Hidden Symmetries Of A Self-Dual Monopole

Abstract

A self-dual (SD) SU(2) monopolé is a static solution of the first order Bogomolny¹ equations

$$ {{D}_{k}}\Phi = {{B}_{k}}\left( { = \frac{1}{2}{{ \in }_{{kij}}}{{F}_{{ij}}}} \right) $$

((1))

The monopole’s field can be identified with a pure Yang-Mills configuration A₀ = 0, A _k , A ₄ = Φ in (1+4)-dimensional flat space which does not depend on the coordinates x and x ⁴. The Dirac equation

$$ i\frac{{\partial \Phi }}{{\partial {{x}^{0}}}} = i{{\gamma }^{0}}{{\not{D}}_{4}}\Psi $$

((2))

plays a decisive role^2,3,4 in describing the fluctuations around a SD monopole. Here we are interested in the symmetries of the 4-dimensional, Euclidean Dirac operator

$$ {{\not{D}}_{4}} = {{\gamma }^{k}}{{D}_{k}} + {{\gamma }^{4}}{{D}_{4}} = \frac{{\sqrt {2} }}{i}\left( {\begin{array}{*{20}{c}} 0 & Q \\ {{{Q}^{ + }}} & 0 \\ \end{array} } \right), $$

(3)

where

$$ Q = \frac{1}{{\sqrt {2} }}\left( {\Phi \cdot {{1}_{2}} - i\vec{\pi }\cdot \vec{\sigma }} \right),\quad \vec{\pi } = - i\vec{D} = - i\vec{\nabla } - {{\vec{A}}^{a}}{{I}_{a}},\quad {{D}_{4}} = \frac{\partial }{{\partial {{x}^{4}}}} - i{{\Phi }^{a}}{{I}_{a}} $$

((4))

We use the following γ-matrices:

$$ {{\gamma }^{k}} = \left( {\begin{array}{*{20}{c}} 0 & {i{{\sigma }^{k}}} \\ { - i{{\sigma }^{k}}} & 0 \\ \end{array} } \right),{{\gamma }^{4}} = \left( {\begin{array}{*{20}{c}} 0 & {{{1}_{2}}} \\ {{{1}_{2}}} & 0 \\ \end{array} } \right),{{\gamma }^{0}} = i{{\gamma }^{5}} = i\left( {\begin{array}{*{20}{c}} {{{1}_{2}}} & 0 \\ 0 & { - {{1}_{2}}} \\ \end{array} } \right). $$

((5))

The I_a (a = 1,2,3) are the standard isospin matrices in some representation and we suppose that nothing depends on the extra coordinate x⁴.