Geometric Models of Matter Conference

This conference is in part being held to celebrate the 50th anniversary of the seminal papers by ‘t Hooft and Polyakov introducing monopoles in gague theory. Here is Michael Atiyah discussing magnetic monopoles and their dynamics (left) and their continued importance in mathematics and physics (right) while being interviewed by Nigel Hitchin in 1997.

Milestones in the history of monopoles

2024 marks fifty years since the discovery of smooth monopoles in nonabelian gauge theories. Below is a history of milestones in the development of the subject since its inception. This page is not large enough to accommodate all of the many contributions on this topic; the authors have endeavoured to highlight developments that had a lasting influence on the subject.

't Hooft and Polyakov independently discover smooth monopoles 1974

't Hooft and Polyakov independently discover smooth monopoles in nonabelian gauge theories. Polyakov's paper placed them in a broader framework of topological solitons, while 't Hooft's paper highlighted their similarity with Dirac's magnetic monopoles.
G. 't Hooft, Magnetic monopolesin unified gauge theories, Nuclear Physics B79 (1974) 276-284
A. M. Polyakov, Particle spectrum in quantum field theory, ZhETF Pis. Red. 20, No. 6, 430-433

Prasad and Sommerfield discover an explicit solution 1975

Prasad and Sommerfield discover an explicit solution for the monopole in the limit of vanishing potential.
M. K. Prasad and C. M. Sommerfield, Exact Classical Solution for the 't Hooft Monopole and the Julia-Zee Dyon, Phys. Rev. Lett. 35, 760

Bogoml'nyi introduces the BPS equation 1976

Bogomol'nyi shows that, in the same limit, the second order Euler-Lagrange equations can be reduced to first order equations, now known as the Bogomolny equations or BPS equations.
E. B. Bogomol'nyi, The Stability of Classical Solutions, Sov. J. Nucl. Phys., Vol 35, No. 4

The link between the BPS equation and ASD equation is established 1977

Cervero (Harvard University preprint) and Lohe independently establish that the BPS equation on R³ is a reduction of the ASD equation on R⁴.
M. A. Lohe, Two- and Three-Dimensional Instantons, Phys. Lett. 70B, No. 3, 325-328

Coulomb cancellation 1977

Manton shows that the scalar and electromagnetic forces between monopoles cancel to leading order, suggesting the existence of static multimonopoles.
N. S. Manton, The Force Between 't Hooft-Polyakov Monopoles, Nucl. Phys. B. 126, 525-541

Existences of multimonopoles 1980

Jaffe and Taubes prove the existence of multimonopoles using PDE methods.
A. Jaffe and C. Taubes, Vortices and monopoles, Birkhäuser

An axial two-monopole 1981

Ward and, independently, Forgacs--Horvath--Palla give the first direct construction of a charge 2 monopole, using integrable systems and twistor techniques. This monopole is axially-symmetric.
Peter Forgács, Zalán Horváth and Lásló Palla, Exact multimonopole solutions in the Bogomolny-Prasad-Sommerfied limit, Physics Letters B 99 232-236
R. S. Ward, A Yang-Mills-Higgs Monopole of Charge 2, Commun. Math. Phys. 79, 317-325

Nahm's equations 1981

Nahm introduces a construction of monopoles from solutions of and ODE, now called the Nahm equation, by adapting the ADHM technique.
W. Nahm, All Self-Dual Multimonopoles for Arbitrary Gauge Groups, Cern preprint TH 3172-CERN

The geodesic approximation 1981

Manton introduced the idea that the low-energy dynamics of magnetic monopoles is approximated by geodesics in the moduli space of static monopoles with the L² metric. This would later be proved by Stuart.
N. S. Manton, A remark on the scattering of BPS monopoles, Physics Letters 110B, 1, 54-56

Spectral curves appear 1982

Hitchin introduces the spectral curve associated to a monopole (introducing minitwistor space in the process) and shows it in turn determines the monopole.
N. J. Hitchin, Monopoles and Geodesics, Commun. Maths. Phys. 83, 579-602

Nahm transform for SU(2) 1983

Hitchin relates Nahm's equation to the monopole spectral curves and thereby shows that Nahm's transform constructs all SU(2) BPS monopoles.
N. J. Hitchin, On the Construction of Monopoles, Commun. Math. Phys. 89, 145-190

Hyperbolic monopoles and rational maps 1984

Atiyah introduces the study of monopoles on hyperbolic space, and relates these to instantons and rational maps. Following this, Donaldson and then Hurtubise describe the moduli space of charge-k monopoles in terms of degree-k rational maps.
M. F. Atiyah, Magnetic monopoles in hyperbolic spaces, in: Vector bundles on algebraic varieties, Tata Institute of Fundamental Research, Bombay.
S. K. Donaldsson, Nahm's equations and the classification of monopoles, Commun. Math. Phys. 96, 387-407
Jacques Hurtubise, Monopoles and rational maps: a note on a theorem of Donaldson, Commun. Math. Phys. 100, 191-196

The 2-monopole metric 1985

Atiyah and Hitchin explicitly construct the hyperkähler metric on the moduli space of centred charge 2 monopoles.
M. F. Atiyah, N. J. Hitchin, Low energy scattering of non-abelian monopoles, Physics Letters A 107.1 (1985).
M. F. Atiyah, N. J. Hitchin, Low-energy scattering of non-Abelian magnetic monopoles, Philosophical Transactions of the Royal Society A 315.1533 (1985).

