Gauge fields are ubiquitous in nature. In the context of quantum electrodynamics, you may be most familiar with the photon, which represents the gauge field mediating electromagnetic forces. But there are also gluons, which mediate strong forces, and the W and Z particles, which mediate the weak forces. According to the standard model, those few gauge bosons, in fact, mediate all elementary interactions.

In materials physics, applied gauge fields in the form of laboratory magnetic fields are essential for realizing exotic quantum phenomena, such as the quantum Hall effect (see the article by Joseph Avron, Daniel Osadchy, and Ruedi Seiler, Physics Today, August 2003, page 38). In a two-dimensional lattice, for instance, electrons that move through a periodic potential and a strong magnetic field are expected to exhibit a recursive, fractal energy spectrum known as Hofstadter’s butterfly, pictured here.

Furthermore, strongly correlated systems can host new gauge fields...

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