Characters of the positive energy UIRs of D = 4 conformal supersymmetry
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haracters of the Positive Energy UIRs of D = 4 Conformal Supersymmetry¶ V. K. Dobrev School of Informatics, University of Northumbria, Ellison Building, Newcastle upon Tyne, NE1 8ST, UK and Institute of Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria (permanent address) Abstract—We give character formulas for the positive energy unitary irreducible representations of the N-extended D = 4 conformal superalgebras su(2, 2/N). Using these, we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4/N). Our derivations use results from the representation theory of su(2, 2/N) developed as early as the 1980s. PACS numbers: 02.20.Qs, 11.25.Hf, 11.30.Pb DOI: 10.1134/S1063779607050024
1. INTRODUCTION Recently, superconformal field theories in various dimensions are attracting greater and greater interest (cf. [1–96] and references therein). Particularly important are those for D ≤ 6 since in these cases the relevant superconformal algebras satisfy the Haag–Lopuszanski–Sohnius theorem [98]. This makes the classification of the UIRs of these superalgebras very important. Until recently, such classification was known only for the D = 4 superconformal algebras su(2, 2/1) [99] and su(2, 2/N) [100–103] (for arbitrary N). Recently, the classification for D = 3 (for even N), D = 5, and D = 6 (for N = 1, 2) was given in [104] (some results being conjectural), and then the D = 6 case (for arbitrary N) was finalized in [105]. Once we know the UIRs of a (super-)algebra, the next task is to find their characters, since these give the spectrum which is important for applications. Some results on the spectrum were given in the early papers [102, 106–108], but it is necessary to have systematic results, for which the character formulas are needed. This is the question we address in this paper for the UIRs of D = 4 conformal superalgebras su(2, 2/N). From the mathematical point of view, this question is clear only for representations with conformal dimension above the unitarity threshold viewed as irreps of the corresponding complex superalgebra sl(4/N). But for su(2, 2/N), even the UIRs above the unitarity threshold are truncated for small values of spin and isospin. Moreover, in applications the most important role is played by the representations with “quantized” conformal dimensions at the unitarity threshold and at discrete points below. In the quantum field or string theory ¶ The
framework, some of these correspond to operators with “protected” scaling dimension and therefore imply “non-renormalization theorems” at the quantum level (cf., e.g., [22, 23]). Thus, we need detailed knowledge about the structure of the UIRs from the representation-theoretical point of view. Fortunately, such information is contained in [100–103]. Following these papers in Section 2, we recall the basic ingredients of the representation theory of the D = 4 superconformal algebras
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