Contents

mapping space

# Contents

## Idea

For $G$ a topological group (or Lie group) and $P \to X$ a $G$-principal bundle, we have that forming mapping spaces out of the circle yields a free loop group-principal bundle over the free loop space of $X$:

$\array{ \mathcal{L}G &\to& \mathcal{L} P \\ && \downarrow \\ && \mathcal{L} X } \,.$

In the special case that $X = Y \times S^1$ is a Cartesian product with the circle, then one can consider the subspace of the free loop space of $X$ on those loops whose projection on the $S^1$-factor is the identity. This subspace is of course equivalent to $Y$, giving a canonical inclusion

$i \;\colon\; Y \hookrightarrow \mathcal{L} (Y \times S^1) \,.$

(Abstractly, this is the adjunct of the identity under the internal hom-adjunction.)

Along this inclusion one can pull back the $\mathcal{L}G$-principal bundle over $\mathcal{L}X$. The caloron correspondence is the statement that if

$\Omega_{Y} P \hookrightarrow \mathcal{L} P$

in turn is the subspace on those loops in $P$ which map $0 \in S^1$ to a chosen section of $P$ over $Y \times \{0\}$, then forming the pullback $i^\ast \Omega_Y P$ in

$\array{ i^\ast \Omega_Y P &\to& \Omega_Y P \\ \downarrow && \downarrow \\ Y &\stackrel{i}{\longrightarrow}& \mathcal{L}(Y \times S^1) }$

constitutes an equivalence of groupoids between that of $G$-principal bundles over $Y \times S^1$ and loop group-principal bundles over $Y$.

The term “caloron correspondence” originates in

• H. Garland, Michael Murray, Kac-Moody monopoles and periodic instantons. Comm. Math. Phys., 120(2):335–351, 1988.

A review and further developments are in