# Professor Nicholas Manton

## Career

- Postdoctoral Appointments: ENS, Paris; MIT; ITP Santa Barbara; St John's College, Cambridge (1978-87).
- University Appointments in DAMTP: University Lecturer (1987-94), Reader (1994-98), Professor of Mathematical Physics (1998-present).
- Head, High Energy Physics group, DAMTP (2002-present).
- Adjudicator of the Adams Prize (2000-02, 2004-present).
- Member of Council, School of the Physical Sciences (2007).
- Fellow of St John's College (1997-present), Director of Studies in Applied Mathematics (1997-98), Member of College Council (2006-09).
- Junior Whitehead Prize, London Mathematical Society, 1991.
- Fellow of the Royal Society, 1996.

## Research

Nick Manton's research interests cover broad areas of theoretical and mathematical physics, in particular, classical and quantum field theory applied to particle and nuclear physics. The majority of his work has been on topological solitons in field theory, which include vortices, monopoles, instantons and Skyrmions. He proposed and developed the theory of non-relativistic soliton dynamics based on the geometry of soliton moduli spaces; this is called the geodesic approximation to soliton dynamics. Some recent work, using a variant of Ricci flow, has elucidated the K"ahler geometry of vortex moduli spaces.

A major recent interest has been to develop Skyrme's idea from the 1960s that atomic nuclei can be modelled as quantized solitons in a nonlinear field theory of pions. These solitons are called Skyrmions, and they have very interesting polyhedral shapes and symmetries. Larger Skyrmions are not just clusters of touching basic Skyrmions (which model protons and neutrons), but the basic Skyrmions deform and merge to some extent, as one expects physically. Improvements in computational power as well as mathematical insight has made it possible to construct and quantize Skyrmions as needed to model Carbon-12 and Oxygen-16. The spins and energies of nuclear ground states and excited states are described quite well by Skyrmions. Compared with other nuclear models, the Skyrme model manages to unify into the spectrum of a single "rigid-body" Hamiltonian the spin excitations of a given nucleus like Carbon-12 with its isospin excitations which include Boron-12 and Nitrogen-12. This work has been in collaboration with Sir Michael Atiyah (Edinburgh), Paul Sutcliffe (Durham), Richard Battye (Manchester) and several graduate students.

About 30 years ago, while a postdoc at ITP Santa Barbara, and in collaboration with Frans Klinkhamer (now at Karlsruhe), he found unstable solitons in the Standard Model of elementary particles. Klinkhamer and Manton called these sphalerons, a name that has caught on and now has a Wikipedia entry. Sphalerons require a dynamical Higgs field, so the recent discovery of the Higgs boson confirms their mathematical validity, and allows one to estimate that the sphaleron energy is about 8 TeV. The probability of producing a sphaleron in particle collisions is thought to be very small, but it will be interesting to investigate this in the light of forthcoming experiments at the CERN LHC on Higgs and multiple W- and Z-boson processes, when LHC achieves collisions at 14 TeV, which it hopes to after the 2013-14 upgrade. Potentially, sphalerons are a source of baryon number violation, important for understanding the matter-antimatter asymmetry of the universe.

## Selected Publications

- Rational Maps, Monopoles and Skyrmions (with C.J.Houghton and P.M. Sutcliffe), Nuclear Physics B510 (1998) 507-537.
- Volume of Vortex Moduli Spaces (with S.M. Nasir), Communications in Mathematical Physics 199 (1999) 591-604.
- Conservation Laws in a First-Order Dynamical System of Vortices (with S.M. Nasir), Nonlinearity 12 (1999) 851-865.
- Asymptotic Interactions of Critically Coupled Vortices (with J.M. Speight), Communications in Mathematical Physics 236 (2003) 535-555.
- The Interaction Energy of Well-separated Skyrme Solitons (with B.J. Schroers and M.A. Singer), Communications in Mathematical Physics 245 (2004) 123-147.
- The Kähler Potential of Abelian Higgs Vortices (with H.-Y. Chen), Journal of Mathematical Physics 46 (2005) 052305.
- Reduced Dynamics of Ward Solitons (with M. Dunajski), Nonlinearity 18 (2005) 1677-1689.
- Superevolution, Journal of Physics A38 (2005) 6065-6079.
- Skyrmions and the ∝-Particle Model of Nuclei (with R.A. Battye and P.M. Sutcliffe), Proceedings of the Royal Society A463 (2007) 261-279.
- Skyrmions and Nuclei, in The Many Facets of Geometry: A tribute to Nigel Hitchin, eds. O. García-Prada, J.P. Bourguignon and S. Salamon, Oxford University Press, 2010.
- One-Vortex Moduli Space and Ricci Flow, Journal of Geometry and Physics 58 (2008) 1772-1783.
- Solitons as Elementary Particles: A Paradigm Scrutinized, Nonlinearity 21 (2008) T221-T232.
- Skyrmions and Nuclei (with R.A. Battye and P.M. Sutcliffe), in The Multifaceted Skyrmion, eds. G.E. Brown and M. Rho, World Scientific, Singapore, 2010.
- Light Nuclei of Even Mass Number in the Skyrme Model (with R.A. Battye, P.M. Sutcliffe and S.W. Wood), Physical Review C80 (2009) 034323.
- Vortices on Hyperbolic Surfaces (with N.A. Rink), Journal of Physics A43 (2010) 434024 (also in IOP Select).
- Vortices and Jacobian Varieties (with N.M. Romão), Journal of Geometry and Physics 61 (2011) 1135-1155.
- Classical Skyrmions -- Static Solutions and Dynamics, Mathematical Methods in the Applied Sciences 35 (2012) 1188-1204.
- Geometric Models of Matter (with M.F. Atiyah and B.J. Schroers), Proceedings of the Royal Society A468 (2012) 1252-1279.
- Monopole Planets and Galaxies, Physical Review D85 (2012) 045022.
- Platonic Hyperbolic Monopoles (with P.M. Sutcliffe), arXiv:1207.2636 (accepted, Communications in Mathematical Physics 2013).
- Skyrmions up to Baryon Number 108 (with D.T.J. Feist and P.H.C. Lau), Physical Review D87 (2013) 085034.
- Vortex Solutions of the Popov Equations, Journal of Physics A46 (2013) 145402.
- Vortex Motion on Surfaces of Small Curvature (with D. Dorigoni and M. Dunajski), arXiv:1308.3088 (submitted, Annals of Physics).

**Book**: Topological Solitons (with P. Sutcliffe), Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2004; paperback edition (with minor corrections), 2007.**Editorial**: Special issue on Integrability, Topological Solitons and Beyond (coedited with A.S. Fokas), Journal of Mathematical Physics 44 (8) (2003) 3147-3673.

## External Activities

- Programme Committee of the International Centre for Mathematical Sciences, Edinburgh (2005-08).
- Chair, Research Meetings Committee, London Mathematical Society (2008-11).
- Management Committee of the Isaac Newton Institute (2011-present).
- Editorial Boards: Nonlinearity (1997-99), Royal Society A-side (2004-08).