Magnetic impurities, integrable vortices and the Toda equation

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Gudnason, Sven Bjarke
Ross, Calum
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The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong–Wong type. Under certain conditions, these generalizations remain integrable. We further set up a gauge theory with a product gauge group, two complex scalar fields and a general charge matrix. The second species of vortices, when frozen, are interpreted as the magnetic impurity for all five vortex equations. We then give a geometric compatibility condition, which enables us to remove the constant term in all the equations. This is similar to the reduction from the Taubes equation to the Liouville equation. We further find a family of charge matrices that turn the five vortex equations into either the Toda equation or the Toda equation with the opposite sign. We find exact analytic solutions in all cases and the solution with the opposite sign appears to be new.
Vortex equations , Integrable vortices , Toda equation , Field theory on curved spaces , Impurities in field theory
Gudnason, S.B., and Ross, C. (2021) 'Magnetic impurities, integrable vortices and the Toda equation', Letters in Mathematical Physics, 111, 100 (20pp). doi: 10.1007/s11005-021-01444-8
© 2021, the Authors, under exclusive licence to Springer Nature B.V. This is a post-peer-review, pre-copyedit version of an article published in Letters in Mathematical Physics. The final authenticated version is available online at: