  We consider a primordial SU(2) vector multiplet during inflation in models
where quantum fluctuations of vector fields are involved in producing the
curvature perturbation. Recently, a lot of attention has been paid to models
populated by vector fields, given the interesting possibility of generating
some level of statistical anisotropy in the cosmological perturbations. The
scenario we propose is strongly motivated by the fact that, for non-Abelian
gauge fields, self-interactions are responsible for generating extra terms in
the cosmological correlation functions, which are naturally absent in the
Abelian case. We compute these extra contributions to the bispectrum of the
curvature perturbation, using the delta N formula and the Schwinger-Keldysh
formalism. The primordial violation of rotational invariance (due to the
introduction of the SU(2) gauge multiplet) leaves its imprint on the
correlation functions introducing, as expected, some degree of statistical
anisotropy in our results. We calculate the non-Gaussianity parameter f_{NL},
proving that the new contributions derived from gauge bosons self-interactions
can be important, and in some cases the dominat ones. We study the shape of the
bispectrum and we find that it turns out to peak in the local configuration,
with an amplitude that is modulated by the preferred directions that break
statistical isotropy.
