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-References and Footnotes-
M.F. ATIYAH, V.G. DRINFELD, N.J. HITCHIN, Yu.I. KANIN, Phys. Lett. 65A (1978), 185
V.G. DRINFELD, Yu.I. MANIN, Uspehi Mat. Nauk 33: 3 (1978), 241
V.G. DRINFELD, Yu.I. MANIN,-Funct. An. Appl. 12: 2 (1978), 81
Preprint ITEP N 72 (1978)
Preprints Moscow University and Steklov Institute, May 1978, Commun.math. Phys. 63, 177 (1978)
W. BARTH, K. HULEK, Manuscripta Mathematica 25, 323 (1978)
G. 't HOOFT, unpublished
R. JACKIW, C. NOHL, C. REBBI, Phys. Rev. D 15, 1642 (1977)
R. JACKIW, C. REBBI, Phys. Rev. Lett. B 67, 189 (1977)
A.S. SCHWARTZ, Phys. Lett. B 67, 172 (1977) Commun.math.Phys., to appear
M.F. ATIYAH, N.J. HITCHIN, I.M. SINGER, Proc. Nat. Acad. Sci. USA, 74 (1977) Proc. Lond. Math. Soc. (1978)
A. BELAVIN, A. POLYAKOV, A.S. SCHWARZ, Y. TYUPKIN, Phys. Lett. B 59, 85 (1975)
R.S. WARD, Phys. Lett. A 61, 81 (1977)
M.F. ATIYAH, R.S. WARD, Commun.math.Phys. 55 (1977), 117
P.B. GILKEY, The Index Theorem and the Heat Equation. Mathematical Lecture Series, 4, Publish or Perish, Boston (1974)
R. PALAIS, Seminar on the Atiyah-Singer Index Theorem, Ann. Math. Stud. no 57, Princeton University Press, Princeton (1965)
H. CARTAN, L. SCHWARTZ, Ed., Séminaire Henri CARTAN, E.N.S. Paris, 1963–1964, Secretariat Mathematique, 11, rue Pierre Curie, 75005 Paris
R. HARTSHORNE, Algebraic Geometry, Graduate Texts in Math. 52, Springer Verlag, New York (1977); see, in particular, p. 250.
R. HARTSHORNE, Commun.math. Phys. 59, 1 (1978)
Ref. [5], [3]-R. STORA in International School of Mathematical Physics, Ettore Majorana, Erice Sicily 1977, Invariant Wave Equations, G. Velo, A.S. Wightman Ed., Springer Lecture Notes in Physics, Vol. 73, Berlin(1978)
-J. MADORE, J.L. RICHARD, R. STORA in Meeting on Solitons, Instantons and Turbulence, Les Houches 1978, E. Brezin, J.L. Gervais Ed., Physics Reports, to be published.
R.O. WELLS, Differential Analysis on Complex Manifolds, Prentice Hall Series in Modern Analysis (1973)
“Twistor News Letters”
F. HIRZEBRUCH, Topological Methods in Algebraic Geometry, Third Edition, Springer Verlag, New York (1966).
That the Euler characteristics of E( (l) is the index of an AHS complex was noticed by Yu. M. Malyuta (Kiev) in November 1977. This author however did not show why this is so. See also R. HARTSHORNE, Stable Vector Bundles of Rank 2 over P3, Berkeley preprint. We wish to thank I.M. Singer for pointing out this reference.
The derivation of H1(P 3, E(−2)) = 0 has been given by several authors, see ref. [1], Dolbeaut cohomology is usually preferred rather than Cečh cohomology (see e.g. J.H. Rawnsley: “Differential Geometry of Instantons”, “On the Atiyah Hitchin Drinfeld Manin vanishing theorem”, Dublin Institute for Advanced Study preprint). A spectral sequence argument has been given by J.L. Verdier in Séminaire ENS 1977–1978, A. Douady, J.L. Verdier Ed. to be published in Astérisque, and communication in “Journées sur les champs de Yang et Mills”, S.M.F. May 24–26, (1978). One can prove [1] by recursion that if H°(P3, E(−1)) = 0, H1(P3, E(−2)) = 0, H°(P3, E(−l)) = H1(P3, E (−l −1)) = 0, for 1 > 1 by using the fact that on a two plane P2 c P3 containing a real line, E is trivial on almost all straightlines in P2, hence H°(P3, Elp2(− e)) = 0 Actually, H°(P3, E(−1)) = 0 follows from H°(P3, E) = 0. Let P2 be the equation of the two plane (a section of O(1)), where O p2 is the sheaf of germs of functions holomorphic in P2. It follows : Hence from which the recursion is established.
N.J. HITCHIN, Communication presented at the “Journëes sur les Champs de Yang et Mills, Société Mathématique de France, Paris, May 24, 25, 26, 1978, and to be published : write tensorize with E : tensorize with Ě :
M.F. ATIYAH, Pisa Lectures and Private Communication. Consider : where T is the tangent bundle, A = −AT a non degenerate correlation : The dual sequence gives rise after tensorization by E to : Hence Now, = N*(1) = V°* dual and thus isomorphic with the null correlation subbundle Vo of T(−1) which is the P3(C) version of the k = 1 instanton. The computation of H1 (E) holds with however the extra index pertaining to the fiber of and the connection α altered by the corresponding connection for which is nothing else than the spinor connection for the invariant metric on P1(H)! A complete description of V° can be found in W. Barth, Math. Ann. 226, 125 (1977). See also ref. [12], p. 166
This can also be seen directly from algebraic geometry and Serre duality, this [7] [10], [12]: From the sequence it follows see e.g. J.L. Verdier in Séminaire ENS 1977–1978, A. Douady, J.L. Verdier, Ed.
E.F. CORRIGAN, D.B. FAIRLIE, P. GODDARD, S. TEMPLETON, Nucl. Phys. B 140 (1978), 31–44
N.H. CHRIST, E.J. WEINBERG, N.K. STANTON, Columbia Preprint (1978)
R.J. CREWTHER, CERN-TH 2522 (1978)
C. MEYERS, M. de ROO, CERN-TH Aug. 1978, Nucl. Phys. to be published
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Madore, J., Richard, J.L., Stora, R. (1979). F = * F , A review. In: Albeverio, S., et al. Feynman Path Integrals. Lecture Notes in Physics, vol 106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09532-2_84
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