Spontaneous symmetry breaking in a non-Abelian topological gauge theory

TL;DR

This work shows how spontaneous symmetry breaking can occur in a non-Abelian topological gauge theory by extending the twisted $\mathcal{N}=2$ SYM with a Fujikawa-type BRST-exact potential. The introduced vacuum yields simultaneous breaking of gauge symmetry and fermionic scalar supersymmetry, releasing local degrees of freedom and producing mass poles for both bosons and fermions that are all set by the same energy scale $v$. A concrete $SO(5) \to SO(2)$ example demonstrates that $m_B^2 = m_F^2 = v^2$ for selected fields, while certain components remain massless, reflecting the topological constraints. These results suggest a mechanism for mass generation in non-Abelian topological phases and motivate further study of renormalizability, unitarity, and possible connections to gravity and holography. The framework preserves topological structure prior to SSB and provides a controlled way to probe local dynamics through the Fujikawa scale.

Abstract

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 $G = SO(N)$, to obtain a gauge theory at the end of the process, we must have $N \geq 5$. We study a particular symmetry breaking of the type $SO(5) \rightarrow SO(2)$, 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.

Figures (2)

• Figure 1: Propagators.
• Figure 2: Feynman diagram with a fermionic loop.