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Different approaches to the complex of three Dirac operators

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Abstract

An attempt to study the compatibility conditions, and the general free resolution, for the system associated with the Dirac operator in \(k\) vector variables appeared already in Sabadini et al. (Math Z 239: 293–320, 2002), from the point of view of Clifford analysis, and in Sabadini et al. (Exp Math 12: 351–364, 2003) using the tool of megaforms. Other studies have been carried out in other papers, like Krump (Adv Appl Clifford Alg 19: 365–374, 2009), Krump and Souček (17: 537–548, 2007), Salač (The generalized Dolbeault complexes in Clifford analysis, Praha 2012), using methods of representation theory. In this paper, we restrict our attention to the case of three variables and we describe the free resolution associate to the module from various different angles. The comparison has several noteworthy consequences. In particular, it gives the explicit description of all the maps contained in the algebraic resolution and shows that they are invariant with respect to the action of \(SL(3)\times SO(m)\). We also discuss how the methods used in this paper can be generalized to the case of \(k>3\) Dirac operators.

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Acknowledgments

The authors acknowledge the support of the grant P201/12/G028 of the Grant Agency of the Czech Republic.

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Authors and Affiliations

  1. Deledda International School, Corso Mentana 27, 16128 , Genova, Italy

    Alberto Damiano

  2. Dipartimento di Milano, Politecnico di Milano, Via Bonardi 9, 20133 , Milan, Italy

    Irene Sabadini

  3. Mathematical Institute, Charles University, Sokolovská 83, Prague, Czech Republic

    Vladimir Souček

Authors
  1. Alberto Damiano
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  2. Irene Sabadini
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  3. Vladimir Souček
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Correspondence to Irene Sabadini.

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Damiano, A., Sabadini, I. & Souček, V. Different approaches to the complex of three Dirac operators. Ann Glob Anal Geom 46, 313–334 (2014). https://doi.org/10.1007/s10455-014-9425-1

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