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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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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
2. Cherkis S.A., “Octonions, Monopoles, and Knots”, Lett. Math. Phys., 105:5 (2015), 641–659
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
4. Dey R., Thakre V., “Generalized Seiberg-Witten Equations on a Riemann Surface”, J. Geom. Symmetry Phys., 45 (2017), 47–66
5. Biswas I., Thakre V., “Generalised Monopole Equations on Kahler Surfaces”, J. Math. Phys., 59:4 (2018), 043503
6. Thakre V., “Generalised Seiberg-Witten Equations and Almost-Hermitian Geometry”, J. Geom. Phys., 134 (2018), 119–132
7. Doan A., “Seiberg-Witten Monopoles With Multiple Spinors on a Surface Times a Circle”, J. Topol., 12:1 (2019), 1–55
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