488
E. Rabinovici
8.2.5.6 More General Matter Composition of SUSY Gauge Theories
One can enrich the structure of the theory by adding N a particles in the adjoint
representation. At first one has no matter in the fundamental representation and
scalar multiplets which are adjoint valued. The potential for the scalars, ϕ i is given
by:
V = ([ϕ, ϕ])
2 .
(8.108)
This potential obviously has a flat direction for diagonal ϕ. The gauge invariant
macroscopic moduli would be Trϕ k . Consider the non-generic example of N C = 2
and N a = 1, the supersymmetry is now increased to N = 2. There is a single complex
modulus, Trϕ 2 . Classically, SU(2) is broken to U(1) for Trφ 2 = 0. One would expect
a singularity at Trϕ 2 = 0. The exact quantum potential vanishes in this case.
Naively, one could have expected that when Trϕ 2 is of order or smaller, one
would expect that the strong infra-red fluctuations would wash away the expectation
value for Trϕ 2 and the theory would be confining. The surprising thing is that
when SU(2) breaks down to U(1), because of the very strong constraints that
supersymmetry imposes on the system, there are only two special points in moduli
space and even there the theory is only on the verge of confinement. Everywhere
else the theory is in the Coulomb phase. At the special points in the moduli space,
new particles will become massless.
One can examine the effective theory at a generic point in moduli space where
the theory is broken down to U(1). The Lagrangian is given by,
L =
d
2 θ Im
τ eff
Trφ
2 , g, Λ
W α W
α
(8.109)
The τ eff is the effective complex coupling which is a function of the modulus,
Trϕ 2 , the original couplings and the scale, . This theory has an SL(2,Z) duality
symmetry. The generators of the SL(2,Z) act on τ , defined by (8.76), as follows:
τ → −
1
τ
, τ → τ + 1
(8.110)
This is a generalization of the usual U(1) duality that occurs with electromagnetism to the case of a complex coupling. Recall the usual electromagnetic duality
for Maxwell theory in the presence of charged matter is:
E → B, B → −E, e → m, m → −e.
(8.111)
This generalizes to a U(1) symmetry by defining:
E + iB, e + im.
(8.112)
E. Rabinovici
8.2.5.6 More General Matter Composition of SUSY Gauge Theories
One can enrich the structure of the theory by adding N a particles in the adjoint
representation. At first one has no matter in the fundamental representation and
scalar multiplets which are adjoint valued. The potential for the scalars, ϕ i is given
by:
V = ([ϕ, ϕ])
2 .
(8.108)
This potential obviously has a flat direction for diagonal ϕ. The gauge invariant
macroscopic moduli would be Trϕ k . Consider the non-generic example of N C = 2
and N a = 1, the supersymmetry is now increased to N = 2. There is a single complex
modulus, Trϕ 2 . Classically, SU(2) is broken to U(1) for Trφ 2 = 0. One would expect
a singularity at Trϕ 2 = 0. The exact quantum potential vanishes in this case.
Naively, one could have expected that when Trϕ 2 is of order or smaller, one
would expect that the strong infra-red fluctuations would wash away the expectation
value for Trϕ 2 and the theory would be confining. The surprising thing is that
when SU(2) breaks down to U(1), because of the very strong constraints that
supersymmetry imposes on the system, there are only two special points in moduli
space and even there the theory is only on the verge of confinement. Everywhere
else the theory is in the Coulomb phase. At the special points in the moduli space,
new particles will become massless.
One can examine the effective theory at a generic point in moduli space where
the theory is broken down to U(1). The Lagrangian is given by,
L =
d
2 θ Im
τ eff
Trφ
2 , g, Λ
W α W
α
(8.109)
The τ eff is the effective complex coupling which is a function of the modulus,
Trϕ 2 , the original couplings and the scale, . This theory has an SL(2,Z) duality
symmetry. The generators of the SL(2,Z) act on τ , defined by (8.76), as follows:
τ → −
1
τ
, τ → τ + 1
(8.110)
This is a generalization of the usual U(1) duality that occurs with electromagnetism to the case of a complex coupling. Recall the usual electromagnetic duality
for Maxwell theory in the presence of charged matter is:
E → B, B → −E, e → m, m → −e.
(8.111)
This generalizes to a U(1) symmetry by defining:
E + iB, e + im.
(8.112)