## Nagoya Mathematical Journal

### Gluing an infinite number of instantons

Masaki Tsukamoto

#### Abstract

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an "infinite energy" situation. We show that we can glue an infinite number of instantons, and that the resulting ASD connections have infinite energy in general. Moreover they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson's "alternating method".

#### Article information

Source
Nagoya Math. J., Volume 188 (2007), 107-131.

Dates
First available in Project Euclid: 17 December 2007

https://projecteuclid.org/euclid.nmj/1197908745

Mathematical Reviews number (MathSciNet)
MR2371770

Zentralblatt MATH identifier
1147.53023

#### Citation

Tsukamoto, Masaki. Gluing an infinite number of instantons. Nagoya Math. J. 188 (2007), 107--131. https://projecteuclid.org/euclid.nmj/1197908745

#### References

• S. Angenent, The shadowing lemma for elliptic PDE, Dynamics of Infinite Dimensional Systems, NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 37, Springer, Berlin, 1987, pp. 7–22.
• S. K. Donaldson, An application of gauge theory to four dimensional topology, J. Differential Geometry, 18 (1983), 279–315.
• S. K. Donaldson, Connections, cohomology and the intersection forms of $4$-manifolds, J. Differential Geometry, 24 (1986), 275–341.
• S. K. Donaldson, M. Furuta and D. Kotschick, Floer homology groups in Yang-Mills theory, Cambridge University Press, Cambridge, 2002.
• S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, New York, 1990.
• D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Second edition, Springer-Verlag, New York, 1991.
• M. Macr\`i, M. Nolasco and T. Ricciardi, Asymptotics for selfdual vortices on the torus and on the plane: a gluing technique, SIAM J. Math. Anal., 37 (2005), 1–16.
• C. H. Taubes, Self-dual Yang-Mills connections on non-self-dual $4$-manifolds, J. Differential Geometry, 17 (1982), 139–170.
• C. H. Taubes, Self-dual connections on $4$-manifolds with indefinite intersection matrix, J. Differential Geometry, 19 (1984), 517–560.