nLab caloron

Idea

Calorons are topologically non-trivial field-configurations in Yang-Mills theory at 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 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 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 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

$$S^2 \times S^1 \longrightarrow G$$

hence

$$S^3 \times S^1 \longrightarrow B G$$

For the case $G =$ SU(2) $\simeq S^3$ and via the usual math/physics translation of terminology (here) this was first considered in Harrington-Shepard 77, p. 3, and baptized a “caloron”-configuration in 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 KvBLL caloron (Kraan-van Baal 98a, Kraan-van Baal 98b, 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. Larsen-Shuryak 14).

Properties

Relation to confinement

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

Greensite 11, section 8.5:

it is natural to wonder if confinement could be derived from some semiclassical treatment of Yang–Mills theory based on the instanton solutions of non-abelian gauge theories. The standard instantons, introduced by Belavin et al. (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 finite temperature, known as calorons, might do the job. These caloron solutions were introduced independently by Kraan and van Baal (41, 42) and Lee and Lu (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 gauges) a superposition of monopoles with spherically symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.

Relation to heterotic string theory

The gauge solution of the NS5-brane in heterotic string theory contains a gauge field on the transverse space $\mathbb{R}^4$ which is a BPST-instanton. Then, if we consider an array of NS5-branes at fixed distance along the circle of a compactified transverse space $\mathbb{R}^3\times S^1$, we obtain a Harrington-Shepard caloron (see e.g. SasYat16, pag.12). If we send the radius to zero $R\rightarrow 0$ we obtain a smeared NS5-brane in heterotic string theory.

History

The paper Harrington-Shepard 1978 considers the contribution from periodic solutions at 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.

Nahm (Nahm 1984) continued the study of calorons, linking them with self-dual monopoles and the ADHM construction.

Related concepts

• instanton, soliton, vortex

• thermal field theory

• caloron correspondence

• Nahm transform

References

General

The concept was introduced in

• Barry J. Harrington, Harvey K. Shepard, Euclidean solutions and finite temperature gauge theory, Nuclear Physics B Volume 124, Issue 4, 27 June 1977, Pages 409-412 (doi:10.1016/0550-3213(77)90413-8)

• Barry J. Harrington, Harvey K. Shepard, Periodic Euclidean solutions and the finite-temperature Yang-Mills gas, Phys. Rev. D 17, 2122 – April 1978 (doi:10.1103/PhysRevD.17.2122)

Further development includes

• W. Nahm, Self-dual monopoles and calorons, in Group Theoretical Methods in Physics, Ed. G. Denardo, G. Ghirardi, and T. Weber, Lecture Notes in Physics 201 (1984) pp 189-200. doi:10.1007/BFb0016145

• Thomas C. Kraan, Pierre van Baal, Periodic Instantons with non-trivial Holonomy, Nucl.Phys. B533 (1998) 627-659 (arXiv:hep-th/9805168)

• Thomas C. Kraan, Pierre van Baal, Exact T-duality between Calorons and Taub-NUT spaces, Phys.Lett.B428:268-276, 1998 (arXiv:hep-th/9802049)

• Kimyeong Lee, Changhai Lu, $SU(2)$ Calorons and Magnetic Monopoles, Phys. Rev. D 58, 025011 (1998) (arXiv:hep-th/9802108)

See also

• Wikipedia, Caloron

Discussion in lattice gauge theory:

• P. Gerhold, E.‐M. Ilgenfritz, M. Müller‐Preussker, B. V. Martemyanov, A. I. Veselov, Topology and confinement at $T \neq 0$: calorons with non‐trivial holonomy, AIP Conference Proceedings 892, 213 (2007) (doi:10.1063/1.2714375)

Relation to skyrmions, instantons, monopoles

The construction of Skyrmions from instantons is due to

• Michael Atiyah, N S Manton, Skyrmions from instantons, Phys. Lett. B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

The relation between skyrmions, instantons, calorons, solitons and monopoles is usefully reviewed and further developed in

• Josh Cork, Calorons, symmetry, and the soliton trinity, PhD thesis, University of Leeds 2018 (web)

• Josh Cork, Skyrmions from calorons, J. High Energ. Phys. (2018) 2018: 137 (arXiv:1810.04143)

In confinement/mass gap

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

• Jeff Greensite, An Introduction to the Confinement Problem, Lecture Notes in Physics, Volume 821, 2011 (doi:10.1007/978-3-642-14382-3)

• P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, An $SU(2)$ KvBLL caloron gas model and confinement, Nucl.Phys.B760:1-37, 2007 (arXiv:hep-ph/0607315)

• Rasmus Larsen, Edward Shuryak, Classical interactions of the instanton-dyons with antidyons, Nucl. Phys. A 950, 110 (2016) (arXiv:1408.6563)

• Rasmus Larsen, Edward Shuryak, Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory, Phys. Rev. D 92, 094022, 2015 (arXiv:1504.03341)

Relation to NS5-brane in heterotic string theory

• Shin Sasaki and Masaya Yata, Non-geometric five-branes in heterotic supergravity, JHEP 1611, 064, 2016 (arXiv:1608.01436)