nLab
Pfaffian line bundle

Contents

Contents

Idea

A BB-parameterized family of Dirac operators on dd-dimensional space gives rise to their determinant line bundle on BB. If d=8k+2d = 8k+2 for kk \in \mathbb{N} this has a canonical square root line bundle, the Pfaffian line bundle.

As quantum anomaly

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.

The following table lists classes of examples of square roots of line bundles

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

References

The general notion of Pfaffian line bundle is described in section 3 of

  • Dan Freed, On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)

  • D. Borthwick, The Pfaffian line bundle, Commun. Math. Phys. 149 (1992) (3) 463–493 doi euclid

  • Section 1.3 in: Ema Previato, Mauro Spera, Isometric embeddings of infinite-dimensional Grassmannians, Regul. Chaot. Dyn. (2011) 16: 356 doi

The string worldsheet Green-Schwarz mechanism which trivializes the worldsheet Pfaffian line bundle, and its relation to string structures, that goes back to Killingback and Edward Witten, has been formalized in

  • Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle (arXiv)

Last revised on October 25, 2019 at 06:00:26. See the history of this page for a list of all contributions to it.