We show that there exists a special compactification of QCD on R3×S1 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 mπ2fπ2=-mq qq. 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 S1, and yield strong support for adiabatic continuity between the small-S1 and large-S1 regimes. We also find concrete microscopic connections between N=1 and N=2 supersymmetric gauge theory dynamics and nonsupersymmetric QCD dynamics.
Bibliographical noteFunding Information:
This work was supported in part by the U.S. Department of Energy under Grants No. DE-FG02-00ER-41132 (A.C.), No. DE-FG02-03ER41260 (T.S.), and No. DE-SC0013036 (M.U.).