  The celebrated LG/CY correspondence asserts that the Gromov-Witten theory of
a Calabi-Yau (CY) hypersurface in weighted projective space is equivalent to
its corresponding FJRW-theory (LG) via analytic continuation. It is well known
that this correspondence fails in non-Calabi-Yau cases. The main obstruction is
a collapsing or dimensional reduction of the state space of the Landau-Ginzburg
model in the Fano case, and a similar collapsing of the state space of
Gromov-Witten theory in the general type case. We state and prove a modified
version of the cohomological correspondence that describes this collapsing
phenomenon at the level of state spaces. This result confirms a physical
conjecture of Witten-Hori-Vafa. The main purpose of this article is to provide
a quantum explanation for the collapsing phenomenon. A key observation is that
the corresponding Picard-Fuchs equation develops irregular singularities
precisely at the points where the collapsing occurs. Our main idea is to
replace analytic continuation with asymptotic expansion in this non-Calabi-Yau
setting. The main result of this article is that the reduction in rank of the
Gromov-Witten I-function due to power series asymptotic expansions matches
precisely the dimensional reduction of the corresponding state space.
Furthermore, asymptotic expansion under a different asymptotic sequence yields
a different I-function which can be considered as the mathematical counterpart
to the additional "massive vacua" of physics.
