nLab caloron correspondence

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

1. $G$-principal bundles over $Y \times S^1$ with a trivialization over $Y \times \{0\}$

2. loop group-principal bundles over $Y$.

Properties

Relation to $Ext/Cyc$ adjunction

On the level of homotopy theory and isomorphism classes of principal bundles, the above Caloron correspondence is a special case of the Ext/Cyc-adjunction (BMSS19 §2.2):

The latter applies to all circle principal bundles $X \longrightarrow Y$, classified by a possibly nontrivial first Chern class $c_1$, and gives that:

(1)$$\big( X \longrightarrow B G \big) \;\; \leftrightarrow \;\; \left( \begin{array}{ccc} Y &&\longrightarrow&& (L B G)\sslash S^1 \\ & \mathllap{_{c_1}}\searrow & & \swarrow\mathrlap{_{(L B G \to \ast) \sslash S^1 }} \\ && B S^1 \end{array} \right) \mathrlap{\,,}$$

as the special case of the general statement for the classifying space $B G$ replaced by any space $\mathcal{A}$.

In the case of connected $G$ we have

$$L B G \,\simeq\, B L G$$

and hence

$$(B L G) \sslash S^1 \,\simeq\, B( S^1 \ltimes L G ) \mathrlap{\,.}$$

In this generality the caloron correspondence appears (not under this name, though) in Bergman & Varadarajan 2005.

Specialized to trivial circle bundles $c_1 = 0$, the above correspondence (1) becomes the internal hom-adjointnessunctor#InTermsOfHomIsomorphism):

$$\big( Y \times S^1 \longrightarrow B G \big) \;\; \leftrightarrow \;\; \big( Y \longrightarrow L B G \big)$$

that the above discussion started with. At the level of principal $\infty$-bundles (ignoring geometric strcture) we may conclude from this the Caloron correspondence more abstractly:

Observe that the condition for $Y \times S^1 \to B G$ to trivialize on $Y \simeq Y \times \{0\}$ means equivalently that $Y \longrightarrow L B G$ factors through the homotopy fiber

$$\Omega B G \xrightarrow { hofib_{ev_0} } L B G \overset { ev_0 } {\longrightarrow} B G$$

of the basepoint evaluation map, being the based loop space of the classifying space.

But that is

$$\Omega B G \underset{hmpt}{\simeq} G \underset{hmpt}{\simeq} B \Omega G$$

(cf. at May recognition theorem).

Related concepts

• caloron

• Nahm transform

• double dimensional reduction

References

The term “caloron correspondence” originates in:

• H. Garland, Michael Murray: Kac-Moody monopoles and periodic instantons. Comm. Math. Phys. 120 2 (1988) 335–351 [doi:10.1007/BF01217968]

Application (not under this term, though) to duality between M-theory and type IIA string theory:

• Varghese Mathai, Hisham Sati: Some Relations between Twisted K-theory and $E_8$ Gauge Theory, JHEP 0403:016 (2004) [doi:10.1088/1126-6708/2004/03/016, arXiv:hep-th/0312033]

The general case (not assuming trivial circle bundles), but with focus on $G =$ E8:

• Aaron Bergman, Uday Varadarajan: Loop Groups, Kaluza-Klein Reduction and M-Theory, JHEP0506 043 (2005) [arXiv:hep-th/0406218, doi:10.1088/1126-6708/2005/06/043]

recalled in

• Hisham Sati: $E_8$ Gauge Theory and Gerbes in String Theory, Adv. Theor. Math. Phys. 14 (2010) 1–39 [arXiv:hep-th/0608190, doi:10.4310/ATMP.2010.v14.n2.a2]

• Hisham Sati: The Loop Group of $E_8$ and Targets for Spacetime, Mod. Phys. Lett. A 24 (2009) 25–40 [arXiv:hep-th/0701231, doi:10.1142/S0217732309028746]

Review and further developments:

• Michael Murray, Raymond Vozzo: The caloron correspondence and higher string classes for loop groups, J. Geom. Phys. 60 9 (2010) 1235–1250 [arXiv:0911.3464 math.DG, doi:10.1016/j.geomphys.2010.05.003]

See also:

• Pedram Hekmati, Michael Murray, Raymond Vozzo, The general caloron correspondence, J. Geom. Phys. 62 2 (2012) 224–241 [arXiv:1105.0805, doi:10.1016/j.geomphys.2011.10.015]

The Ext/Cyc-adjunction:

• Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber; §2.2 of: Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory, Communications in Mathematical Physics 371 197 (2019) [doi:10.1007/s00220-019-03441-4, arXiv:1806.01115]

• Hisham Sati, Urs Schreiber; §2.2 of: Cyclification of Orbifolds, Comm. Math. Phys. 405 67 (2024) [doi:10.1007/s00220-023-04929-w, arXiv:2212.13836 math.AT, talk]