BRST Charge for the Orthogonal Series of Bershadsky - Knizhnik Quasi

The quantum BRST charges for Bershadsky-Knizhnik orthogonal quasisuperconformal algebras are constructed. These two-dimensional superalgebras have the N -extended non-linearly realised supersymmetry and the SO(N) internal symmetry. The BRST charge nilpotency conditions are shown to have a unique solution at N > 2, namely, N = 4 and k = −2, where k is central extension parameter of the Kač-Moody subalgebra. We argue about the existence of a new string theory with the non-linearly realised N = 4 world-sheet supersymmetry and negative ‘critical dimension’. Supported in part by the ‘Deutsche Forschungsgemeinschaft’ and the NATO grant CRG 930789 On leave of absence from: High Current Electronics Institute of the Russian Academy of Sciences, Siberian Branch, Akademichesky 4, Tomsk 634055, Russia 1. Any known critical N -extended fermionic string theory with N ≤ 4 world-sheet supersymmetries is based on a two-dimensional (2d) linear N -extended superconformal algebra which is gauged [1]. When a number of world-sheet supersymmetries exceeds two, there are more opportunities to build up new string theories, namely, by utilyzing 2d non-linear quasi-superconformal algebras (QSCAs) which are known to exist for an arbitrary N > 2. The QSCAs can be considered on equal footing with the W algebras without, however, having currents of spin higher than two. In the past, only one string theory for N > 2 was actually constructed by gauging the ‘small’ linear N = 4 SCA with SU(2) internal symmetry [1, 2]. Still, it is of interest to know how many different N = 4 string theories exist at all. Any N = 4 string constraints are going to be very strong, so that their explicit realisation should always imply non-trivial interplay between geometry, conformal invariance and extedned supersymmetry. The N = 4 strings are also going to be relevant in the search for the ‘universal string theory’ [3]. In addition, strings with N = 4 supersymmetry are expected to have deep connections with integrable models [4, 5], so that we believe they are worthy to be studied. The full classification of QSCAs is known due to by Fradkin and Linetsky [6]. In particular, the osp(N |2;R) and su(1, 1|N) series of QSCAs with ̂ SO(N) and ̂ U(N) Kač-Moody (KM) symmetries, respectively, were discovered before by Knizhnik [7] and Bershadsky [8]. It has been known for some time that the unitary series of Bershadsky-Knizhnik QSCAs does not admit nilpotent quantum BRST charges for any N > 2 [9], so that we are going to concentrate on the orthogonal series of QSCAs having the SO(N) internal symmetry. 3 2. The current contents of the 2d Bershadsky-Knizhnik orthogonal QSCA [7, 8] is given by the holomorphic fields T (z), G(z) and J(z), all having the standard mode expansions T (z) = ∑ n Lnz −n−2 , G(z) = ∑ r Grz −r−3/2 , J(z) = ∑