8 Beyond the Standard Model
489
The duality symmetry now acts by:
E + iB → exp (iα) (E + iB) , e + im → exp (iα) (e + im) .
(8.113)
For the SU(2) case the moduli are given by u = Trϕ 2 , for SU(N C ) the moduli are
given by u k = Trϕ k , k = 2, . . . ,N C . The classical moduli space is singular at times,
there are no perturbative or nonperturbative corrections.
The dependence of τ on the moduli coordinate u was found. The vast number of
results and literature on this will not be described here. Briefly:
The following complex equation,
y
2
= ax
3
+ bx
2
+ cx + d
(8.114)
determines a torus. The complex structure of the torus, τ torus will be identified with
the complex coupling τ eff . a,b,c,d are known holomorphic functions of the moduli,
couplings, and scale, and so will implicitly determine τ torus .
When y(x) and y
(x) vanish, for some value of x, τ is singular. Therefore,
τ eff = i∞, g
2
eff = 0
(8.115)
and the effective coupling vanishes. This reflects the presence of massless charged
objects. This occurs for definite values of u in the moduli space. These new massless
particles are monopoles or dyons. The theory is on the verge of confinement. For
N = 2 supersymmetry that is the best one can do. The monopoles are massless but
they have not condensed. For condensation to occur the monopoles should become
tachyonic indicating an instability that produces a condensation. One can push this
to confinement by adding a mass term: ˜
mTrϕ 2 , or generally for SU(N C ) the term:
δW = g k u k .
(8.116)
This breaks N = 2 supersymmetry down to N = 1. The effective prepotential is
now:
W = M (u k ) q ˜
q + g k u k
(8.117)
then
∂W
∂ u k
= 0,
∂W
∂ (q ˜
q)
= 0 ⇒ M (< u k >) = 0, ∂ u k M (< u k >) < q ˜
q >= −g k
(8.118)
Since generically,
∂ u k M (< u k >) = 0
(8.119)
489
The duality symmetry now acts by:
E + iB → exp (iα) (E + iB) , e + im → exp (iα) (e + im) .
(8.113)
For the SU(2) case the moduli are given by u = Trϕ 2 , for SU(N C ) the moduli are
given by u k = Trϕ k , k = 2, . . . ,N C . The classical moduli space is singular at times,
there are no perturbative or nonperturbative corrections.
The dependence of τ on the moduli coordinate u was found. The vast number of
results and literature on this will not be described here. Briefly:
The following complex equation,
y
2
= ax
3
+ bx
2
+ cx + d
(8.114)
determines a torus. The complex structure of the torus, τ torus will be identified with
the complex coupling τ eff . a,b,c,d are known holomorphic functions of the moduli,
couplings, and scale, and so will implicitly determine τ torus .
When y(x) and y
(x) vanish, for some value of x, τ is singular. Therefore,
τ eff = i∞, g
2
eff = 0
(8.115)
and the effective coupling vanishes. This reflects the presence of massless charged
objects. This occurs for definite values of u in the moduli space. These new massless
particles are monopoles or dyons. The theory is on the verge of confinement. For
N = 2 supersymmetry that is the best one can do. The monopoles are massless but
they have not condensed. For condensation to occur the monopoles should become
tachyonic indicating an instability that produces a condensation. One can push this
to confinement by adding a mass term: ˜
mTrϕ 2 , or generally for SU(N C ) the term:
δW = g k u k .
(8.116)
This breaks N = 2 supersymmetry down to N = 1. The effective prepotential is
now:
W = M (u k ) q ˜
q + g k u k
(8.117)
then
∂W
∂ u k
= 0,
∂W
∂ (q ˜
q)
= 0 ⇒ M (< u k >) = 0, ∂ u k M (< u k >) < q ˜
q >= −g k
(8.118)
Since generically,
∂ u k M (< u k >) = 0
(8.119)