Chiral Lagrangian from Duality and Monopole Operators in Compactified QCD

TL;DR

It is shown that there exists a special compactification of QCD on R^{3}×S^{1} in which the theory has a domain where continuous chiral symmetry breaking is analytically calculable, and there are concrete microscopic connections between N=1 and N=2 supersymmetric gauge theory dynamics and nonsupersymmetric QCD dynamics.

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 $m_π^2 f_π^2 = m_q \langle \bar{q} q \rangle$. 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.

Figures (2)

• Figure 1: Eigenvalue distributions for the gauge holonomy $\Omega$ (blue dots) $N_c = 4$ and flavor holonomy $\Omega_F$ (red dots) with $N_f =4$ (on the left) and $N_f=3$ (on the right). The $U(1)_Q$ holonomy $\Omega_Q = e^{i\theta}$ rotates $\Omega_F$ and $\Omega$ eigenvalues relative to each other. The green arrows represent the four monopole-instanton events, which are labeled by the affine roots $\vec{\alpha}_{i}$. 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 $\vec{\alpha}_{1}$ has no zero modes while $\vec{\alpha}_{2,3,4}$ 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 ${\cal N}=1$ SYM on ${\mathbb R}^3 \times S^1$ in which monopole-instantons saturate the chiral condensate.