nLab caloron correspondence




For GG a topological group (or Lie group) and PXP \to X a GG-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 XX:

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

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

i:Y(Y×S 1). 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 G\mathcal{L}G-principal bundle over X\mathcal{L}X. The caloron correspondence is the statement that if

Ω YPP \Omega_{Y} P \hookrightarrow \mathcal{L} P

in turn is the subspace on those loops in PP which map 0S 10 \in S^1 to a chosen section of PP over Y×{0}Y \times \{0\}, then forming the pullback i *Ω YPi^\ast \Omega_Y P in

i *Ω YP Ω YP Y i (Y×S 1) \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 GG-principal bundles over Y×S 1Y \times S^1 and loop group-principal bundles over YY.


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

See also

Last revised on October 14, 2019 at 11:07:24. See the history of this page for a list of all contributions to it.