Monopoles, affine algebras and the gluino condensate

TL;DR

This paper calculates the gluino condensate in 4D supersymmetric gauge theories using monopole contributions on a compactified space, confirming previous weak coupling results and providing new predictions for exceptional groups.

Contribution

It introduces a semi-classical approach using affine Toda potentials to determine gluino condensates across all simple gauge groups, including exceptional ones.

Findings

Gluino condensate values match weak coupling instanton results for classical groups.

Number of supersymmetric vacua equals the dual Coxeter number c_2.

Monopoles carry fractional topological charge 1/c_2 in each vacuum.

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.