Non-Abelian Vortex in Four Dimensions as a Critical String on a Conifold

25 Pages·2016·0.09 MB·English

Non-Abelian Vortex in Four Dimensions as a
Critical String on a Conifold
P. Koroteev, M. Shifman and A. Yung
1 Introduction
Shifman and Yung, 2015:
Non-Abelian vortex in = 2 supersymmetric QCD can behave as a
N
critical superstring
4D = 2 QCD: U(N = 2) gauge group and N = 4 quark flavors.
f
N
Non-Abelian vortex strings
Non-Abelian strings were suggested in = 2 U(N) QCD
N
Hanany, Tong 2003
Auzzi, Bolognesi, Evslin, Konishi, Yung 2003
Shifman Yung 2004
Hanany Tong 2004
Z Abelian string: Flux directed in the Cartan subalgebra, say for
N
SO(3) = SU(2)/Z
2
flux τ
3
∼
Orientational zero modes
Non-Abelian string :
Rotation of color flux inside SU(N).
S
Idea:
Non-Abelian string has more moduli then Abrikosov-Nielsen-Olesen
(ANO) string.
It has translational + orientaional moduli
We can fulfill the criticality condition:
The solitonic non-Abelian vortex have six orientational moduli,
•
which, together with four translational moduli, form a
ten-dimensional space (N = 2, N = 4).
f
For N = 2N 2D world sheet theory on the string is conformal.
f
•
Most of solitonic strings are ”thick”.
Transverse size = 1 , where m is the typical mass of bulk excitations.
m
ANO string: Nambu-Goto action
∂n
2 √
S = T d σ h + O
NG
mn
( !)
Z
where T is string tension and
µ
h = det(∂ x ∂ x )
α β µ
Polchinski-Strominger, 1991: Without higher derivative terms
the world sheet theory is not UV complete
Higher derivative terms at weak coupling, g 1
≪
∂n
√
O , m g T
mn ∼
!
At J 1 ∂ √T
∼ →
Thus higher derivative terms
n
T
→ m2
(cid:18) (cid:19)
blow up at weak coupling!
Polyakov: string surface become ”crumpled”.
String grows short and thick.
QUESTION:
Can we find any example of a 4D field theory which supports thin
vortex strings?
Non-Abelian vortex in = 2 QCD with U(2) gauge group and N = 4
f
N
flavors is critical.
Given that for non-Abelian vortex low energy world sheet theory is
critical
Shifman, Yung 2015: Conjecture that
Thin string regime
2
T m
≪
is actually satisfied at strong coupling g2 1.
c
∼
2 2
m(g) , g g
c
→ ∞ →
Higher derivative corrections can be ignored
2 Non-Abelian vortex strings
Bulk theory: 4D = 2 QCD with Fayet-Iliopoulos term.
N
For U(N) gauge group in the bulk we have 2D CP(N 1) model on the
−
string
CP(N 1) == U(1) gauge theory in the strong coupling limit
−
e2
2 2
2 P P 2
S = d x n + n β ,
CP(N−1) α
∇ 2 | | −
( )
Z
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
where nP are complex fields P =(cid:12) 1, ...,(cid:12)N,
Condition
4π
P 2
n = β = ,
| | g2
imposed in the limit e2
→ ∞
More flavors semilocal non-Abelian string
⇒
The orientational moduli described by a complex vector nP (here
P = 1, ..., N),
N˜ = (N N) size moduli are parametrized by a complex vector ρK
f
−
(K = N + 1, ..., N ).
f
The effective two-dimensional theory is the = (2, 2) weighted CP
N
model
e2
2 2 2
2 P ˜ K P 2 K 2
S = d x n + ρ + n ρ β ,
WCP α α
∇ ∇ 2 | | − | | −
( )
Z
(cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
P = 1, ..., N , K = N + 1, ..., N .
f
The fields nP and ρK have charges +1 and 1 with respect to the
−
auxiliary U(1) gauge field
e2
→ ∞