2. Conformal supersymmetry in 4D

Abstract

Recently, superconformal field theories in various dimensions have been attracting ever more interest, cf. (for references up to year 2000): [306, 170, 248, 313, 89, 186, 233, 355, 215, 138-140, 49, 193, 141, 142, 347, 72, 160, 168, 258, 273, 309, 9, 58, 102, 189, 171, 177, 311, 317, 4, 76, 101, 103, 164, 167, 172, 174, 173, 175, 105] and the references therein. Particularly important are those for D ≤ 6, since in these cases the relevant superconformal algebras satisfy Nahm’s classification [306] based on the Haag- Lopuszanski-Sohnius theorem [218]. This makes the classification of the UIRs of these superalgebras very important. First such classification was given for the D = 4 superconformal algebras su(2, 2/1) [182] and su(2, 2/N) [138-142] (for arbitrary N). Then the classification for D = 3 (osp(N/4) for even N), D = 5, and D = 6 (osp(8∗/2N) for N = 1, 2) was given in [302] (some results being conjectural), and then the D = 6 case (for arbitrary N) was finalized in [122] (see Section 3.1). Once we know the UIRs of a (super-) algebra the next question is to find their characters, since these give the spectrum which is important for the applications. Some results on the spectrum were given in the early papers [233, 355, 215, 140] but it is necessary to have systematic results for which the character formulas are needed. This is the question we address in this chapter 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 the 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 framework some of these correspond to fields with “protected” scaling dimension and therefore imply “non-renormalization theorems” at the quantum level, cf., e. g., [228, 174]. This is intimately related to the super-invariant differential operators and equations satisfied by the superfields at these special representations. Thus, we need detailed knowledge about the structure of the UIRs from the representationtheoretical point of view. Fortunately, such information is contained in [138-142]. Following these papers we first recall the basic ingredients of the representation theory of the D = 4 superconformal algebras. In particular we recall the structure of Verma modules and UIRs. First the general theory for the characters of su(2, 2/N) is developed in great detail. For the general theory we use the (generalized) odd reflections introduced in [139] (see also [349]).1 We also pin-point the difference between character formulas for sl(4, N) and su(2, 2/N); for the latter we need to introduce and use the notion of counterterms in the character formulas. The general formulas are valid for arbitrary N and are given for the so-called bare characters (or superfield decompositions). We also summarize our results on the decompositions of long superfields as they descend to the unitarity threshold. To give the character formulas explicitly we need to recall also the character formulas of su(2, 2) and su(N), for which we give explicitly all formulas that we need. Finally, we give the explicit complete character formulas for N = 1 and for a number of important examples for N = 2, 4. In this chapter we mostly follow the papers [123, 129, 127, 128] and also using essentially the results of [138-142].