nLab fractional superstring

Idea

The fractional superstrings are supposed to be a family of generalizations of the superstring that exhibit parafermion statistics. They were first described in Argyres and Henry Tye (1991) in an attempt to find other kinds of string theories with lower critical dimension (see string theory).

A string theory (as a sigma-model) defines in particular a two-dimensional worldvolume (i.e. worldsheet) theory with specific symmetry: a conformal field theory in the bosonic case, and a superconformal field theory in the super- case. The idea is to start from the worldsheet theory and impose particular transformations as symmetries of the theory, motivated by the assumption that any theory defined as such corresponds to a worldsheet theory. These symmetry transformations include generators, or currents, with fractional spins, of which the $\frac{3}{2}$-spin current of the superstring case is an example.

Some consistency checks are described in (Henry Tye (1993)).

The relevant geometric structure on the worlsheet appears to be an r-spin structure (see e.g. (Randal-Williams (2010)) (Runkel and Szegedy (2018))).

References

General

• Philip Argyres, and S.H. Henry Tye?. Fractional Superstrings with Space-Time Critical Dimensions Four and Six. (1991). arXiv:hep-th/9109001v2

• S.H. Henry Tye?. Fractional Superstrings with Space-Time Critical Dimensions Four and Six. (1993). arXiv:hep-th/9311021

• Philip Argyres. A three-dimensional fractional string. (2005) doi

On the heterotic parafermionic superstring and its potential anomalies

• Paul Frampton, and James Liu. Heterotic Parafermionic Superstring. (1992). arXiv:hep-th/9210049

Further developments

• Ahmed El Fallah, El Hassane Saidi and Rainer Dick. On generalized fractional superstring models and the associative division algebras. (2000). doi

On r-spin structures

• Oscar Randal-Williams. Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces. (2010) arXiv:1001.5366

• Ingo Runkel, Lóránt Szegedy. Topological field theory on r-spin surfaces and the Arf invariant. (2018). arXiv:1802.09978