nLab Pfaffian line bundle

Idea

A $B$-parameterized family of Dirac operators on $d$-dimensional space gives rise to their determinant line bundle on $B$. If $d = 8k+2$ for $k \in \mathbb{N}$ 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.

Related entries

• Pfaffian

• Pfaffian line bundle

• fermionic path integral

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

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

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: Emma 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 $V$ of dimension $N$, embeds the Grassmannian $Gr^0_V(V+V^*)$ of maximal isotropic subspaces of $V+V^*$, with respect to the natural scalar product, into the projectivization of the exterior space $\Lambda(V)$, and the Plücker map, which embeds the Grassmannian $Gr_V(V+V^*)$ of all N-planes in $V+V^*$ into the projectivization of $\Lambda^N(V+V^*)$. The Plücker coordinates on $Gr^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle $Pf^*\to Gr^0_V(V+V^*,Q)$. 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)