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Dear fellow mathematicians, during some gauge-theoretic explorations
I've stumble upon a recent paper by Nick Manton on Abelian higgs vortices where he studies Abelian higgs "vortices" on hyperbolic surfaces.
Does anyone have any historical knowledge of whether the word vortex in this context have any connection with point vortices from fluid dynamics?

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You may want to take a look at Jaffe and Taubes, Vortices and Monopoles, Birkhauser. –  Willie Wong Oct 18 '10 at 11:11
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The vortices are not in a fluid, but in the magnetic field in a superconducting magnet. The vortex equations describe such a field at a critical value of a certain coupling constant (and are therefore of limited physical significance). Besides Jaffe-Taubes' classic, take a look at Witten's article in the AMS Bulletin: ams.org/journals/bull/2007-44-03 –  Tim Perutz Oct 18 '10 at 15:22
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2 Answers 2

The study of vortices in Abelian Higgs models in the context of relativistic QFT goes back to the paper by Nielsen and Olesen, Nucl.Phys.B61:45-61,1973. They in turn were directly inspired by the study of vortices carrying quantized magnetic flux in type II superconductors which goes back to Abrikosov (see his Nobel lecture here: http://nobelprize.org/nobel_prizes/physics/laureates/2003/abrikosov-lecture.html). In that context the Abelian gauge field is identified with the electromagnetic field and the Higgs field with the Cooper pair of electrons. I have not gone back to the original papers by Abrikosov (which are probably in Russian) but I would guess that the term vortex is based on an analogy to fluid mechanics with the vorticity of the fluid vortex (curl of the velocity field) being analogous to the magnetic field of the Abelian Higgs vortex (curl of the vector potential).

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I expect that this terminology came from an analogy of an analogy.

Quoting from Quantum Field Theory of Anyons, by Fröhlich and Marchetti (1988), "Particles with real spin and intermediate statistics have been called 'anyons' by Wilczek [1]. He suggested that electrically charged magnetic vortices, occuring in certain Higgs models, may be examples of such particles."

So certain types of anyons can reasonably be, and have been, called vortices. I looked at the Manton paper, but didn't understand it at all; however, my guess is that the things Manton is calling vortices are related to the electrically charged magnetic vortices mentioned above (and if you consider these just mathematically and not physically, there is no apparent reason that they should be called vortices).

I would welcome corrections from anyone with better knowledge of the historical record (my answer came from knowing about the anyonic vortex terminology and Google).

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Thanks Peter, that certainly helps. –  swallowtail Oct 19 '10 at 20:51
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