"There is 1-to-1 correspondence between pin$^−$ structures on a vector bundle V and spin structures on V ⊕ det V."
In which context, is this relation between pin$^−$ structure and spin structure true? How to show the above is true?
Is the det V (of the vector bundle V) a determinant bundle that always has to be a determinant line bundle? (are there any kinds of determinant line bundle which are not determinant line bundle? )