Harmonic superspaces forN= (1,1), 6DSYM theory

Abstract

Abstract. This is a short account of the off-shell N = (1, 0) and on-shell N = (1, 1), 6D harmonic superspace formalism and its applications for the analysis of higher-dimension invariants in N = (1, 1) SYM theory. Motivations and contentsFor last years, there is a permanent interest in the maximally extended (with 16 supercharges) supersymmetric gauge theories in diverse dimensions (see, e.g., [1]),The famous N = 4, 4D SYM theory was the first example of an UV finite theory. Perhaps, it is also completely integrable [2]. The N = (1, 1), 6D SYM is not renormalizable by formal counting (the coupling constant is dimensionful) but it is also expected to possess unique properties. In particular, it respects the so called "dual conformal symmetry", like its 4D counterpart [3]. It provides the effective theory descriptions of some particular low energy sectors of string theory, such as D5-brane dynamics. The full effective action of D5-brane, generalizing the N = (1, 1) SYM action, was conjectured to be of non-abelian Born-Infeld type [4,5]. The N = (1, 1) SYM is anomaly free, as distinct from N = (1, 0) SYM.The N = (1, 1) and N = (1, 0) SYM theories can be viewed as a laboratory for studying N = 8 supergravity and its some lower N analogs, which are also non-renormalizable.The recent perturbative calculations in N = (1, 1) SYM show a lot of unexpected cancelations of the UV divergencies. The theory is UV finite up to 2 loops, while at 3 loops only a single-trace (planar) counterterm of canonical dim 10 is required. The allowed double-trace (non-planar) counterterms do not appear [6] - [8]. Various arguments to explain this were put forward [9] - [12], though the complete understanding is still lacking. Some new non-renormalization theorems could be expected in this connection.The maximal off-shell supersymmetry one can gain in 6D is N = (1, 0) supersymmetry. The natural off-shell formulation of N = (1, 0) SYM theory is achieved in harmonic N = (1, 0), 6D superspace [13,14] as a generalization of the harmonic N = 2, 4D one [15,16]. This harmonic 6D formalism was further developed in [17] -[20] and [21]