Wu–Yang dictionary

In topology and high energy physics, the Wu–Yang dictionary refers to the mathematical identification that allows to translate back and forth between the concepts of gauge theory and those of differential geometry. It was devised by Tai Tsun Wu and C. N. Yang in 1975 when studying the relation between electromagnetism and fiber bundle theory. This dictionary has been credited as bringing mathematics and theoretical physics closer together. A crucial example of the success of the dictionary is that it allowed to understand Paul Dirac's monopole quantization in terms of Hopf fibrations. In 1975, theoretical physicists Tsun Wu and C. N. Yang working in Stony Brook University, published a paper on the mathematical framework of electromagnetism and the Aharonov–Bohm effect in terms of fiber bundles. A year later, mathematician Isadore Singer came to visit and brought a copy back to the University of Oxford. Singer showed the paper to Michael Atiyah and other mathematicians, sparking a close collaboration between physicists and mathematicians. Yang also recounts a conversation that he had with one of the mathematicians that founded fiber bundle theory, Shiing-Shen Chern: In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of Shiing-Shen Chern in El Cerrito, near Berkeley. (I had taken courses with him in the early 1940s when he was a young professor and I an undergraduate student at the National Southwest Associated University in Kunming, China. That was before fiber bundles had become important in differential geometry and before Chern had made history with his contributions to the generalized Gauss–Bonnet theorem and the Chern classes.) We had much to talk about: friends, relatives, China. When our conversation turned to fiber bundles, I told him that I had finally learned from Jim Simons the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world.