Spontaneous symmetry breaking in a non-Abelian topological gauge theory

We study the spontaneous symmetry breaking mechanism in a non-Abelian topological gauge field theory, built from the twisted $\mathcal{N} = 2$ super-Yang-Mills theory in the presence of a Fujikawa-type potential. Specifically, by employing Fujikawa's Becchi-Rouet-Stora-Tyutin method, local degrees of freedom are released from the introduction of a potential in the trivial sector of equivariant cohomology. Such a potential displays a nontrivial vacuum solution, which induces the spontaneous symmetry breaking of the gauge symmetry together with the original fermionic scalar supersymmetry of the topological action. In this case, not only massive vector bosons emerge, but also fermionic fields with massive poles. This result shows that the introduction of a topological phase in non-Abelian gauge theories could provide a mechanism of mass generation for fermions with their masses correlated to the mass of Higgs gauge bosons ($m_B$). For the SSB in the topological case, three different vacuum directions are required. Otherwise, the supersymmetry could not be broken, and mass generation for fermions will not occur. Starting with a theory with symmetry $G = SU(N)$, to obtain a gauge theory at the end of the process, we must have $N \geq 3$. We study a maximal symmetry breaking of the type $SU(3) \rightarrow U(1) \times U(1)$, and obtain their fermionic propagators with mass poles $m^2_F = m^2_B = v^2$ after SSB, being $v^2 $ the energy scale introduced by the Fujikawa-type potential.