Gravity and Strings

276 The Taub–NUT solution
become the components of the three-dimensional YM vector field. The Fi0 components of
the field strength are
Fi0 = ∂i − [Ai,] = Di, (9.35)
i.e. the three-dimensional YM covariant derivative of the scalar . After integrating over
the redundant coordinate τ (which we take to be periodic with period 2π), the Euclidean
YM action becomes
SEYM = 2π
d 3 x Tr 1
4 FijFij + 1
2 DiDi , (9.36)
and the (anti-)self-duality equation for F becomes the Bogomol’nyi equation [163, 248]
Fij =∓ɛijkDk. (9.37)
Let us now consider the (four-dimensional, Lorentzian) Georgi–Glashow model [425]
which consists of an SU(2) gauge field A coupled to a triplet of Higgs fields with a
potential V () = 1
2λ[Tr(2 ) − 1] 2
SGG = d 4 x − 1
4Tr F 2 + 1
2Tr(D)2 − 1
2λ[Tr(2 ) − 1] 2 . (9.38)
’t Hooft [890] and Polyakov [783] found a magnetic-monopole solution of this model that
generalizes Dirac’s. In the λ = 0 limit (the Bogomol’nyi–Prasad–Sommerfield (BPS) limit),
the solution takes an especially simple form [788] and has special properties that can also
be related to supersymmetry (see Chapter 13).
Let us focus on purely magnetic (i.e. A0 = 0) and static (∂0Aµ = ∂0 = 0) field configurations.
Their energy (taking λ = 0) is given precisely by [1/(2π)]SEYM in Eq. (9.36). It is
not surprising that, therefore, the energy of these configurations is bounded: the manifestly
positive integral
d 3 x Tr Fij ± ɛijkDk
2
= 8E ±
d 3 xɛijkTr(FijDk) ≥ 0. (9.39)
On integrating by parts and using the three-dimensional Bianchi identity, we find that
d 3
xɛijkTr(FijDk) = d 3
x∂i(ɛijkFij) = 4 Tr(F) =−4p, (9.40)
where we have used Stokes’ theorem and where p is the SU(2) magnetic charge. Thus,
S 2 ∞
E ≥ 1
|p|, (9.41)
2
which is the Bogomol’nyi or BPS bound.Weknow that p is quantized (for g = 1), p = 2πn.
Using this fact and the relation E = [1/(2π)]SEYM, this relation is completely equivalent
to Eq. (9.20). On the other hand, the configurations that minimize the energy E = 1
2 |p|
(saturate the BPS bound) are those satisfying the first-order Bogomol’nyi equation and it is
easy to prove that these configurations also solve all the (second-order) equations of motion
of the λ = 0Georgi–Glashow model.