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\section*{caloron}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToConfinement}{Relation to confinement}\dotfill \pageref*{RelationToConfinement} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_skyrmions_instantons_monopoles}{Relation to skyrmions, instantons, monopoles}\dotfill \pageref*{relation_to_skyrmions_instantons_monopoles} \linebreak
\noindent\hyperlink{in_confinementmass_gap}{In confinement/mass gap}\dotfill \pageref*{in_confinementmass_gap} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Calorons are topologically non-trivial [[field (physics)|field]]-configurations in [[Yang-Mills theory]] at [[thermal field theory|positive temperature]] $T \gt 0$. They correspond to what at vanishing [[temperature]] $T =  0$ are called [[instantons]], specifically [[BPST instantons]] (but there is no continuous limit relating the two).

While the physical reality of [[instantons]] is at best subtle (they are interpreted as witnessing [[quantum tunneling]] between actual [[vacua]]) calorons are meant to correspond to actual [[vacua]] of [[Yang-Mills theory]] at [[thermal field theory|positive temperature]], reflecting the general statement that [[Wick rotation]] has good physical meaning at positive temperature.

More in detail:

Upon [[Wick rotation]], $G$-[[Yang-Mills theory]] on [[Minkowski spacetime]] $\mathbb{R}^{3,1}$ in a [[KMS state]] of [[positive number|positive]] [[temperature]] $T$ is expressed by [[Euclidean field theory]] on the [[Riemannian manifold]] $\mathbb{R}^3 \times S^1_{\beta}$, where the first factor is 3-dimensional [[Euclidean space]] and the second factor is a [[circle]] of [[length]] $\beta \coloneqq 1/T$ (proportional to) the inverse [[temperature]].

A \emph{caloron} is a [[gauge field]]-configuration in this Euclidean Yang-Mills theory on $\mathbb{R}^3 \times S^1_\beta$ [[vanishing at infinity]]. By the [[clutching construction]] the topological class of these bundles (their [[second Chern class]] for [[gauge group]] a [[simple Lie group]]) is given by [[homotopy classes]] of maps

\begin{displaymath}
S^2 \times S^1 \longrightarrow G
\end{displaymath}
hence

\begin{displaymath}
S^3 \times S^1 \longrightarrow B G
\end{displaymath}
For the case $G =$ [[special unitary group|SU(2)]] $\simeq S^3$ and via the usual math/physics translation of terminology (\href{BPTS-instanton#FromTheMathsToThePhysicsStory}{here}) this was first considered in \hyperlink{HarringtonShepard77}{Harrington-Shepard 77, p. 3}, and baptized a ``caloron''-configuration in \hyperlink{HarringtonShepard78}{Harrington-Shepard 78, p. 1}.

A further refinement of the construction included the nonzero [[vacuum expectation value]] $\langle (A_4)^3\rangle$ of the ``time'' component of the [[vector potential]], called the \emph{KvBLL caloron} (\hyperlink{KraanVanBaal98a}{Kraan-van Baal 98a}, \hyperlink{KraanVanBaal98b}{Kraan-van Baal 98b}, \hyperlink{LeeLu98}{Lee-Lu 98}).

This solution revealed the substructure: it gets split into $N_c$ (number of colors) separate (anti)self- dual 3d solitons with nonzero (Euclidean) electric and magnetic charges (see e.g. \hyperlink{LarsenShuryak14}{Larsen-Shuryak 14}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToConfinement}{}\subsubsection*{{Relation to confinement}}\label{RelationToConfinement}

As part of a possible solution to the [[confinement]]/[[mass gap]]-problem:

\hyperlink{Greensite11}{Greensite 11, section 8.5}:

\begin{quote}%
it is natural to wonder if [[confinement]] could be derived from some [[semiclassical approximation|semiclassical]] treatment of [[Yang–Mills theory]] based on the [[instanton]] solutions of [[nonabelian group|non-abelian]] [[gauge theories]]. The [[BPST instanton|standard]] [[instantons]], introduced by Belavin et al. (\href{BPTS-instanton#BelavinPolyakovSchwartzTyupkin75}{40}), do not seem to work; their [[field strengths]] fall off too rapidly to produce the desired magnetic disorder in the vacuum.

In recent years, however, it has been realized that instanton solutions at [[thermal field theory|finite temperature]], known as \emph{calorons}, might do the job. These caloron solutions were introduced independently by Kraan and van Baal (\hyperlink{KraanVanBaal98}{41}, \hyperlink{KraanVanBaal98b}{42}) and Lee and Lu (\hyperlink{LeeLu98}{43}) (KvBLL), and they have the remarkable property of containing [[monopole]] constituents which may, depending on the type of caloron, be widely separated.

$[...]$

The caloron idea is probably the most promising current version of [[monopole]] [[confinement]] in pure non-abelian gauge theories, but it is basically (in certain [[gauge fixing|gauges]]) a superposition of [[monopoles]] with spherically symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.


\end{quote}
\hypertarget{history}{}\subsection*{{History}}\label{history}

The paper \hyperlink{Harrington-Shepard_78}{Harrington-Shepard 1978} considers the contribution from periodic solutions at \emph{positive, finite} temperature to the [[Yang-Mills equations]] to the action of a `Yang-Mills gas': an equilibrium ensemble of YM fields. They give explicit solutions to the classical equations of motion `of charge 1' i.e. representing a generator in the [[homotopy group]] classifying topologically inequivalent solutions.

