Gauge theory on nonassociative spaces

Abstract

We show how to do gauge theory on the octonions and other nonassociative algebras such as `fuzzy $R^4$' 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 usual gauge theory on the underlying classical space. We give a general U(1)-Yang-Mills example for any quasi-algebra and a full description of the moduli space of flat connections in this theory for the cube $Z_2^3$ and hence for the octonions. We also obtain further results about the octonions themselves; an explicit Moyal-product description of them as a nonassociative quantisation of functions on the cube, and a characterisation of their cochain twist as invariant under Fourier transform.

24 pages latex, two .eps figures

References in corpus (8)

• Nonassociative Star Product Deformations for D-brane Worldvolumes in Curved Backgrounds
• The Eight Dimensional Quantum Hall Effect and the Octonions
• On spherical harmonics for fuzzy spheres in diverse dimensions
• Higher Dimensional Geometries from Matrix Brane constructions
• Braided Cyclic Cocycles and Non-Associative Geometry
• Quantization by cochain twists and nonassociative differentials
• Electromagnetism and Gauge Theory on the Permutation Group $S_3$
• Noncommutative physics on Lie algebras, Z_2^n lattices and Clifford algebras

Cited by in corpus (25)

• Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics
• Quantization by cochain twists and nonassociative differentials
• Homotopical Quantum Field Theory
• Nonassociative Riemannian Geometry by Twisting
• Superconducting currents and charge gradients in the octonion spaces
• Nonassociativity and Integrable Hierarchies
• Gauging the twisted Poincare symmetry as noncommutative theory of gravitation
• Colorless operators in a non-associative quantum theory
• Torques and angular momenta of fluid elements in the octonion spaces
• Gauge field theories with covariant star-product
• Some spin textures relevant to magnetic moments in the octonion spaces
• Conserved quantities of vectorial magnitudes within the material media
• One underlying mechanism for two piezoelectric effects in the octonion spaces
• On a class of gauge theories
• One-loop unitarity of scalar field theories on Poincare invariant commutative nonassociative spacetimes
• Two incompatible types of invariants in the octonion spaces
• Gauge fields and four interactions in the trigintaduonion spaces
• Eight Equilibrium and continuity equations within the material media
• Quasi-Quantum Planes and Quasi-Quantum Groups of Dimension $p^3$ and $p^4$
• The Majid-Ruegg model and the Planck scales
• On quasi-Hopf smash products and twisted tensor products of quasialgebras
• Some field equations with composite operators in octonion spaces
• Quasi-Quantum Linear Spaces
• Alternative twisted tensor products and Cayley algebras
• Basic postulates of some coordinate transformations within material media