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 of a given foliation by Lagrangian submanifolds (polarization). This allows to form a square root line bundle of the canonical bundle of (a “Theta characteristic”, see below) 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).
Let be a compact symplectic manifold equipped with a Kähler polarization hence a Kähler manifold structure . A metaplectic structure is now a choice of square root of the canonical line bundle (a Theta characteristic for the complex analytic space ). This is equivalently a spin structure on (see the discussion at spin structure – over Kähler manifolds).
This appears as (Paradan 09, prop. 2.2).
|line bundle||square root||choice corresponds to|
|canonical bundle||Theta characteristic||over Riemann surface and Hermitian manifold (e.g.Kähler manifold): 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
Alexander Cardona, Geometric and metaplectic quantization (pdf)
Relation to spin^c quantization is discussed in
Discussion with an eye towards Theta characteristics is in
Further references include