Determinant, Wall Monodromy and Spherical Functor

We apply the definition of determinant in the compactified moduli space as a generalization of the discriminant. We study the relationship between the wall monodromy and the determinant in the GIT wall crossing. The wall monodromy is an EZ-spherical functor in the sense of Horja. By constructing a fibration structure on Z, we obtain a semi-orthogonal decomposition of the derived category of coherent sheaves of Z, hence decompose the EZ-spherical functor into a sequence of its subfunctors. We also show that the intersection multiplicity of the discriminant and the exponent of the discriminant in the determinant both have their correspondences in this decomposition.