**Authors:** Carlos Castro

A ternary gauge field theory is explicitly constructed based on a totally
antisymmetric ternary-bracket structure associated with a 3-Lie algebra.
It is shown that the ternary infinitesimal gauge transformations do obey
the key closure relations [δ_{1}, δ_{2}] = δ_{3}.
Invariant actions for the 3-Lie
algebra-valued gauge fields and scalar fields are displayed. We analyze
and point out the difficulties in formulating a nonassociative Octonionic
ternary gauge field theory based on a ternary-bracket associated with the
octonion algebra and defined earlier by Yamazaki. It is shown that a
Yang-Mills-like quadratic action is invariant under global (rigid) transformations
involving the Yamazaki ternary octonionic bracket, and that
there is closure of these global (rigid) transformations based on constant
antisymmetric parameters Λ^{ab} = -Λ^{ba}. Promoting the latter parameters
to spacetime dependent ones Λ^{ab}(x^{μ}) allows to build an octonionic ternary
gauge field theory when one imposes gauge covariant constraints on the
latter gauge parameters leading to field-dependent gauge parameters and
nonlinear gauge transformations. In this fashion one does not spoil the
gauge invariance of the quadratic action under this restricted set of gauge
transformations and which are tantamount to spacetime-dependent scalings
(homothecy) of the gauge fields.

**Comments:** 17 pages, submitted to the Int. J. Mod. Phys. A

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[v1] 14 May 2011

[v2] 20 Jun 2011

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