Japanese version
Higgs field is also important in Mathematics. Essays have been published in the Notices of the AMS Sep 2007. In Physics the Higgs field is a scalar field, while in mathematics 1-form. It appears in the application to number theory.
The original article:
What is a Higgs bundle? (in English)
A translation into Japanese:
What is a Higgs bundle? (in Japanese)
In the paper by E. Witten titled "Mirror Symmetry, Hitchin's Equations, And Langlands Duality" there are some comments between the Higgs field in particle physics and one in Mathematical context.
ReplyDeleteRemark 2.1. As an aside, one may ask how closely related φ, known in the present context as the Higgs field, is to the Higgs fields of particle physics. Thus, to what extent is the terminology that was introduced in Hitchin (1987a) actually justified? The main difference is that Higgs fields in particle physics are scalar fields, while φ is a one-form on C (valued in each case in some representation of the gauge group). However, although Hitchin’s equations were first written down and studied directly, they can be obtained from N = 4 supersymmetric gauge theory via a sort of twisting procedure (similar to the procedure that leads from N = 2 supersymmetric gauge theory to Donaldson theory). In this twisting procedure, some of the Higgs-like scalar fields of N = 4 super Yang-Mills theory are indeed converted into the Higgs field that enters in Hitchin’s equations. This gives a reasonable justification for the terminology.