# Contents

## Idea

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

## Definition

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

###### Definition

The Pfaffian $\mathrm{Pf}\left(A\right)\in k$ is the element

$\frac{1}{{2}^{n}n!}\sum _{\sigma \in {S}_{2n}}\mathrm{sgn}\left(\sigma \right)\prod _{i=1}^{n}{A}_{\sigma \left(2i-1\right),\sigma \left(2i\right)}\phantom{\rule{thinmathspace}{0ex}},$\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;

• $\mathrm{sgn}\left(\sigma \right)$ is the signature of a permutation.

## Properties

### In terms of Berezinian integrals

###### Proposition

Let ${\Lambda }_{2n}$ be the Grassmann algebra on $2n$ generators $\left\{{\theta }_{i}\right\}$, which we think of as a vector $\stackrel{⇀}{\theta }$

Then the Pfaffian $\mathrm{Pf}\left(A\right)$ is the Berezinian integral

$\mathrm{Pf}\left(A\right)=\int \mathrm{exp}\left(⟨\stackrel{⇀}{\theta },A\cdot \stackrel{⇀}{\theta }⟩\right)d{\theta }_{1}d{\theta }_{2}\cdots d{\theta }_{2n}\phantom{\rule{thinmathspace}{0ex}}.$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

$\mathrm{det}\left(A\right)\propto \int \mathrm{exp}\left(⟨\stackrel{⇀}{\theta },A\cdot \stackrel{⇀}{\psi }⟩\right)d{\theta }_{1}d{\theta }_{2}\cdots d{\theta }_{2n}d{\psi }_{1}d{\psi }_{2}\cdots d{\psi }_{2n}\phantom{\rule{thinmathspace}{0ex}}.$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 }_{\mathrm{Pf}}\left({z}_{1},\dots ,{z}_{N}\right)=\mathrm{pfaff}\left(\frac{1}{{z}_{k}-{z}_{l}}\right)\prod _{i\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 $\mathrm{pfaff}\left({M}_{kl}\right)$ 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
• J.-G. Luque, J.-Y. Thibon, Pfaffian and hafnian identities in shuffle algebras, math.CO/0204026