On Nahm's transformation with twisted boundary conditions

TL;DR

This paper extends Nahm's transformation to self-dual fields on a 4D torus with non-zero twist, showing it remains an involution and results in a new self-dual gauge field on a different torus.

Contribution

It introduces a new definition of Nahm's transformation applicable to twisted boundary conditions on the 4D torus, maintaining self-duality and involution properties.

Findings

Transformation preserves self-duality.

Results in a new gauge field on a different torus.

Maintains involution property.

Abstract

Following two different tracks, we arrive at a definition of Nahm's transformation valid for self-dual fields on the 4-dimensional torus with non-zero twist tensor.The transform is again a self-dual gauge field defined on a new torus and with non-zero twist tensor. It preserves the property of being an involution.