# Hidden Symmetries Of A Self-Dual Monopole

• L. Fehér
• P. Horváthy
• L. O’Raifeartaigh

## 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 role2,3,4 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 Ia (a = 1,2,3) are the standard isospin matrices in some representation and we suppose that nothing depends on the extra coordinate x4.

## Keywords

Dirac Equation Zero Mode Dynamical Symmetry Hide Symmetry Gravitational Instanton
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Plenum Press, New York 1989

## Authors and Affiliations

• L. Fehér
• 1
• 2
• P. Horváthy
• 2
• 3
• L. O’Raifeartaigh
• 2
1. 1.Bolyai Institute, JateSzegedHungary
2. 2.Dublin Institute for Advanced StudiesDublinIreland
3. 3.Maths. Dept., UniversitéMetzFrance