Moduli Space for Invariant Solutions of Seiberg-Witten Equations

Year 2024, Volume: 19 Issue: 1, 8 - 17, 27.05.2024

Muhiddin Uğuz

https://doi.org/10.29233/sdufeffd.1407647

Abstract

In this work we study the G-invariant solutions of the Seiberg-Witten equations when G is a cyclic group acting on a manifold M, preserving the metric and the orientation. G is assumed to have a lift to principle 〖Spin〗^c bundle which gives rise to Seiberg-Witten equations in question. In this work, we prove that when the dimension b_+^G of the G-fixed points of harmonic two forms is positive, for a generic choice of an element in this fixed point set, the moduli space of invariant solutions of Seiberg-Witten equations is a compact, smooth and oriented manifold of dimension d^G=ind D_A^G-b_+^G-1.

Keywords

Gauge Theory, Equivariant Seiberg-Witten theory, Equivariant Seiberg-Witten moduli space.

Ethical Statement

As the author of this study, I declare that I do not have any ethics committee approval and/or informed consent statement.

References

• J. H. C. Whitehead, “On simply-connected 4-dimensional polyhedral”,Comment. Math.Helv., 22:48–92, 1949.
• S.K. Donaldson and P.B. Kronheimer, The Geometry of Four-Manifolds, Clarendon Press- Oxford, 1990.
• M.Freedman, “The topology of four dimensional manifolds”, J. Diff. Geo., 17:357–454, 1982.
• J.W. Milnor and D. Husemoller, “Symmetric Bilinear Forms”, Ergebnisse der Mathematik und ihrer Grezgebiete, Volume 73. Springer Verlag, New York-Heidelberg-Berlin, 1973.
• John D. Moore, Lectures on Seiberg-Witten Invariants. Springer Verlag, 1996.
• Ted Petrie and John Randall. Connections,Definite Forms, and Four-Manifolds. Clarendon Press Oxford, 1990.
• John W. Morgan. The Seiberg-Witten Equations and Application to the Topology of Smooth four-Manifolds. Princeton University Press, 1996.
• Daniel S. Freed, Karen K. Uhlenbeck Instantons and 4-Manifolds. Springer-Verlag, 1984.