Weyl Equation and (Non)-Commutative SU(n+1) BPS Monopoles

We apply the ADHMN construction to obtain the SU(n+1)(for generic values of n) spherically symmetric BPS monopoles with minimal symmetry breaking. In particular, the problem simplifies by solving the Weyl equation, leading to a set of coupled equations, whose solutions are expressed in terms of the Whittaker functions. Next, this construction is generalized for non-commutative SU(n+1) BPS monopoles, where the corresponding solutions are given in terms of the Heun B functions.

💡 Research Summary

The paper presents a systematic construction of spherically symmetric BPS monopoles in the gauge group SU(n + 1) with minimal symmetry breaking, using the ADHMN (Atiyah‑Drinfeld‑Hitchin‑Manin‑Nahm) formalism, and then extends the construction to the non‑commutative setting.
The authors begin by recalling that the ADHMN method reduces the self‑dual Yang‑Mills equations in four dimensions to a one‑dimensional Nahm equation together with a Weyl equation for an auxiliary spinor‑valued vector. For minimal symmetry breaking the Nahm data simplify dramatically: the Nahm matrices are proportional to the generators of an (n + 1)‑dimensional irreducible representation of SU(2). Consequently the Nahm equation is automatically satisfied, and the problem reduces to solving the Weyl equation
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