Exact T-duality between Calorons and Taub-NUT spaces

TL;DR

This work constructs explicit SU(2) caloron solutions with topological charge $k=1$ and nontrivial holonomy $\omega$, using the Nahm transform and ADHM data. The moduli space is shown to factorize as $\mathbb{R}^3\times S^1$ times a Taub–NUT space with mass $M=1/\sqrt{8\omega(1-2\omega)}$, revealing an exact classical T-duality between H-monopoles and Kaluza–Klein monopoles. The construction hinges on an infinite periodic array of instantons, Fourier transform to Nahm data, and a Green’s-function formulation that yields an explicit self-dual gauge field. These results connect finite-temperature gauge theory, caloron structure, and string-duality frameworks, offering precise geometric insight into monopole dualities.

Abstract

We determine all SU(2) caloron solutions with topological charge one and arbitrary Polyakov loop at spatial infinity (with trace 2.cos(2.pi.omega)), using the Nahm duality transformation and ADHM. By explicit computations we show that the moduli space is given by a product of the base manifold R^3 X S^1 and a Taub-NUT space with mass M=1/sqrt{8.omega(1-2.omega)}, for omega in [0, 1/2], in units where S^1=R/Z. Implications for finite temperature field theory and string duality between Kaluza-Klein and H-monopoles are briefly discussed

Figures (1)

• Figure 1: Profiles for calorons at $\omega=0$, 0.125, 0.25 (from top to bottom) with $\rho=1$. The axis connecting the lumps, separated by a distance $\pi$ (for $\omega\neq0$), corresponds to the direction of $\hat{\omega}$. The other direction indicates the distance to this axis, making use of the axial symmetry of the solutions. Vertically is plotted the action density, at the time of its maximal value, on equal logarithmic scales for the three profiles. The profiles were cut off at an action density below $1/e$. The mass ratio of the two lumps is approximately $\omega/\bar{\omega}$, i.e. zero (no second lump), a third and one (equal masses), for the respective values of $\omega$.