Universal String and Small N = 4 Superstring

It was previously shown that most of the superstrings can be obtained from those with higher world-sheet supersymmetry as spontaneously broken phases. In this paper, we show that the small N = 4 superstring, which was left out of this hierarchy of the universal string, can be obtained from the large N = 4 strings. We also show that the N = 2 string is a special vacuum of small N = 4 string. Thus all the known superstring theories can be derived from a universal string by spontaneous breakdown of supersymmetry. e-mail address: [email protected] e-mail address: [email protected] Recently it has been shown that any N = 0 (N = 1) string can be obtained from N = 1 (N = 2) superstring as a spontaneously broken phase [1]. This remarkable discovery has opened the possibility of constructing a universal string from which all known string theories can be derived just by selecting different vacua. Indeed this has been further generalized, and it is now known that an arbitrary N -extended superstring can be regarded as a special class of vacua for (N + 1) superstring [2]. At the moment, this hierarchy has been successfully formulated for N -extended superstrings based on linear superconformal algebras (SCAs) with SO(N) current algebra [3]. However there is an important class of superstrings which is usually termed the ‘small’ N = 4 superstring with only SU(2) current algebra. This class of superstrings is the largest extended one that has been formulated by lagrangian and also constitutes an interesting theory from the point of view of topological string. It is thus important to examine if this class of superstrings also belongs to the hierarchy of the universal string or if it is completely decoupled from such a unified approach. The purpose of this paper is to show that it is in fact possible to embed the small N = 4 superstring into the hierarchy. We will show that the N = 2 superstring is realized as a special vacuum of the small N = 4 superstring, which in turn is realized in the large N = 4 superstring. We also point out that the large N = 4 superstring with an arbitrary parameter x (the ratio of the levels of SU(2) current algebras contained in the theory) is realized in the already existing embeddings. This completes the program of the universal string concerning the linear SCA [2]. In order to discuss the embeddings of some superstring theory into the small N = 4, we must first ask what theory can be embedded into the small N = 4 string. This can be easily answered if one notices that the N = 3 superstring contains a symmetry generator of dimension 1 2 which is not present in the small N = 4 superstring. Consequently the largest super-subalgebra of the small N = 4 SCA is the N = 2 SCA. We thus conclude that it is only possible to embed the N = 2 superstring into the small N = 4 string. Let us first discuss the embedding of N = 2 superstrings into the small N = 4 one. For this purpose, we use N = 2 superfields. Thus the small N = 4 SCA can be written