hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
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 $X$:
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
(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
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
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
A review and further developments are in
See also
Last revised on October 14, 2019 at 07:07:24. See the history of this page for a list of all contributions to it.