### Browsed byCategory: Mathematical Physics

Flat Space as a Gravitational Instanton

## Flat Space as a Gravitational Instanton

This post is about $M = \mathbb{H}$ with metric $g = dq d\bar{q}$. The Gibbons-Hawking ansatz is an ansatz for $4$-dimensional gravitational instantons with a $1$ parameter group of symmetries (think of $U(1)$ or $\mathbb{R}$). One may construct such an instanton from a connection $1$-form $\omega$ for a circle bundle $P$ over an open subset $B$ of $\mathbb{R}^3$ (assuming the group is $U(1)$) and from a smooth function $\Phi$ on $B$, thought of as a smooth section of the adjoint…

The Atiyah-Sutcliffe determinant

## The Atiyah-Sutcliffe determinant

This post is a continuation of my series of posts, which includes previous posts such as “the spin-statistics theorem and the Berry-Robbins problem” and “the Atiyah problem on configurations”. I will make use of notation introduced there, particularly in the latter post. I will explain in this post, how to define the Atiyah-Sutcliffe normalized determinant function, which is a smooth complex-valued function on $C_n(\mathbb{R}^3)$. Given a configuration $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n) \in C_n(\mathbb{R}^3)$ of $n$ distinct points in $\mathbb{R}^3$,…

The Atiyah Problem on Configurations

## The Atiyah Problem on Configurations

This post is a continuation of my previous post “the spin-statistics theorem and the Berry-Robbins problem”. The reader is referred to that post to understand the origins of the Atiyah problem on configurations and the Atiyah-Sutcliffe conjectures. Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^3$. Given $\mathbf{x} = (\mathbf{x}_1,\ldots,\mathbf{x}_n) \in C_n(\mathbb{R}^3)$, so that each $\mathbf{x}_i \in \mathbb{R}^3$, for $i = 1, \ldots, n$ and the $\mathbf{x}_i$ are distinct, we will associate to the configuration $\mathbf{x}$ $n$…

The spin-statistics theorem and the Berry-Robbins problem

## The spin-statistics theorem and the Berry-Robbins problem

In ​1​ , Berry and Robbins propose an interesting way to obtain the spin-statistics theorem, which is close to the famous belt trick, though expressed more mathematically. They completely explain their construction for $2$ particles, but while attempting to extend their construction to $n$ particles, they faced a technical obstruction. This led to the Berry-Robbins problem. Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct particles. Then the symmetric group $S_n$ acts on $C_n(\mathbb{R}^3)$ by permuting the components of any…