Gravity and electroweak sector from symmetry breaking of an $SO(3,3)$ BF theory

An $SO(3,3)$ BF-type gauge theory is formulated on a six-dimensional spacetime of split signature $(3,3)$, interpreted as the pre-electroweak-symmetry-breaking phase. A MacDowell--Mansouri-type symmetry breaking to $SU(2)\times SU(2)$ is implemented, and the corresponding stabilizer and coset structures are computed. The curvature decomposes into chiral sectors, and effective tetrads are introduced using components of the higher-dimensional connection. The resulting left and right sectors are formulated as constrained BF/Plebanski-like theories with appropriate simplicity and reality conditions. The six-dimensional theory yields two overlapping four-dimensional Lorentzian sectors of opposite signature, related via gluing constraints across their intersection. In the first sector, the selfdual two-forms ($\Sigma^{(+)}$) satisfy simplicity constraints that select the non-degenerate branch and reproduce Einstein gravity. Subsequently, the $SU(2)_R\times U(1)_{Y{\rm dem}}\to U(1)_{\rm dem}$ breaking pattern is outlined which admits an ultra-soft regime consistent with current phenomenological bounds under sufficiently suppressed couplings. In the second sector, the antiself dual two-forms ($\Sigma^{(-)}$) satisfy analogous simplicity constraints, realizing weak gauge dynamics as gravity on the opposite-signature sector. Subsequently, the $SU(2)_L\otimes U(1)_Y$ electroweak symmetry is realized within the Yang--Mills branch of the BF theory which incorporates the standard Higgs mechanism $SU(2)_L\otimes U(1)_Y \to U(1)_{\mathrm{EM}}$, recovering the conventional electroweak $W^\pm$, $Z$, and photon spectrum.