In 2007, a quantum theory for quasi-homogeneous polynomial singularities was developed by Fan, Jarvis, and Ruan, based on ideas of Witten, and now called FJRW theory. It should be seen as the counterpart of Gromov-Witten (GW) theory for hypersurfaces in weighted projective spaces via the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence.
In 2003, Lee established a K-theoretic version of GW theory. However, some aspects of GW theory, such as mirror symmetry, were still missing until last year. Indeed, Givental gave in 2015 a refined version called permutation equivariant GW K-theory, and proved some mirror symmetry statements in this new context, e.g. for the quintic hypersurface in P^4.
In this talk, I will describe a joint work with Valentin Tonita and Yongbin Ruan, in which we define a K-theoretic version of FJRW theory and we study its permutation equivariant part. I will focus on the quintic polynomial and explain how to prove mirror symmetry and the LG/CY correspondence. Speaker(s): Jeremy Guere (Berlin)