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Tr. Mat. Inst. Steklova, 2004, Volume 246, Pages 263–276 (Mi tm159)  

This article is cited in 7 scientific papers (total in 7 papers)

Hyperkähler Manifolds and Seiberg–Witten Equations

V. Ya. Pidstrigach

Mathematisches Institut, Georg-August-Universität Göttingen

Abstract: The mathematical properties of the so-called gauged nonlinear $\sigma$-model in dimension 4 are studied. An important element of the construction is a nonlinear generalization of the Dirac operator on a 4-manifold such that the fiber of the spinor vector bundle, a copy of quaternions $\mathbb H$, is replaced by a hyperkähler manifold endowed with a hyperkähler Lie group action and an additional symmetry. This Dirac operator is used to define Seiberg–Witten moduli spaces. An explicit Weitzenböck formula for such a Dirac operator is derived and applied to describe some properties of the Seiberg–Witten moduli spaces.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 246, 249–262

Bibliographic databases:
UDC: 514.7+514.8
Received in February 2004

Citation: V. Ya. Pidstrigach, “Hyperkähler Manifolds and Seiberg–Witten Equations”, Algebraic geometry: Methods, relations, and applications, Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 246, Nauka, MAIK Nauka/Inteperiodika, M., 2004, 263–276; Proc. Steklov Inst. Math., 246 (2004), 249–262

Citation in format AMSBIB
\Bibitem{Pid04}
\by V.~Ya.~Pidstrigach
\paper Hyperk\"ahler Manifolds and Seiberg--Witten Equations
\inbook Algebraic geometry: Methods, relations, and applications
\bookinfo Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences
\serial Tr. Mat. Inst. Steklova
\yr 2004
\vol 246
\pages 263--276
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm159}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2101297}
\zmath{https://zbmath.org/?q=an:1101.53026}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2004
\vol 246
\pages 249--262


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Haydys A., “Nonlinear Dirac operator and quaternionic analysis”, Communications in Mathematical Physics, 281:1 (2008), 251–261  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Cherkis S.A., “Octonions, Monopoles, and Knots”, Lett. Math. Phys., 105:5 (2015), 641–659  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Nakajima H., “Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, I”, Adv. Theor. Math. Phys., 20:3 (2016), 595–669  crossref  mathscinet  zmath  isi  elib  scopus
    4. Dey R., Thakre V., “Generalized Seiberg-Witten Equations on a Riemann Surface”, J. Geom. Symmetry Phys., 45 (2017), 47–66  crossref  mathscinet  zmath  isi  scopus
    5. Biswas I., Thakre V., “Generalised Monopole Equations on Kahler Surfaces”, J. Math. Phys., 59:4 (2018), 043503  crossref  mathscinet  zmath  isi  scopus
    6. Thakre V., “Generalised Seiberg-Witten Equations and Almost-Hermitian Geometry”, J. Geom. Phys., 134 (2018), 119–132  crossref  mathscinet  zmath  isi  scopus
    7. Doan A., “Seiberg-Witten Monopoles With Multiple Spinors on a Surface Times a Circle”, J. Topol., 12:1 (2019), 1–55  crossref  mathscinet  zmath  isi  scopus
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