It is called a correction mostly for historical reasons, since it was not included in all constructions from the beginning.
A metaplectic structure on a symplectic manifold induces a metalinear structure on each Lagrangian submanifold . This allows to form a square root line bundle of the canonical bundle of and hence induces an inner product on sections of the tensor product with the restriction of any line bundle on (a prequantum line bundle, notably).
|line bundle||square root||choice corresponds to|
|canonical bundle||Theta characteristic||over Riemann surface: spin structure|
|density bundle||half-density bundle|
|canonical bundle of Lagrangian submanifold||metalinear structure||metaplectic correction|
|determinant line bundle||Pfaffian line bundle|
|quadratic secondary intersection pairing||partition function of self-dual higher gauge theory||integral Wu structure|
For general discussion see the references listed at geometric quantization, for instance the introduction in section 7.2 of
Discussion with an eye towards Theta characteristics is in
Further references include