On the Mathematics and Physics of Mixed Spin P-fields

We outline various developments of affine and general Landau Ginzburg models in physics. We then describe the A-twisting and coupling to gravity in terms of Algebraic Geometry. We describe constructions of various path integral measures (virtual fundamental class) using the algebro-geometric technique of cosection localization, culminating in the theory of ``Mixed Spin P (MSP) fields" developed by the authors.