Chiral Lagrangian from Duality and Monopole Operators in Compactified QCD

We show that there exists a special compactification of QCD on ${\mathbb{R}}^3\times S^1$ in which the theory has a domain where continuous chiral symmetry breaking is analytically calculable. We give a microscopic derivation of the chiral Lagrangian, the chiral condensate, and the Gell-Mann–Oakes–Renner relation $m_{\pi }^2{f}_{\pi }^2=-m_q⟨\overline{q}q⟩$. Abelian duality, monopole operators, and flavor-twisted boundary conditions play the main roles. The flavor twisting leads to the new effect of fractional jumping of fermion zero modes among monopole instantons. Chiral symmetry breaking is induced by monopole-instanton operators, and the Nambu-Goldstone pions arise by color-flavor transmutation from gapless “dual photons.” We also give a microscopic picture of the “constituent quark” masses. Our results are consistent with expectations from chiral perturbation theory at large $S^1$, and yield strong support for adiabatic continuity between the small-$S^1$ and large-$S^1$ regimes. We also find concrete microscopic connections between $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric gauge theory dynamics and nonsupersymmetric QCD dynamics.

Figure 1

Eigenvalue distributions for the gauge holonomy $\mathrm{\Omega }$ (blue dots) $N_c=4$ and flavor holonomy ${\mathrm{\Omega }}_F$ (red dots) with $N_f=4$ (on the left) and $N_f=3$ (on the right). The $U(1{)}_Q$ holonomy ${\mathrm{\Omega }}_Q=e^{i\theta }$ rotates ${\mathrm{\Omega }}_F$ and $\mathrm{\Omega }$ eigenvalues relative to each other. The green arrows represent the four monopole-instanton events, which are labeled by the affine roots ${\overrightarrow{\alpha }}_i$. Each time an arrow “passes” a flavor holonomy eigenvalue, the associated monopole-instanton operator picks up two fermion zero modes. So in the left $N_f=4$ figure, all monopole instantons have two zero modes, while in the right $N_f=3$ figure ${\overrightarrow{\alpha }}_1$ has no zero modes while ${\overrightarrow{\alpha }}_{2,3,4}$ each have two zero modes.

Figure 2

The distribution of fundamental fermion zero modes for the monopole instantons with a trivial flavor holonomy (top row) and a ${\mathbb{Z}}_{N_f}$-symmetric flavor holonomy (bottom row) for $N_f=N_c$. The latter distribution is identical to $\mathcal{N}=1$ SYM theory on ${\mathbb{R}}^3\times S^1$ in which monopole instantons saturate the chiral condensate.