For three Lie-algebra-valued functions of one variable, the Nahm equations areThey arise by taking a solution of the ASDYM equations with orientation , imposing independence of , and setting by a local Yang-Mills gauge transformation. Then , , and anti-self-duality of gauge curvature gives the displayed equations. The opposite orientation reverses their sign.
Put , and , and defineDirect expansion gives . The Nahm equations imply , and , so . In any finite-dimensional matrix representation, cyclicity of the matrix trace givesThus every coefficient of the degree-at-most- matrix trace polynomial is conserved. At projective infinity the polynomial is interpreted as a section of , not a globally holomorphic scalar function on the complex projective line.
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The Nahm equations are a set of differential equations that describe the behavior of certain types of mathematical and physical objects, particularly in the context of supersymmetry and gauge theory. They were introduced by Werner Nahm in the context of solitons and are particularly relevant in the study of BPS (Bogomolny-Prasad-Sommerfield) states in supersymmetric theories.