Topologically non-trivial solutions of quantum field theory have always been a theoretically “elegant” subject, covering all sorts of interesting and physically relevant field configurations, such as magnetic monopoles, sphalerons and black holes. These objects have played an important role in shaping quantum field theories and have provided important physical insights into cosmology, particle colliders and condensed-matter physics.
In layman’s terms, a field configuration is topologically non-trivial if it exhibits the topology of a “mathematical knot” in some space, real or otherwise. A mathematical knot (or a higher-dimensional generalisation such as a Möbius strip) is not like a regular knot in a piece of string: it has no ends and cannot be continuously deformed into a topologically trivial configuration like a circle or a sphere.
One of the most conceptually simple non-trivial configurations arises in the classification of solitons, which are finite-energy extended configurations of a scalar field behaving like the Higgs field. Among the various finite-energy classical solutions for the Higgs field, there are some that cannot be continuously deformed into the vacuum without an infinite cost in energy, and are therefore “stable”. For finite-energy configurations that are spherically symmetric, the Higgs field must map smoothly onto its vacuum solution at the boundary of space.
The ’t Hooft–Polyakov monopole, which is predicted to exist in grand unified theories, is one such finite-energy topologically non-trivial solitonic configuration. The black hole is an example from general relativity of a singular space–time configuration with a non-trivial space–time topology. The curvature of space–time blows up in the singularity at the centre, and this cannot be removed either by continuous deformations or by coordinate changes: its nature is topological.
Such configurations constituted the main theme of a recent Royal Society Hooke meeting “Topological avatars of new physics”, which took place in London from 4–5 March. The meeting focused on theoretical modelling and experimental searches for topologically important solutions of relativistic quantum field theories in particle physics, general relativity and cosmology, and quantum gravity. Of particular interest were topological objects that could potentially be detectable at the Large Hadron Collider (LHC), or at future colliders.
Gerard ’t Hooft opened the scientific proceedings with an inspiring talk on formulating a black hole in a way consistent with quantum mechanics and time-reversal symmetry, before Steven Giddings described his equally interesting proposal. Another highlight was Nicholas Manton’s talk on the inevitability of topological non-trivial unstable configurations of the Higgs field – “sphalerons” – in the Standard Model. Henry Tye said sphalerons can in principle be produced at the (upgraded) LHC or future linear colliders. A contradictory view was taken by Sergei Demidov, who predicted that their production will be strongly suppressed at colliders.
One of the exemplars of topological physics receiving significant experimental attention is the magnetic monopole
A major part of the workshop was devoted to monopoles. The theoretical framework of light monopoles within the Standard Model, possibly producible at the LHC, was presented by Yong Min Cho. These “electroweak” monopoles have twice the magnetic charge of Dirac monopoles. Like the ’t Hooft–Polyakov monopole, but unlike the Dirac monopole, they are solitonic structures, with the Higgs field playing a crucial role. Arttu Rajantie considered relatively unsuppressed thermal production of generic monopole–antimonopole pairs in the presence of the extreme high temperatures and strong magnetic fields of heavy-ion collisions at the LHC. David Tong discussed the ambiguities on the gauge group of the Standard Model, and how these could affect monopoles that are admissible solutions of such gauge field theories. Importantly, such solutions give rise to potentially observable phenomena at the LHC and at future colliders. Anna Achucaro and Tanmay Vachaspati reported on fascinating computer simulations of monopole scattering, as well as numerical studies of cosmic strings and other topologically non-trivial defects of relevance to cosmology.
One of the exemplars of topological physics currently receiving significant experimental attention is the magnetic monopole. The MoEDAL experiment at the LHC has reported world-leading limits on multiply magnetically charged monopoles, and Albert de Roeck gave a wide-ranging report on the search for the monopole and other highly-ionising particles, with Laura Patrizii and Adrian Bevan also reporting on these searches and the machine-learning techniques employed in them.
Supersymmetric scenarios can consistently accommodate all the aforementioned topologically non-trivial field theory configurations. Doubtless, as John Ellis described, the story of the search for this beautiful – but as yet hypothetical – new symmetry of nature, is a long way from being over. Last but not least, were two inspiring talks by Juan Garcia Bellido and Marc Kamionkowski on the role of primordial black holes as dark matter, and their potential detection by means of gravitational waves.
The workshop ended with a vivid round-table discussion of the importance of a new ~100 TeV collider. The aim of this machine is to explore beyond the historic watershed represented by the discovery of the Higgs boson, and to move us closer to understanding the origin of elementary particles, and indeed space–time itself. This Hooke workshop clearly demonstrated the importance of topological avatars of new physics to such a project.