On the equivalence of fermionic string to bosonic string in two dimensions

Two-dimensional fermionic string theory is shown to have a structure of topological model, which is isomorphic to a tensor product of two topological ghost systems independent of each other. One of them is identified with c = 1 bosonic string theory while the other has trivial physical contents. This fact enables us to regard two-dimensional fermionic string theory as an embedding of c = 1 bosonic string theory in the moduli space of fermionic string theories. Upon this embedding, the discrete states of c = 1 string theory are mapped to those of fermionic string theory, which is considered to be the origin of the similarity between the physical spectra of these two theories. We also discuss a novel BRST operator associated with this topological structure. E-mail address: [email protected] JSPS Research Fellow One of the most significant developments in bosonic string theory in less than two dimensions is the discovery of the topological description. The exact results by means of the matrix models are successfully reproduced in terms of a certain kind of topological theory [1]-[5]. In particular, the integrable structure that has been first observed in the matrix models [6, 7] is clearly understood from the point of view of topological theory [4, 5, 8]. In a sense, the integrability of bosonic string theory is a natural consequence of the underlying topological theory. This integrable structure is also derived [9] within the continuum approach by making use of the w∞-currents associated with the discrete states [10]-[13]. The use of the continuum approach is inevitable for the case of fermionic, or NeveuSchwarz-Ramond, string theory, since there are no matrix models relevant to fermionic string theory. As is well known, the continuum approach is a hard scheme to perform direct calculation. However, for the case of less than two dimensions, fermionic string theory has several common features with bosonic one, such as the existence of the discrete states [14, 15]. This fact makes us expect that fermionic string theory has a structure analogous to bosonic one and can be treated in a way parallel to the bosonic case. Actually, it has been shown [16] that fermionic sting theory in less than two dimensions has an integrable structure similar to that observed in the bosonic case. Having an integrable structure in fermionic string theory, it is natural to ask what kind of topological theory is behind it. Since the obtained structure heavily relies on the existence of the discrete states [16], the relevant topological model will be also closely related with the discrete states. In the bosonic (or c = 1) case, it has been pointed out [17] that we can regard string theory as a bosonization of a topological model, in which the fundamental fields are realized by the discrete states. The extension of this result to the fermionic case will be a first step toward the full understanding of topological structure in fermionic string theory. An analysis along this direction has been performed in ref.[18]. However, the obtained results is incomplete and the further investigation is necessary. In this article, we put forward the analysis in ref.[18] and find a complete topological structure in two-dimensional fermionic string theory, which we call as ĉ = 1 string theory. The structure we have found consists of two parts, both of which can be regarded as a topological ghost system. One of them contains the discrete operators, in particular the ground ring generator [13, 15], as the fundamental field, and can be identified with the structure observed in c = 1 bosonic string theory. This fact enables us to regard