ar X iv : h ep - t h / 03 11 24 7 v 1 2 6 N ov 2 00 3 On representations of the exceptional superconformal algebra

A superconformal algebra is a simple complex Lie superalgebra g spanned by the coefficients of a finite family of pairwise local fields a(z) = ∑ n∈Z a(n)z , one of which is the Virasoro field L(z), [3, 8, 11]. Superconformal algebras play an important role in the string theory and conformal field theory. The Lie superalgebras K(N) of contact vector fields with Laurent polynomials as coefficients (with N odd variables) is a superconformal algebra which is characterized by its action on a contact 1-form, [3, 6, 8, 12]. These Lie superalgebras are also known to physicists as the SO(N) superconformal algebras, [1]. Note that K(N) is spanned by 2 fields. It is simple if N 6= 4, if N = 4, then the derived Lie superalgebra K (4) is simple. The nontrivial central extensions of K(1), K(2) and K (4) are well-known: they are isomorphic to the so-called Neveu-Schwarz superalgebra, “the N = 2”, and “the big N = 4” superconformal algebra, respectively, [1]. It was discovered independently in [3] and [17] that the Lie superalgebra of contact vector fields with polynomial coefficients in 1 even and 6 odd variables contains an exceptional simple Lie superalgebra (see also [6, 9, 10, 18, 19]). In [3] a new exceptional superconformal algebra spanned by 32 fields was constructed as a subalgebra of K(6), and it was denoted by CK6. It was proven that CK6 has no nontrivial central extensions. It was also pointed out that CK6 appears to be the only new superconformal algebra, which completes their list (see [11, 12]).