Gravity and Strings

9.2 The Euclidean Taub–NUT solution 277
The immediate conclusion of this discussion is that, if we take SU(2) (anti-)self-dual
instantons that do not depend on the τ coordinate, we have automatically a magnetic
monopole solution of the Georgi–Glashow model with λ = 0 satisfying the Bogomol’nyi
bound. In particular, the BPS limit of the ’t Hooft–Polyakov SU(2) is obtained using the
’t Hooft Ansatz with the harmonic function
V = 1 + λ
. (9.42)
|x3|
9.2.4 The BPST instanton and the KK monopole
We are now ready to establish a relation between the Euclidean Taub–NUT solution (KK
monopole) and the BPST instanton. We are going to see that the spin-connection frame
components ωmnp of the KK monopole are identical to the SO(4)-embedded components
of the BPST instanton connection Amnp ˜ with the harmonic function V identical to the
harmonic function H of the KK monopole, depending on just three coordinates x3.
In the simplest frame,
e 0 =H − 1 2 [dτ + Aidx i ], e0=H 1 2 ∂τ ,
e i =H 1 2 dx i , ei=H − 1 2 [∂i − Ai∂τ ],
the frame components of the spin connection (which is just an SO(4) connection) are
ω0 i0(e)=− 1
2 ∂i ln H, ωi 0 j(e)=H −1 ∂[i A j],
ω0 ij(e)=H −1 ∂[i A j], ωijk(e)=−δi[ j∂k] ln H.
(9.43)
(9.44)
Here it is important to observe that all partial derivatives in this expression have frame
indices. Using the Dirac-monopole equation for the 1-form A,
the KK-monopole spin connection becomes
ɛijk∂[i A j] =±∂k H, (9.45)
ω (±) 1
0 i0 (e)=− 2∂i ln H, ω (±)
i 0 j (e)=±ɛijk∂k ln H,
ω (±)
0 ij (e)=±ɛijk∂k ln H, ω (±)
ijk (e)=−δi[ j∂k] ln H,
(9.46)
which is identical to the connection à in Eq. (9.34). It is, therefore, (anti-)self-dual and has
SU(2) holonomy.
9.2.5 Bianchi IX gravitational instantons
In [449] the class of gravitational instantons with an SU(2) or SO(3) isometry group acting
transitively (Bianchi IX metrics) was studied, with special emphasis on those with selfdual
curvature. This class includes some of the gravitational instantons that we have studied,
namely Taub–NUT, Taub-bolt, and Eguchi–Hanson instantons, and its discussion will
provide us with some further interesting examples.