Real instantons, Dirac operators and quaternionic classifying spaces

Authors:
Paul Norbury and Marc Sanders

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2193-2201

MSC (1991):
Primary 53C07, 55P38; Secondary 55R45

DOI:
https://doi.org/10.1090/S0002-9939-96-03358-8

MathSciNet review:
1327032

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the moduli space of based gauge equivalence classes of instantons on principal bundles over with first Pontryagin class . In this paper, we use a monad description (Y. Tian, *The Atiyah-Jones conjecture for classical groups*, preprint, S. K. Donaldson, Comm. Math. Phys. **93** (1984), 453--460) of these moduli spaces to show that in the limit over , the moduli space is homotopy equivalent to the classifying space . Finally, we use Dirac operators coupled to such connections to exhibit a particular and quite natural homotopy equivalence.

**[ADHM]**M.F. Atiyah, V. G. Drinfeld, N.J. Hitchin and Y. I. Manin,*Construction of instantons*, Phys. Lett. A, 65 (1978), 185-187. MR**82g:81049****[AJ]**M.F. Atiyah and J.D.S. Jones,*Topological aspects of Yang-Mills theory*, Comm. Math. Phys. 61 (1978), 97-118. MR**80j:58021****[AHS]**M.F. Atiyah, N.J. Hitchin, and I.M. Singer,*Self-duality in four dimensional Riemannian Geometry*, Proc. of Royal Society London, A 362 (1978), 425-461. MR**80d:53023****[D]**S.K. Donaldson,*Instantons and geometric invariant theory*, Comm. Math. Phys., 93 (1984), 453-460. MR**86m:32043****[K]**F. Kirwan,*Geometric Invariant Theory and the Atiyah-Jones conjecture*, Proceedings of the Sophus Lie Memorial Conference (Oslo, 1992), editors O.A. Laudal and B. Jahren, Scandinavian University Press, 1994, 161-188.**[S]**M. Sanders,*Classifying spaces and Dirac operators coupled to instantons*, Transactions of the American Mathematical Society**347**(1995), 4037--4072. CMP**95:15****[Ti]**Y. Tian,*The Atiyah-Jones Conjecture for Classical Groups*, preprint.

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Additional Information

**Paul Norbury**

Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Email:
norbs@maths.warwick.ac.uk

**Marc Sanders**

Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, Minneapolis, Minnesota 55455

Email:
sanders@math.umn.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03358-8

Keywords:
Instantons,
Dirac operators,
classifying space

Received by editor(s):
January 13, 1995

Communicated by:
Ronald Stern

Article copyright:
© Copyright 1996
American Mathematical Society