Seven-Sphere and the Exceptional N=7 and N=8 Superconformal Algebras

TL;DR

This paper explores realizations of exceptional N=7 and N=8 superconformal algebras with Spin(7) and G_2 symmetries, linking them to seven-sphere geometries and octonion algebra, with implications for string theory.

Contribution

It provides new realizations of exceptional superconformal algebras using coset spaces related to seven-spheres and octonions, highlighting their algebraic and geometric structures.

Findings

Unitary highest-weight representations with specific central charges.

Coset spaces correspond to seven-sphere geometries with torsion.

Connections between octonions, triality, and superconformal algebras.

Abstract

We study realizations of the exceptional non-linear (quadratically generated, or W-type) N=8 and N=7 superconformal algebras with Spin(7) and G_2 affine symmetry currents, respectively. Both the N=8 and N=7 algebras admit unitary highest-weight representations in terms of a single boson and free fermions in 8 of Spin(7) and 7 of G_2, with the central charges c_8=26/5 and c_7=5, respectively. Furthermore, we show that the general coset Ans"atze for the N=8 and N=7 algebras naturally lead to the coset spaces SO(8)xU(1)/SO(7) and SO(7)xU(1)/G_2, respectively, as the additional consistent solutions for certain values of the central charge. The coset space SO(8)/SO(7) is the seven-sphere S^7, whereas the space SO(7)/G_2 represents the seven-sphere with torsion, S^7_T. The division algebra of octonions and the associated triality properties of SO(8) play an essential role in all these realizations. We also comment on some possible applications of our results to string theory.