# nLab Pfaffian

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

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

## Definition

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

###### Definition

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

$\frac{1}{2^n n!} \sum_{\sigma \in S_{2n}} sgn(\sigma) \prod_{i = 1}^n A_{\sigma(2i -1), \sigma(2i)} \,,$

where

• $\sigma$ runs over all permutations of $2n$ elements;

• $sgn(\sigma)$ is the signature of a permutation.

## Properties

### In terms of Berezinian integrals

###### Proposition

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

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

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

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

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

## Pfaffian state

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

$\Psi_{Pf}(z_1,\ldots,z_N) = pfaff\left(\frac{1}{z_k-z_l}\right)\prod_{i\lt j}(z_i-z_j)^q exp(-\frac{1}{4}\sum |z|^2)$

where $pfaff(M_{k l})$ denotes the Pfaffian of the matrix whose labels are $k,l$ and $q= 1/\nu$ 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

### 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 $\tau$-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
• M. Spera, A C–algebraic approach to determinants and Pfaffians, Acta Cosmologica Fasc. XXI-2, (1995) 203–208.

### Euler forms

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