nLab Pfaffian line bundle






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


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

  • F. Balogh, J. Harnad, J. Hurtubise, Isotropic Grassmannians, Plücker and Cartan maps, arxiv/2007.03586

This work is motivated by the relation between the KP and BKP integrable hierarchies, whose τ-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map which, for a vector space VV of dimension NN, embeds the Grassmannian Gr V 0(V+V *)Gr^0_V(V+V^*) of maximal isotropic subspaces of V+V *V+V^*, with respect to the natural scalar product, into the projectivization of the exterior space Λ(V)\Lambda(V), and the Plücker map, which embeds the Grassmannian Gr V(V+V *)Gr_V(V+V^*) of all N-planes in V+V *V+V^* into the projectivization of Λ N(V+V *)\Lambda^N(V+V^*). The Plücker coordinates on Gr V 0(V+V *)Gr^0_V(V+V^*) are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle Pf *Gr V 0(V+V *,Q)Pf^*\to Gr^0_V(V+V^*,Q). In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet type, expressing the determinants of square submatrices of a skew symmetric N×N matrix as bilinear sums over the Pfaffians of their principal minors

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 February 3, 2021 at 09:10:26. See the history of this page for a list of all contributions to it.