Spontaneous symmetry breaking in pure 2D Yang-Mills theory

We consider purely topological $2$d Yang-Mills theory on a torus with the second Stiefel-Whitney class added to the Lagrangian in the form of a $θ$-term. It will be shown, that at $θ=π$ there exists a class of $SU(2 N)/\mathbb{Z}_2$ ($N>1$) gauge theories with a two-fold degenerate vacuum, which spontaneously breaks the time reversal and charge conjugation symmetries. The corresponding order parameter is given by the generator $\mathcal{O}$ of the $\mathbb{Z}_N$ one-form symmetry.

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