The worldsheet conformal field theory of the fractional superstring

Abstract

Two of the important unresolved issues concerning fractional superstrings have been the appearance of new massive sectors whose spacetime statistics properties are unclear, and the appearance of new types of "internal projections" which alter or deform the worldsheet conformal field theory in a highly non-trivial manner. In this paper we provide a systematic analysis of these two connected issues, and explicitly map out the effective post-projection worldsheet theories for each of the fractional-superstring sectors. In this way we determine their central charges, highest weights, fusion rules, and characters, and find that these theories are isomorphic to those of free worldsheet bosons and fermions. We also analyze the recently-discovered parafermionic "twist current" which has been shown to play an important role in reorganizing the fractional-superstring Fock space, and find that this current can be expressed directly in terms of the primary fields of the post-projection theory. This thereby enables us to deduce some of the spacetime statistics properties of the surviving states. * E-mail address: dien@hep.physics.mcgill.ca. * Note that since this worldsheet theory is a tensor product, spacetime Lorentz invariance implies invariance under permutations of the transverse dimensions.

Aspects of fractional superstrings

“…[7] and [11] argue that the corresponding string functions should be interpreted as "internal" projections, i.e., cancellations of degrees of freedom in the fractional superstring models. See also [14], [15], and [13]). Relatedly, the authors show that when the A K -sector is written as A boson K − A fermion K , as done above, the q-expansions of both A boson K and A fermion K are all positive.…”

“…They have, in fact, found thetafunction expressions for the B boson K -and B fermion K -subsectors. [15,14] compactified directions to correspond to the last 2 strings functions in a term. Thus, the spin-statistics of the physical states of the D = 6 model as observed in four-dimensional uncompactified spacetime is determined by the (matching) n subscripts of the first two string functions (corresponding to the two uncompactified transverse dimensions) in each term of four string functions, c However, the C K -sectors are harder to interpret.…”

“…In the last few years, several generalizations of standard (supersymmetric) string theory have been proposed. [20,28,25,22] One of them [16,7,10,6,9,14,15,13] uses the (fractional spin) parafermions introduced from the perspective of 2-D conformal field theory (CFT) by Zamolodchikov and Fateev [31] in 1985 and further developed by Gepner and Qiu [18] . 1 In a series of papers, possible new string theories with local parafermionic world sheet currents (of fractional conformal spin) giving critical dimensions D = 6, 4, 3, and 2 have been proposed.…”

Fractional supersymmetric Liouville theory and the multi-cut matrix models

“…One might be a phenomenological reason to have a model of lower (or reasonable) critical dimensions less than ten [9,10]. Another can be a possibility of extending the spacetime spectrum to include different types of statistics [11][12][13] (This feature might be good for applying to some systems with different kinds of statistics). In addition, it is also interesting to deepen our understanding of the RNS superstring formulation from the viewpoint of this generalization of worldsheet conformal field theory.…”

“…Roughly speaking, fractional superstring theory is obtained by replacing the worldsheet fermions in the superstring theory with the Zamolodchikov-Fateev parafermions [14]. Even though there is much progress in studying this system [1,[9][10][11][12][13][15][16][17][18][19][20], several difficulties have prevented us from revealing its whole body and structure. The main difficulty comes from the special feature of this fractional supersymmetry: spin of the current is equal to some fractional number, (k + 4)/(k + 2) (k = 2, 3, · · · ), and the algebra of this current is of the non-local non-Abelian braiding type [15].…”