Topological Structure in ĉ = 1 Fermionic String Theory

ĉ = 1 fermionic string theory, which is considered as a fermionic string theory in two dimension, is shown to decompose into two mutually independent parts, one of which can be viewed as a topological model and the other is irrelevant for the theory. The physical contents of the theory is largely governed by this topological structure, and the discrete physical spectrum of ĉ = 1 string theory is naturally explained as the physical spectrum of the topological model. This topological structure turns out to be related with a novel hidden N = 2 superconformal algebra (SCA) in the enveloping algebra of the N = 3 SCA in fermionic string theories. E-mail address: [email protected] JSPS Research Fellow E-mail address: [email protected] JSPS Research Fellow The remarkable feature of the two-dimensional string theory is that only the tachyon field is left as a continuous physical degree of freedom, while most of the excitation modes of the string are eliminated and there remains an infinite set of discrete physical states as their remnants [1, 2, 3, 4, 5, 6]. The drastic reduction of physical degrees of freedom makes the theory so simple that the two-dimensional string theory may provide a good testing ground for discussing the non-perturbative formulation and the dynamical principle of string theory. The reduction of physical degrees of freedom suggests that the two-dimensional string theory can be described as a topological model, which may give us an alternative formulation of string theory. Indeed, several authors showed the correspondence of c = 1 string theory, which is considered as a bosonic string theory in two-dimensional space-time, to certain topological field theories [7, 8, 9, 10, 11, 12]. However they have not been able to clarify the role of the discrete states which we consider to be deeply connected with the topological nature of two-dimensional string theories. In the recent work [13], a relation between c = 1 string theory and a topological sigma model was suggested, in which the former was shown to be a kind of realization, or ‘bosonization’, of the latter. In this formulation, the existence of the discrete states was crucial. Namely, the target space coordinate and the gauge ghost of the topological sigma model were realized by one of the ground ring generators [4] and a discrete tachyon, respectively. Consequently, all the discrete states in c = 1 string theory were reproduced as the physical states of the topological sigma model. In this letter, we apply the analysis performed on c = 1 string theory [13] to the case of ĉ = 1 fermionic string theory, which is considered as a fermionic string theory in two dimension. We have found that ĉ = 1 string theory contains a subspace which can be viewed as a topological model. This topological model is essentially the same one as obtained in the c = 1 case. The physical spectrum of the topological model turns into the discrete physical spectrum in ĉ = 1 string theory. Therefore, one can say that ĉ = 1 string theory is largely governed by the topological model as for the physical contents of the theory. As we will show, the N = 3 superconformal symmetry in fermionic string theories [14] plays an important role in this topological structure. Namely, we identify the (twisted) N = 2 superconformal algebra (SCA) of the topological model with a hidden N = 2 SCA in the enveloping algebra of the N = 3 SCA. Associated with this fact, we found a novel automorphism of N = 3 SCA in the enveloping algebra.