nLab
Pfaffian

Contents

Contents

Idea

The Pfaffian of a skew-symmetric matrix is a square root of its determinant.

Definition

Let A=(A i,j)A = (A_{i,j}) be a skew-symmetric (2n×2n)(2n \times 2n)-matrix with entries in some field (or ring) kk.

Definition

The Pfaffian Pf(A)kPf(A) \in k is the element

12 nn! σS 2nsgn(σ) i=1 nA σ(2i1),σ(2i), \frac{1}{2^n n!} \sum_{\sigma \in S_{2n}} sgn(\sigma) \prod_{i = 1}^n A_{\sigma(2i -1), \sigma(2i)} \,,

where

Properties

In terms of Berezinian integrals

Proposition

Let Λ 2n\Lambda_{2n} be the Grassmann algebra on 2n2n generators {θ i}\{\theta_i\}, which we think of as a vector θ\vec \theta

Then the Pfaffian Pf(A)Pf(A) is the Berezinian integral

Pf(A)=exp(θ,Aθ)dθ 1dθ 2dθ 2n. Pf(A) = \int \exp( \langle \vec \theta, A \cdot \vec \theta \rangle ) d \theta_1 d \theta_2 \cdots d \theta_{2n} \,.
Remark

Compare this to the Berezinian integral representation of the determinant, which is

det(A)exp(θ,Aψ)dθ 1dθ 2dθ 2ndψ 1dψ 2dψ 2n. det(A) \propto \int \exp( \langle \vec \theta, A \cdot \vec \psi \rangle ) d \theta_1 d \theta_2 \cdots d \theta_{2n} d \psi_1 d \psi_2 \cdots d \psi_{2n} \,.

Pfaffian state

Pfaffians appear in the expression of certain multiparticle wave functions. Most notable is the pfaffian state of NN spinless electrons

Ψ Pf(z 1,,z N)=pfaff(1z kz l) i<j(z iz j) qexp(14|z| 2) \Psi_{Pf}(z_1,\ldots,z_N) = pfaff\left(\frac{1}{z_k-z_l}\right)\prod_{i\lt j}(z_i-z_j)^q exp(-\frac{1}{4}\sum |z|^2)

where pfaff(M kl)pfaff(M_{k l}) denotes the Pfaffian of the matrix whose labels are k,lk,l and q=1/νq= 1/\nu is the filling fraction, which is an even integer. For Pfaffian state see

  • Gregory Moore, N. Read, Nonabelions in the fractional quantum hall effect, Nucl. Phys. 360B(1991)362 pdf

References

General

  • J.-G. Luque, J.-Y. Thibon, Pfaffian and hafnian identities in shuffle algebras, math.CO/0204026
  • Claudiu Raicu, Jerzy Weyman, Local cohomology with support in ideals of symmetric minors and Pfaffians, arxiv/1509.03954
  • Haber, Notes on antisymmetric matrices and the pfaffian, pdf

There is also a deformed noncommutative version of Pfaffian related to quantum linear groups:

  • Naihuan Jing, Jian Zhang, Quantum Pfaffians and hyper-Pfaffians, Adv. Math. 265 (2014), 336–361, arxiv/1309.5530

Pfaffian variety is subject of 4.4 in

  • Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, arxiv/1404.3143

Relation to τ\tau-functions is discussed in

  • J. W. van de Leur, A. Yu. Orlov, Pfaffian and determinantal tau functions I, arxiv/1404.6076

Other articles:

  • András C. Lőrincz, Claudiu Raicu, Uli Walther, Jerzy Weyman, Bernstein-Sato polynomials for maximal minors and sub-maximal Pfaffians, arxiv/1601.06688

Euler forms

Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:

  • Shiing-Shen Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics Second Series, Vol. 45, No. 4 (1944), pp. 747-752 (jstor:1969302)

  • Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

  • Siye Wu, Section 2.2 of Mathai-Quillen Formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)

  • Gerard Walschap, chapter 6.3 of Metric Structures in Differential Geometry, Graduate Texts in Mathematics, Springer 2004

  • Hiro Lee Tanaka, Pfaffians and the Euler class, 2014 (pdf)

  • Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)

Last revised on April 28, 2019 at 12:38:11. See the history of this page for a list of all contributions to it.