Monopoles, affine algebras and the gluino condensate

TL;DR

The paper develops a controlled semi-classical framework for ${\cal N}=1$ SYM by compactifying on ${\mathbb R}^3 \times S^1$ and identifying monopole constituents of instantons. The monopole contributions generate an affine Toda superpotential for the low-energy abelian theory, yielding $c_2$ supersymmetric vacua and fractional monopole charges, with the gluino condensate computable in each vacuum. The analysis, valid for all simple gauge groups, matches weak-coupling instanton results for classical groups and extends to exceptional groups, providing new predictions and resolving longstanding puzzles about the condensate. The work connects instanton physics, affine Lie algebras, and holomorphy, and highlights the role of instanton partons in the nonperturbative dynamics of supersymmetric gauge theories.

Abstract

We examine the low-energy dynamics of four-dimensional supersymmetric gauge theories and calculate the values of the gluino condensate for all simple gauge groups. By initially compactifying the theory on a cylinder we are able to perform calculations in a controlled weakly-coupled way for small radius. The dominant contributions to the path integral on the cylinder arise from magnetic monopoles which play the role of instanton constituents. We find that the semi-classically generated superpotential of the theory is the affine Toda potential for an associated twisted affine algebra. We determine the supersymmetric vacua and calculate the values of the gluino condensate. The number of supersymmetric vacua is equal to c_2, the dual Coxeter number, and in each vacuum the monopoles carry a fraction 1/c_2 of topological charge. As the results are independent of the radius of the circle, they are also valid in the strong coupling regime where the theory becomes decompactified. In this way we obtain values for the gluino condensate which for the classical gauge groups agree with previously known ``weak coupling instanton'' expressions (but not with the ``strong coupling instanton'' calculations). This detailed agreement provides further evidence in favour of the recently advocated resolution of the the gluino condensate puzzle. We also make explicit predictions for the gluino condensate for the exceptional groups.