Abstract
The adiabatic limit procedure associates with every solution of Abelian Higgs model in (2 + 1) dimensions a geodesic in the moduli space of static solutions. We show that the same procedure for Seiberg-Witten equations on 4-dimensional symplectic manifolds introduced by Taubes may be considered as a complex (2+2)-dimensional version of the (2 + 1)-dimensional picture. More precisely, the adiabatic limit procedure in the 4-dimensional case associates with a solution of Seiberg-Witten equations a pseudoholomorphic divisor which may be treated as a complex version of a geodesic in (2+1)-dimensional case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Donaldson S K. The Seiberg-Witten equations and 4-manifold topology. Bull Amer Math Soc, 1996, 33: 45–50
Jaffe A, Taubes C H. Vortices and Monopoles. Boston: Birkhäuser, 1980
Manton N S. A remark on the scattering of BPS monopoles. Phys Lett Ser B, 1982, 110: 54–56
Palvelev R V. Justification of the adiabatic principle in the Abelian Higgs model. Trans Moscow Math Soc, 2011, 72: 219–244
Palvelev R V, Sergeev A G. Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations. Proc Steklov Inst Math, 2012, 277: 191–205
Salamon D. Spin Geometry and Seiberg-Witten Invariants. Warwick: University of Warwick, 1996
Seiberg N, Witten E. Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nuclear Phys B, 1994, 426: 19–22; Erratum: Nuclear Phys B, 1994, 430: 485–486
Seiberg N, Witten E. Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nuclear Phys B, 1994, 431: 484–550
Sergeev A G. Adiabatic limit in Ginzburg-Landau and Seiberg-Witten equations. Proc Steklov Inst Math, 2015, 289: 227–285
Taubes C H. SW ⇒ Gr: From the Seiberg-Witten equations to pseudoholomorphic curves. J Amer Math Soc, 1996, 9: 845–918
Witten E. Monopoles and four-manifolds. Math Res Lett, 1994, 1: 769–796
Acknowledgements
This work was supported by Russian Foundation of Basic Research (Grants Nos. 16-01-00117 and 16-52-12012), the Program of support of Leading Scientific Schools (Grants No. NSh-9110.2016.1) and the Program of Presidium of Russian Academy of Sciences “Nonlinear dynamics”.
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of Professor LU QiKeng (1927–2015)
Rights and permissions
About this article
Cite this article
Sergeev, A. Seiberg-Witten theory as a complex version of Abelian Higgs model. Sci. China Math. 60, 1089–1100 (2017). https://doi.org/10.1007/s11425-016-9030-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-9030-4