On the homology of 𝑆𝑈(𝑛) instantons

Boyer, Charles P.; Mann, Benjamin M.; Waggoner, Daniel

doi:10.1090/S0002-9947-1991-1034658-2

On the homology of $\textrm {SU}(n)$ instantons
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by Charles P. Boyer, Benjamin M. Mann and Daniel Waggoner

Trans. Amer. Math. Soc. 323 (1991), 529-561

DOI: https://doi.org/10.1090/S0002-9947-1991-1034658-2

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Abstract:

In this paper we study the homology of the moduli spaces of instantons associated to principal ${\mathbf {SU}}(n)$ bundles over the four-sphere. This is accomplished by exploiting an "iterated loop space" structure implicit in the disjoint union of all moduli spaces associated to a fixed ${\mathbf {SU}}(n)$ with arbitrary instanton number and relating these spaces to the known homology structure of the four-fold loop space on $B{\mathbf {SU}}(n)$.

References

Tatsuji Kudo and Shôrô Araki, Topology of $H_n$-spaces and $H$-squaring operations, Mem. Fac. Sci. Kyūsyū Univ. A 10 (1956), 85–120. MR 87948, DOI 10.2206/kyushumfs.10.85
M. F. Atiyah, Geometry on Yang-Mills fields, Scuola Normale Superiore, Pisa, 1979. MR 554924
M. F. Atiyah, N. J. Hitchin, V. G. Drinfel′d, and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. MR 598562, DOI 10.1016/0375-9601(78)90141-X
M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
M. F. Atiyah and J. D. S. Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61 (1978), no. 2, 97–118. MR 503187, DOI 10.1007/BF01609489
J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609, DOI 10.1007/BFb0068547
Charles P. Boyer and Benjamin M. Mann, Homology operations on instantons, J. Differential Geom. 28 (1988), no. 3, 423–465. MR 965223
Charles P. Boyer and Benjamin M. Mann, Instantons and homotopy, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 87–102. MR 1000369, DOI 10.1007/BFb0085220
William Browder, Homology operations and loop spaces, Illinois J. Math. 4 (1960), 347–357. MR 120646
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146, DOI 10.1007/BFb0080464
S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), no. 4, 453–460. MR 763753, DOI 10.1007/BF01212289
Eldon Dyer and R. K. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), 35–88. MR 141112, DOI 10.2307/2372804
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and four-manifolds, Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, 1984. MR 757358, DOI 10.1007/978-1-4684-0258-2
David Groisser and Thomas H. Parker, The geometry of the Yang-Mills moduli space for definite manifolds, J. Differential Geom. 29 (1989), no. 3, 499–544. MR 992329
N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. MR 877637, DOI 10.1007/BF01214418
Jacques Hurtubise, Instantons and jumping lines, Comm. Math. Phys. 105 (1986), no. 1, 107–122. MR 847130, DOI 10.1007/BF01212344
H. Blaine Lawson Jr., The theory of gauge fields in four dimensions, CBMS Regional Conference Series in Mathematics, vol. 58, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 799712, DOI 10.1090/cbms/058
J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609, DOI 10.1007/BFb0068547
R. James Milgram, Iterated loop spaces, Ann. of Math. (2) 84 (1966), 386–403. MR 206951, DOI 10.2307/1970453
R. James Milgram, The $\textrm {mod}\ 2$ spherical characteristic classes, Ann. of Math. (2) 92 (1970), 238–261. MR 263100, DOI 10.2307/1970836
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
M. S. Narasimhan and S. Ramanan, Existence of universal connections, Amer. J. Math. 83 (1961), 563–572. MR 133772, DOI 10.2307/2372896
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197
Clifford Henry Taubes, Stability in Yang-Mills theories, Comm. Math. Phys. 91 (1983), no. 2, 235–263. MR 723549, DOI 10.1007/BF01211160
Clifford Henry Taubes, Path-connected Yang-Mills moduli spaces, J. Differential Geom. 19 (1984), no. 2, 337–392. MR 755230
Clifford Henry Taubes, The stable topology of self-dual moduli spaces, J. Differential Geom. 29 (1989), no. 1, 163–230. MR 978084

Loop spaces and the classical unitary groups

George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0