Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions

Abstract

In recent work, we derived the long-distance confining dynamics of certain QCD-like gauge theories formulated on small ${S}^{1}\ifmmode\times\else\texttimes\fi{}{\mathbb{R}}^{3}$ based on symmetries, an index theorem, and Abelian duality. Here, we give the microscopic derivation. The solution reveals a new mechanism of confinement in QCD(adj) in the regime where we have control over both perturbative and nonperturbative aspects. In particular, consider $SU(2)$ QCD(adj) theory with $1\ensuremath{\le}{n}_{f}\ensuremath{\le}4$ Majorana fermions, a theory which undergoes gauge symmetry breaking at small ${S}^{1}$. If the magnetic charge of the Bogomol'nyi-Prasad-Sommerfield (BPS) monopole is normalized to unity, we show that confinement occurs due to condensation of objects with magnetic charge 2, not 1. Because of index theorems, we know that such an object cannot be a two identical monopole configuration. Its net topological charge must vanish, and hence it must be topologically indistinguishable from the perturbative vacuum. We construct such non-self-dual topological excitations, the magnetically charged, topologically null molecules of a BPS monopole and $\overline{\mathrm{KK}}$ antimonopole, which we refer to as magnetic bions. An immediate puzzle with this proposal is the apparent Coulomb repulsion between the BPS-$\overline{\mathrm{KK}}$ pair. An attraction which overcomes the Coulomb repulsion between the two is induced by $2{n}_{f}$-fermion exchange. Bion condensation is also the mechanism of confinement in $\mathcal{N}=1$ SYM on the same four-manifold. The $SU(N)$ generalization hints a possible hidden integrability behind nonsupersymmetric QCD of affine Toda type, and allows us to analytically compute the mass gap in the gauge sector. We currently do not know the extension to ${\mathbb{R}}^{4}$.