Abstract
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.
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Notes
Here, \(M_0\) will be either \(\mathbb {S}^2\), \(\mathbb {H}^2\) or \(\mathbb {R}^2\). In the two latter cases, z is a global coordinate, whereas for \(\mathbb {S}^2\) we need at least two patches to cover the space globally.
This will shove the multivalued \(\alpha \) phase of the Higgs field into the gauge field.
Of course physically, it is semantics whether the VEV \(\uplambda _0\) is physically the VEV or it has further contributions from other “fields,” like \(\uplambda _0-\sigma \), when \(\sigma \) is a constant.
Toda equations occur when K is the Cartan matrix of any simple Lie algebra. Thus, there will also be Toda equations corresponding to the Cartan matrix for \(G_{2}\).
In the general case of the Toda system with a rank R Lie algebra, the matrices are \((R+1)\)-by-A rectangular matrices given by \(M_A {:}{=}(u,\partial _z u,\partial _z^2 u,\ldots ,\partial _z^{A-1} u)\) and u is given by \((1,f_1(z),f_2(z),\ldots ,f_R(z))^{\mathrm{T}}\), where \(A=1,2,\ldots ,R\).
In the general case of the Toda system with a rank R Lie algebra, the matrices are \((R+1)\)-by-A rectangular matrices given by \(M_A {:}{=}(u,\partial _z u,\partial _z^2 u,\ldots ,\partial _z^{A-1} u)\) and \(W_A {:}{=}(v,\partial _z v,\partial _z^2 v,\ldots ,\partial _z^{A-1} v)\) with u given by \((1,f_1(z),f_2(z),\ldots ,f_R(z))^\mathrm{T}\) and v given by \((1,-\uplambda f_1(z),-\uplambda f_2(z),\ldots ,-\uplambda f_R(z))^\mathrm{T}\), where \(A=1,2,\ldots ,R\).
The latter is not really a Toda system of equations, because \(\uplambda =0\) eliminates the mixing of the fields.
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Acknowledgements
C. R. thanks Steffen Krusch for introducing him to the topic of vortices with magnetic impurities.
Funding
S. B. G. thanks the Outstanding Talent Program of Henan University for partial support. The work of S. B. G. was supported by the National Natural Science Foundation of China (Grant No. 11675223 and No. 12071111).
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Gudnason, S.B., Ross, C. Magnetic impurities, integrable vortices and the Toda equation. Lett Math Phys 111, 100 (2021). https://doi.org/10.1007/s11005-021-01444-8
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DOI: https://doi.org/10.1007/s11005-021-01444-8
Keywords
- Vortex equations
- Integrable vortices
- Toda equation
- Field theory on curved spaces
- Impurities in field theory