TY - JOUR

T1 - Magnetic Impurities, Integrable Vortices and the Toda Equation

AU - Gudnason, Sven Bjarke

AU - Ross, Calum

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2021/7/21

Y1 - 2021/7/21

N2 - 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.

AB - 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.

KW - Field theory on curved spaces

KW - Impurities in field theory

KW - Integrable vortices

KW - Toda equation

KW - Vortex equations

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UR - https://www.mendeley.com/catalogue/71ec4971-d566-3a52-bbb5-f9b929764e56/

U2 - 10.1007/s11005-021-01444-8

DO - 10.1007/s11005-021-01444-8

M3 - Article (journal)

SN - 1573-0530

SN - 1573-0530

VL - 111

SP - 1

EP - 20

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

M1 - 100

ER -