Non-Abelian Monopoles in the Higgs Phase

33 Pages·2011·17.57 MB·English

Non-Abelian Monopoles
in the
Higgs Phase
Muneto Nitta and Walter Vinci
arXiv: 1012.4057; Nucl.Phys.B848:121-154,2011
Walter Vinci
Research Fellow
University of Minnesota
Outline
Introduction-Overview
Coulomb Phase
Non-Abelian monopoles
Moduli spaces (Rational Maps)
Higgs Phase
Non-Abelian vortices and kinks
Moduli spaces (Moduli Matrix)
Kinks Monopoles
⇔
Non-Abelian” Monopoles and Kinks
“
Moduli Matrix for Monopoles
Motivations
Non-Perturbative Dynamics of Gauge Field Theories
Monopoles and Vortices
Dual superconductivity model of confinement
Nambu-‘t Hooft-Mandelstam
Supersymmetric Gauge Theories
Reliable control of strong coupling effects Seiberg-Witten
Confinement, dualities
String/Field theory interplay:
D-branes/domain walls, strings/vortices, wall/vortex junction
Monopoles
in the
Coulomb Phase
Non-Abelian Monopoles
Montonen, Olive,
Weinberg et al.
SU(N) Gauge theory:
1 1
2 2
= Tr (F ) + D Φ
µν µ
L 2g2 g2 | |
￿ ￿
µ . . . 0
1
N
. . .
2
Φ = v/N, Φ = . . . , µ = 0
 . . .  i
| | ⇒
∞ ∞
i=1
0 . . . µ
￿
N
 
 
SU (N )
Abelian case: all eigenvalues
N 1 N 1
π = π (U (1) ) = .
Z
2 1 − −
U (1)N 1
are different
−
￿ ￿
SU (N )
Non-Abelian case: some
q 1
π =
Z
2 −
eigenvalues coincide S(U (n ) U (n ) U (n ))
1 2 q
￿ × · · · × ￿
Monopoles in the Coulomb Phase
Non-Abelian Monopoles
Bogomol’nyi completion:
1 1
3 2
E = d x Tr (￿ F D Φ) + ∂ (￿ ΦF )
ijk jk i i ijk jk
4g2 − g2
￿ ￿ ￿
Topological term
Bogomol’nyi equations:
D Φ = ￿ F
i ijk jk
Monopoles in the Coulomb Phase
Moduli Spaces
(A , Φ) satisfying BPS equations and boundary conditions
i
(x) G : (0 , 0, x ) 1 , x .
0 0 3 3
G ∈ G → → ∞
SU(2)
3 1
= (A , Φ) / = S
R
framed i 0
framed
M G M ×
Nham transorm, Spectral curves, Twistors, Rational maps...
Monopoles in the Coulomb Phase
Rational Map Construction
Scattering data of the Hitchin equation along x
3
Donaldson

ψ = (D + Φ)ψ = 0
H 3
∇
µ 0
Φ =
SU(2) case
0 µ
∞
￿ − ￿
0
m/g µ x
ψ(x ) x e 3 x +
3 3 − 3
∼ 1 | | → ∞
￿ ￿
1 0
m/g µ x m/g µ x
x ψ(x ) b x e 3 + a x e 3
3 3 3 3 −
−∞ ← ∼ 0 | | 1 | |
￿ ￿ ￿ ￿
[D , ] = 0 D ψ ∂ ψ = ∂ ψ = 0
z¯ H z¯ H z¯ H H z¯
∇ ∇ ∼ ∇ ∇
a a(z), b b(z)
Holomorphic functions
→ →
Monopoles in the Coulomb Phase
Rational Map Construction
SU(2) Case:
m 1
b(z) q z + + q
1 − m
R (z) = · · ·
m
≡ a(z) zm + p zm 1 + + p
1 − m
· · ·
m : Monopole charge
Holomorphic (based) rational map to
the projective complex line
SU (2)

1 1 1
R : P P P =
C C C
m
→ U (1)
R = (a(z), b(z))
m
α
1
R =
Fundamental SU(2) monopole : α = 0
i
z z
￿
0
−
SU(2),k=1
3 1
= (z ) (α) S
C C R
0 ∗
framed
M × ￿ ×
Monopoles in the Coulomb Phase
Rational Map Construction
Jarvis
General case: flag manifolds
1
R(z) : P lag
C F
n ,...,n
→ 1 q
SU (N )
lag =
F
n ,...,n
1 q S(U (n ) U (n ))
1 q
× · · · ×
Monopoles in the Coulomb Phase