From vortices to instantons on the Euclidean Schwarzschild manifold
Abstract
The first irreducible solution of the $\SU (2)$ self-duality equations on the Euclidean
Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later
than the famous BPST instantons on $\rl^4$ were discovered. While soon after, in 1978,
the ADHM construction gave a complete description of the moduli spaces of instantons
on $\rl^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts
for the past 40 years. By exploring a correspondence between the planar Abelian vortices
and spherically symmetric instantons on ES, we obtain: a complete description of a
connected component of the moduli space of unit energy $\SU (2)$ instantons; new examples
of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete
classification of finite energy, spherically symmetric, $\SU (2)$ instantons. As opposed
to the previously known solutions, the generic instanton coming from our construction
is not invariant under the full isometry group, in particular not static. Hence disproving
a conjecture of Tekin.
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