In the path integral quantization of quantum field theory, the partial path integral of an action functional over just the fermionic fields yields in general not a function on the remaining space of bosonic fields, but a section of a line bundle on this bosonic configuration space, the determinant line bundle of the family of Dirac operators. The Lorentzian metric? assumed in relativistic quantum field theory leads in the Wick-rotated theory to the passage to the square root of this line bundle.
Therefore the nontriviality of the Pfaffian line bundle is in these dimensions the fermionic quantum anomaly.
|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|
The general notion of Pfaffian line bundle is described in section 3 of
The string worldsheet Green-Schwarz mechanism which trivializes the worldsheet Pfaffian line bundle, and its relation to string structures that goes bak to Killingback and Edward Witten has been formalized in