Nahm dualities on the torus - a synthesis

TL;DR

This paper provides a unified analytical and numerical framework for understanding self-dual SU(2) gauge fields on tori, revealing dualities between calorons and instantons through the Nahm transformation with twisted boundary conditions.

Contribution

It extends the Nahm transformation to twisted boundary conditions and demonstrates the duality between calorons and instantons on different torus scales.

Findings

Calorons are Nahm dual to instantons depending on torus size.

Holonomies correspond to constituent locations in the dual picture.

Duality becomes exact as torus dimensions approach infinity.

Abstract

We give a unified description of self-dual SU(2) gauge fields on tori of size lt x ls^3 based on a mixture of analytical and numerical methods using the Nahm transformation, extended to the case of twisted boundary conditions. We show how torus calorons (lt/ls small) are Nahm dual to the torus instantons (lt/ls large). Holonomies are dual to the locations of constituents, this duality becoming exact in the limiting cases ls or lt --> infinity. Implications for the moduli spaces are discussed.