[[Werner Nahm|Nahm]] (\hyperlink{Nahm_84}{Nahm 1984}) continued the study of calorons, linking them with self-dual [[monopoles]] and the [[ADHM construction]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[instanton]], [[soliton]], [[vortex]]


\item [[caloron correspondence]]


\item [[Nahm transform]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The concept was introduced in

\begin{itemize}%
\item Barry J. Harrington, Harvey K. Shepard, \emph{Euclidean solutions and finite temperature gauge theory}, Nuclear Physics B Volume 124, Issue 4, 27 June 1977, Pages 409-412 ()


\item Barry J. Harrington, Harvey K. Shepard, \emph{Periodic Euclidean solutions and the finite-temperature Yang-Mills gas}, Phys. Rev. D 17, 2122 -- April 1978 (\href{https://doi.org/10.1103/PhysRevD.17.2122}{doi:10.1103/PhysRevD.17.2122})



\end{itemize}
Further development includes

\begin{itemize}%
\item W. Nahm, \emph{Self-dual monopoles and calorons}, in Group Theoretical Methods in Physics, Ed. G. Denardo, G. Ghirardi, and T. Weber, Lecture Notes in Physics \textbf{201} (1984) pp 189-200. doi:\href{http://dx.doi.org/10.1007/BFb0016145}{10.1007/BFb0016145}


\item Thomas C. Kraan, Pierre van Baal, \emph{Periodic Instantons with non-trivial Holonomy}, Nucl.Phys. B533 (1998) 627-659 (\href{https://arxiv.org/abs/hep-th/9805168}{arXiv:hep-th/9805168})


\item Thomas C. Kraan, Pierre van Baal, \emph{Exact T-duality between Calorons and Taub-NUT spaces}, Phys.Lett.B428:268-276, 1998 (\href{https://arxiv.org/abs/hep-th/9802049}{arXiv:hep-th/9802049})


\item Kimyeong Lee, Changhai Lu, \emph{$SU(2)$ Calorons and Magnetic Monopoles}, Phys. Rev. D 58, 025011 (1998) (\href{https://arxiv.org/abs/hep-th/9802108}{arXiv:hep-th/9802108})



\end{itemize}
Discussion in [[lattice gauge theory]]:

\begin{itemize}%
\item P. Gerhold, E.‐M. Ilgenfritz, M. Müller‐Preussker, B. V. Martemyanov, A. I. Veselov, \emph{Topology and confinement at $T \neq 0$: calorons with non‐trivial holonomy}, AIP Conference Proceedings 892, 213 (2007) (\href{https://doi.org/10.1063/1.2714375}{doi:10.1063/1.2714375})

\end{itemize}
\hypertarget{relation_to_skyrmions_instantons_monopoles}{}\subsubsection*{{Relation to skyrmions, instantons, monopoles}}\label{relation_to_skyrmions_instantons_monopoles}

The construction of [[Skyrmions]] from [[instantons]] is due to

\begin{itemize}%
\item [[Michael Atiyah]], N S Manton, \emph{Skyrmions from instantons}, Phys. Lett. B, 222(3):438–442, 1989 ()

\end{itemize}
The relation between [[skyrmions]], [[instantons]], [[calorons]], [[solitons]] and [[monopoles]] is usefully reviewed and further developed in

\begin{itemize}%
\item [[Josh Cork]], \emph{Calorons, symmetry, and the soliton trinity}, PhD thesis, University of Leeds 2018 (\href{http://etheses.whiterose.ac.uk/22097/}{web})


\item [[Josh Cork]], \emph{Skyrmions from calorons}, J. High Energ. Phys. (2018) 2018: 137 (\href{https://arxiv.org/abs/1810.04143}{arXiv:1810.04143})



\end{itemize}
\hypertarget{in_confinementmass_gap}{}\subsubsection*{{In confinement/mass gap}}\label{in_confinementmass_gap}

Discussion as part of a solution to the [[confinement]]/[[mass gap]]-problem:

\begin{itemize}%
\item Jeff Greensite, \emph{An Introduction to the Confinement Problem}, Lecture Notes in Physics, Volume 821, 2011 (\href{https://link.springer.com/book/10.1007%2F978-3-642-14382-3}{doi:10.1007/978-3-642-14382-3})


\item P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, \emph{An $SU(2)$ KvBLL caloron gas model and confinement}, Nucl.Phys.B760:1-37, 2007 (\href{https://arxiv.org/abs/hep-ph/0607315}{arXiv:hep-ph/0607315})


\item Rasmus Larsen, [[Edward Shuryak]], \emph{Classical interactions of the instanton-dyons with antidyons}, Nucl. Phys. A \textbf{950}, 110 (2016) (\href{https://arxiv.org/abs/1408.6563}{arXiv:1408.6563})


\item Rasmus Larsen, [[Edward Shuryak]], \emph{Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory}, Phys. Rev. D 92, 094022, 2015 (\href{https://arxiv.org/abs/1504.03341}{arXiv:1504.03341})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Caloron}{Caloron}}

\end{itemize}
[[!redirects calorons]]



\end{document}