islands

a project exploring the Nahm equations, monopoles & more

# The Nahm Equations

## 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

$\frac{d}{ds} T_1 + [T_0, T_1] = [T_2,T_3]$
$\frac{d}{ds} T_2 + [T_0, T_2] = [T_3,T_1]$
$\frac{d}{ds} T_3 + [T_0, T_3] = [T_1,T_2]$

## 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.

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 This page was last modified 00:11, 28 November 2010. - This page has been accessed 5,509 times. - About Islands