Central charge reduction and spacetime statistics in the fractional superstring.

Fractional superstrings in the tensor-product formulation experience “internal projections” which reduce their effective central charges. Simple expressions for the characters of the resulting effective worldsheet theory are found. All states in the effective theory can be consistently assigned definite spacetime statistics. The projection to the effective theory is shown to be described by the action of a dimension-three current in the original tensor-product theory. ∗E-mail address: [email protected]. Address after September 1, 1993: School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540. †E-mail address: [email protected]. Fractional superstrings [1, 2] have been proposed as a possible new class of string theories generalizing superand heterotic strings, and since then there has been considerable effort in understanding their worldsheet properties and spacetime phenomenologies [3, 4, 5, 6, 7, 8]. These strings have the important property that their critical spacetime dimensions and central charges are less than those of the superstring, and this reduction in the critical dimension is accomplished by replacing the worldsheet supersymmetry of the superstring with a “K-fractional supersymmetry” which relates bosons to fields of dimension (spin) 2/(K+2) on the worldsheet, K ≥ 2. (The case K = 2 corresponds to the ordinary superstring.) There are two proposals for the identification of the K-fractional worldsheet supersymmetry. The first proposal [1], which we will refer to as the chiral algebra approach, associates the fractional worldsheet supersymmetry with the Virasoro algebra extended via the inclusion of the fractional supercurrent G, a certain chiral operator of dimension 1 + 2/(K + 2). This algebra is chosen because of its wellbehaved representation theory [9], and in the K = 4 case the associated fractional strings have been shown to have sensible tree-level scattering amplitudes [8]. Demanding the existence of extra null states in these theories [1, 5] indicates that their critical central charges are ccrit = 6K K + 2 + 24 K . (1) However, since explicit representations of the K-fractional chiral algebras at these values of the central charge are not known, one cannot determine the corresponding critical spacetime dimensions for fractional-superstring propagation. The second proposal [2, 3, 4, 5], which we will refer to as the tensor-product approach, is based on the observation that the above K-fractional chiral algebras have special representations with central charges cK ≡ 3K/(K+2) composed of a free boson plus a ZZK-parafermion theory [10]. Interpreting the free boson as a spacetime coordinate of the string, one identifies the D-fold tensor product of the cK theory with itself as the worldsheet conformal field theory (CFT) describing a K-fractional string propagating in D-dimensional spacetime. Since the K-fractional chiral algebra is non-linear, taking tensor products of its representations does not produce another