Gauge theory on nonassociative spaces

Abstract

We show how to do gauge theory on the octonions and other nonassociative algebras such as “quasi-R4” models proposed in string theory. We use the theory of quasialgebras obtained by cochain twist introduced previously. The gauge theory in this case is twisting–equivalent to the usual gauge theory on the underlying classical space. We give a general U(1)-Yang–Mills example for any quasialgebra and a full description of the moduli space of flat connections in this theory for the cube Z23 and hence for the octonions. We also obtain further results about the octonions themselves; an explicit Moyal-product description of them as a nonassociative quantization of functions on the cube, and a characterization of their cochain twist as invariant under Fourier transform.