Complex Analysis and Operator Theory

, Volume 9, Issue 2, pp 431–444 | Cite as

Applications of Hypercomplex Automorphic Forms in Yang–Mills Gauge Theories

  • Rolf Sören KraußharEmail author
  • Jürgen Tolksdorf


In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual \(SU(2)\)-Yang–Mills instanton solutions on certain classes of conformally flat \(4\)-manifolds. We use a hypercomplex argument principle to establish a natural link between the fundamental solutions of \(D \Delta f = 0\) and the second Chern class of the \(SU(2)\) principal bundles over these manifolds. The considered base manifolds of the bundles are not simply-connected, in general. Actually, this paper summarizes an extension of the corresponding results of Gürsey and Tze on a hyper-complex analytical description of \(SU(2)\) instantons. Furthermore, it provides an application of the recently introduced new classes of hypercomplex-analytic automorphic forms.


Yang–Mills gauge theory \(SU(2)\) instantons Quaternionic analyticity Conformally flat manifolds Hypercomplex argument principle Chern numbers Hypercomplex automorphic forms 

Mathematics Subject Classification

30G35 70S15 



The authors are very thankful to Professor John Ryan from the University of Arkansas and to Professor Vladimir Soucek from Charles University of Prague for the very fruitful discussions which lead to a successful development of this paper. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 267087.


  1. 1.
    Ahlfors, L.V.: Möbius transformations in \(\mathbb{R}^{n}\) expressed through \(2\times 2\) matrices of Clifford numbers. Complex Var. 5, 215–224 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Blaschke, W.: Vorlesung über Differentialgeometrie I. Springer, Berlin (1924)CrossRefGoogle Scholar
  3. 3.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Research Notes, vol. 76, Boston (1982)Google Scholar
  4. 4.
    Bulla, E., Constales, D., Kraußhar, R.S., Ryan, J.: Dirac type operators for arithmetic subgroups of generalized modular groups. Journal für die Reine und Angewandte Mathematik (Crelle’s Journal) 643, 1–19 (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Colombo, F., González-Cervantes, J.O., Sabadini, I.: A non constant coefficients differential operator associated to slice monogenic functions. Trans. AMS 365, 303–318 (2013)CrossRefzbMATHGoogle Scholar
  6. 6.
    Colombo, F., Sabadini, I., Sommen, F.: The Fueter mapping in integral form and the F-functional calculus. Math. Methods Appl. Sci. 33(17), 2050–2066 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Constales, D., Grob, D., Kraußhar, R.S.: A new class of hypercomplex analytic cusp forms. Trans. AMS 365(2), 811–835 (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor Valued Functions. Kluwer, Dordrecht (1992)Google Scholar
  9. 9.
    Fueter, R.: Analytische Funktionen einer Quaternionenvariablen. Comment. Helv. Mat. 4, 9–20 (1932)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Fueter, R.: Functions of a Hyper Complex Variable. Lecture notes written and supplemented by E. Bareiss. Math. Inst. Univ. Zürich, Fall Semester (1948/1949)Google Scholar
  11. 11.
    Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Basel (1990)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)zbMATHGoogle Scholar
  13. 13.
    Gürsey, F., Tze, H.: Complex and quaternionic analyticity in chiral and gauge theories I. Ann. Phys. 128, 29–130 (1980)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gürsey, F., Tze, H.: On the role of division, Jordan and related algebras in particle physics. World Scientific, Singapore (1996)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hempfling, T., Kraußhar, R.S.: Order theory for isolated points of monogenic functions. Arch. Math. 80, 406–423 (2003)zbMATHGoogle Scholar
  16. 16.
    Kraußhar, R.S.: Generalized Analytic Automorphic Forms in Hypercomplex Spaces. Frontiers in Mathematics. Birkhäuser, Basel (2004)Google Scholar
  17. 17.
    Kraußhar, R.S., Ryan, J.: Some conformally flat spin manifolds, Dirac operators and automorphic forms. J. Math. Anal. Appl. 325(1), 359–376 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Krieg, A.: Eisenstein series on real, complex and quaternionic half-soaces. Pac. J. Math. 133, 315–354 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kou, K.I., Qian, T., Sommen, F.: Generalizations of Fueter’s theorem. Methods Appl. Anal. 9(2), 273–289 (2002)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Kuiper, N.H.: On conformally flat spaces in the large. Ann. Math. (2) 50, 916–924 (1949)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Laville, G., Ramadanoff, I.: Holomorphic Cliffordian functions. Adv. Appl. Clifford Algebras 8(2), 323–340 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Laville, G., Ramadanoff, I.: Elliptic Cliffordian functions. Complex Var. 45(4), 297–318 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Leutwiler, H.: Modified Clifford analysis. Complex Var. 17, 153–171 (1991)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Qiao, Y., Bernstein, S., Eriksson, S.-L., Ryan, J.: Function theory for Laplace and Dirac–Hodge operators in hyperbolic space. Journal d’Analyse Mathématique 98, 43–64 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Ryan, J.: Conformal Clifford manifolds arising in Clifford analysis. Proc. R. Ir. Acad 85A(1), 1–23 (1985)Google Scholar
  26. 26.
    Ryan, J.: Intertwining operators for iterated Dirac operators over Minkowski-type spaces. J. Math. Anal. Appl. 177(1), 1–23 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Sce, M.: Osservazione sulle serie di potenzi nei moduli quadratici. Lincei Rend. Sci. Fis. Mat. et Nat. 23, 220–225 (1957)MathSciNetGoogle Scholar
  28. 28.
    Zöll, G.: Ein Residuenkalkül in der Clifford-Analysis und die Möbiustransformationen für euklidische Räume, PhD Thesis, RWTH Aachen (1987)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Erziehungswissenschaftliche Fakultät, Fachgebiet MathematikUniversität ErfurtErfurtGermany
  2. 2.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany

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