. Kiem and J. Li. As an application we compare the virtual structure
sheaves of the moduli space of stable quasimaps and stable LG-quasimaps.
