The Nahm Equations
From Islands
What are the Nahm equations?
The Nahm equations involve four matrices, conventionally denoted
- T0,T1,T2,T3
Each of these matrices - or more properly, each component of each matrix - is a function of a variable s. The Nahm equations describe how these matrices depend on s; they are ordinary differential equations in s.
The equations are
![\frac{d}{ds} T_1 + [T_0, T_1] = [T_2,T_3]](/~islands/images/math/0/d/a/0daff7f359e5482a50cee0e714f82974.png)
![\frac{d}{ds} T_2 + [T_0, T_2] = [T_3,T_1]](/~islands/images/math/1/6/3/16319ead5f85ff60d56527041cdebf33.png)
![\frac{d}{ds} T_3 + [T_0, T_3] = [T_1,T_2]](/~islands/images/math/6/2/0/620233c50451bfa3746c725bb1ffcda4.png)
Why are the Nahm equations?
As this website hopes to show, the Nahm equations occur in a variety of different contexts, hidden in different parts of different (but interrelated) theories. They first appeared in a construction of monopoles by Werner Nahm, yet in another sense they can be thought of as a generalisation of Euler's equations for a spinning top.
Perhaps the most natural viewpoint is to view Nahm's equations as a part of the theory of self-dual connections; or maybe as part of string theory, where Nahm's equations describe configurations of extended spatial objects known as branes. One can view all these different topics as a set of islands in an archipelago of ideas - with the Nahm equations being a natural point from which to start a journey of discovery, hopping from island to island. We hope you enjoy your time on our site.
