Abstract: The study of Landau-Ginzburg/Calabi-Yau correspondence using linear sigma models was proposed by Witten in the early 90's of last century. A mathematical version of the correspondence using curve-counting theories has been realized by Chiodo, Ruan, Iritani, and many other people in the past decade. Namely, there is an equivalence between counting stable maps in the CY hypersurface of a weighted polynomial W (i.e. Gromov-Witten theory of the hypersurface) and counting $W$-spin structures (i.e. Fan-Jarvis-Ruan-Witten theory of W). So LG/CY correspondence can be realized as GW/FJRW correspondence.
I will talk about a generalization of such a GW/FJRW correspondence for non-CY hypersurfaces. A key ingredient here is the Gamma structures developed by Iritani, and Katzarkov-Kontsevich-Pantev. For Fano manifolds, Galkin-Golyshev-Iritani's Gamma conjectures predict the Gamma class of the quantum cohomology equals the asymptotic class from an irregular meromorphic connection. The irregular Riemann-Hilbert correspondence of the connection generates Stokes phenomenon. We can view the FJRW theory as a part of the GW theory in the Stoke decomposition. The similar phenomenon works for hypersurfaces of general type as well, where the GW theory is viewed as a part of the FJRW theory. Our GW/FJRW correspondence is compatible with Orlov's semi-orthogonal decomposition.