Chiral Lagrangian from Duality and Monopole Operators in Compactified QCD

Abstract

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 $m_{\pi}^2 f_{\pi}^2 = m_q \langle \bar{q} q \rangle$. Abelian duality, monopole operators, and flavor-twisted boundary conditions, or a background flavor holonomy, 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 ${\cal N}=1$ and ${\cal N}=2$ supersymmetric gauge theory dynamics and non-supersymmetric QCD dynamics.