Developments on Nahm's equations 1989

Ercolani and Sinha show how to solve Nahm's equations with gauge group SU(2) using the Baker-Akhiezer function, and Hurtubise and Murray extend the Nahm transform to all classical Lie groups, under the assumption of maximal symmetry-breaking. The formulation of a Nahm transform for the exceptional Lie groups is to this day an open problem.
N. Ercolani and A. Sinha, Monopoles and Baker Functions, Commun. Math. Phys. 125, 385-416
J. Hurtubise and M. K. Murray, On the Construction of Monopoles for the Classical Groups, Commun. Math. Phys. 122, 35-89

Hyperbolic monopoles are holographic 1990

Braam and Austin exploit the fact that hyperbolic monopoles are determined by their asymptotic connection to give a metric on the space of hyperbolic monopoles.
P. J. Braam and D. M. Austin, Boundary Values of Hyperbolic Monopoles, Nonlin. 3, 809-823

Bound states of monopoles 1994

Sen conjectures the existence of normalisable harmonic forms on monopole moduli spaces, based on dualities of supersymmetric theories.
A. Sen, Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2,Z) invariance in string theory, Physics Letters B 329, 217-221

The Seiberg-Witten solution 1994

Seiberg and Witten derive the quantum moduli space of N=2 super-Yang-Mills theory, and in particular show that monopoles are responsible for confinement in these gauge theories.
N. Seiberg and E. Witten, Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory, Nucl. Phys. B 426:19-52
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B 431:484-550

Geodesic approximation is proved 1994

Stuart proves that the low-energy dynamics of monopoles is approximated by geodesics in the moduli space, confirming a prediction of Manton.
D. Stuart, The geodesic approximation for the Yang-Mills-Higgs equations, Commun. Math. Phys. 166: 149-190

Symmetric Monopoles 1995

Hitchin, Manton and Murray find the Nahm data of some monopoles with platonic symmetry, and use rational maps to describe the scattering of cyclically-symmetric monopoles.
N. J. Hitchin, N. S. Manton and M. K. Murray, Symmetric monopoles, Nonlinearity 8, 661-692

Visualising monopoles 1995

Houghton and Sutcliffe construct explicit Nahm data for many monopoles, and use this to visualise monopoles by plotting iso-energy-density surfaces.
C. J. Houghton and P. M. Sutcliffe, Tetrahedral and cubic monopoles, Commun. Math. Phys. 180, 343-361
C. J. Houghton and P. M. Sutcliffe, Octahedral and dodecahedral monopoles, Nonlinearity 9, 385
C. J. Houghton and P. M. Sutcliffe, Monopole scattering with a twist, Nucl. Phys. B 464, 59-84
C. J. Houghton and P. M. Sutcliffe, Inversion symmetric 3-monopoles and the Atiyah-Hitchin manifold, Nonlinearity 9, 1609-1622

Sen conjecture for 2-monopoles 2000

Hitchin proves Sen's conjectures for the 2-monopole moduli space.
N. Hitchin, L² cohomology of hyperkähler quotients, Commun. Math. Phys. 211, 153-165

Periodic monopoles 2000

Cherkis and Kapustin begin the study of periodic monopoles by formulating the Nahm transform.
S. A. Cherkis and A. Kapustin, Nahm Transform For Periodic Monopoles And N=2 Super Yang-Mills Theory, Commun. Math. Phys. 218, 333-371

The magnetic bag conjecture 2006

Bolognesi conjectures that the smallest monopole of fixed charge is shell-like object called a "magnetic bag".
S. Bolognesi, Multi-monopoles and Magnetic Bags, Nucl. Phys. B 752:93-123

Monopole walls 2012

Cherkis and Ward formulate the Nahm transform and spectral data for doubly-periodic monopoles, also known as monopole walls.
S. A. Cherkis and R. S. Ward, Moduli of Monopole Walls and Amoebas, JHEP 1205 090

G₂ monopoles 2013

Oliveira begins the study of monopoles on G₂-manifolds by constructing the first examples.
Goncalo Oliveira, Monopoles on the Bryant–Salamon G₂-manifolds, J. Geom. Phys. 86 599-632

Framed BPS states 2016

Moore, Royston and Van den Bleeken formulate make predictions for kernels of Dirac operators on moduli spaces of (singular) monopoles, based on an analysis of framed BPS states with defects.
G. W. Moore, A. B. Royston and D. Van den Bleeken, Semiclassical framed BPS states, JHEP07 071
G. W. Moore, A. B. Royston and D. Van den Bleeken, L²-Kernels Of Dirac-Type Operators On Monopole Moduli Spaces, arXiv:1512.08923

Explicit construction of monopoles 2018

Braden and Enolski show how to construct monopoles explicitly using the Nahm transform. In particular, they construct the 2-monopole explicitly.
H. W. Braden and V. Z. Enolski, The Construction of Monopoles, Commun. Math. Phys. 362, 547–570
H. W. Braden and V. Z. Enolski, The Charge 2 Monopole via the ADHMN construction, Advances in Theoretical and Mathematical Physics 25 (2021) 791-956

Sen conjecture for 3-monopoles 2021

Kottke and Rochon prove Sen's conjecture for the 3-monopole moduli space, using a compactification due to Fritzsch-Kottke-Singer.
K. Fritzsch, C. Kottke and M. Singer, Monopoles and the Sen Conjecture: Part I, arXiv:1811.00601
C. Kottke and F. Rochon, L²-cohomology of quasi-fibered boundary metrics, arXiv:2103.16655

50 years of monopoles in gauge theory

Milestones in the history of monopoles