The Gromov-Witten (GW) invariants of the quintic is one of the most famous research topics from the very first day of mirror symmetry to today. There is another physical theory for the quintic polynomial, called Landau-Ginzburg theory. Physicists conjectured that these two theories can be identified via some mysterious transformations. The analytical construction of enumerative invariants in the affine LG-space was given by Fan, Jarvis and Ruan (FJRW invariants). Mixed spin P-field (MSP) theory is a mathematical attempt to unlock the mysterious link between GW-invariants and FJRW-invariants. It uses the newly developed technologies such as the cosection localization by Kiem and J.Li and the P-field theory of Chang and J. Li. Elliptic GW invariants can be calculated using MSP moduli.