Mysteries and beauty of soliton dynamicsinnonAbelian gauge theories

(1)

Mysteries and beauty of soliton dynamics in

nonAbelian gauge theories

T o r i n o 1 5 / 1 1 / 2 0 1 3

K . K o n i s h i ( U n i v . P i s a / I N F N , P i s a )

Thursday, November 14, 2013

(2)

Solitons in physics

• Abrikosov vortices in superconductors

• Solitons (solitary waves on the water) of nonlinear equations ...

• Vortices in plasma

• Magnetic monopoles in electrodynamics (Dirac)

• Condensed matter, QHE, BE condensed cold atoms, ...

• Hydrodynamics, aerodynamics, meteorology, vortices, tornados, ...

• Neutron stars, Cosmic strings...

• Elementary particle physics (Fundamental interactions)

• Nielsen-Olesen vortex

• ‘t Hooft-Polyakov monopoles

• Skyrmion, Instantons, Domain walls, Lumps,...

• NonAbelian monopoles and vortices

• Quantum dynamics, Confinement, XSB

• ... ...

collective modes symmetri es gauge dyna mics

topology dimensio nality

Preludes :

(3)

Plan of the talk

Magnetic monopoles

Dirac --- ‘t Hooft/Polyakov --- Seiberg-Witten

’31 - ’74 - ’94

IFP - CFT Confinement

NonAbelian monopoles SCFT

NonAbelian vortices

Weakly gauged

nonAbelian vortices:

2D-4D coupled gauge systems

IFP - CFT Confinement IFP - CFT Confinement

Strongly-coupled monopoles and confinement

①

②

③

④

⑤ ⑥

’74 - ’81 - ’00 -

’08

’81

’03 - ’11

’12-’13 ’09- ’12- ’13

(4)

Dirac - W

u-Yang - ‘t Ho

oft-Polyakov - Seiberg

-Witten

① Magnetic monopoles

• Inv. under EM duality of the Maxwell eqs

Electro-Magnetic Duality

∇ · (E + i B) = 0; ∇ × (E + i B) = i ∂

∂t (E + i B).

• Invariant under

(E + i B) → e

^{i φ}

(E + i B) (1)

• Include E → B, B → −E ;

• EM duality broken by charges

∇ · E = g δ

³

(r); ∇ · B = 0;

g + i g

_m

→ e

^{i φ}

(g + i g

_m

) (2)

• Magnetic monopoles possible if (Dirac, 1931) g g

_m

= n

2 , n = 0, ±1, ±2, . . . (cfr. Eq.(2)) (3)

• Origin of the electric charge quantization ?

• Dyons (g

m 1

, g

_{e 1}

) , (g

_{m 2}

, g

_{e 2}

) are also possible if

g

_{e 1}

g

_{m 2}

− g

^{e 2}

g

_{m 1}

= n 2 .

4

broken by charge

∇ · E = g δ

³

(r); ∇ · B = 0

• Mag. monopoles OK if

Electro-Magnetic Duality

∇ · (E + i B) = 0; ∇ × (E + i B) = i ∂

∂t (E + i B).

• Invariant under

(E + i B) → e

^{i φ}

(E + i B) (1)

• Include E → B, B → −E ;

• EM duality broken by charges

∇ · E = g δ

³

(r); ∇ · B = 0;

g + i g

_m

→ e

^{i φ}

(g + i g

_m

) (2)

• Magnetic monopoles possible if (Dirac, 1931) g g

_m

= n

2 , n = 0, ±1, ±2, . . . (cfr. Eq.(2)) (3)

• Origin of the electric charge quantization ?

• Dyons (g

m 1

, g

_{e 1}

) , (g

_{m 2}

, g

_{e 2}

) are also possible if

g

_{e 1}

g

_{m 2}

− g

^{e 2}

g

_{m 1}

= n 2 .

4

Dirac 1931

• Mag. monopoles in spontaneously broken gauge theories as solitons

SU(2) ➞ U(1)

‘t Hooft, Polyakov ‘ 74

monopole solution [19], embedded in an appropriate corner of SU (N + 1) gauge group. By choosing an SU (2) ⊂ SU(N + 1) group generated by

S₁ = 1 2





0 1

0_{N −1}

1 0



 , S₂ = 1 2





0 −i

0_{N −1}

i 0



 , S₃ = 1 2





1 0

0_{N −1}

0 −1



 , (2.30) (which is broken to U (1) by the VEV of φ) a monopole solution can be constructed explicitly as [20]:

A_i(r) = A^a_i(r) S_a; φ(r) = (N + 1) v₁r^aS_a

r χ(r) + v₁





−^{N −1}₂

1_{N −1}

−^{N −1}₂



 , (2.31)

where³

A^a_i(r) = #_ajir^j

r²a(r) , (2.32)

is the standard ’t Hooft-Polyakov-BPS solution with a(r) = 1− gv₁r

sinh(gv₁r) ; χ(r) = coth(gv₁r)− 1

gv₁r , (2.33)

the latter behaving asymptotically as

a(r) → 1 , χ(r) → sign(v1) , r → ∞ , (2.34)

(for the antimonopole, χ(r) → −χ(r)). The constant in the φ(r) field is added so that it reduces asymptotically to the vacuum expectation value of Eq. (2.14), in a fixed (here chosen as (0, 0,∞)) direction.

The “magnetic flux” emanated from the magnetic monopole is given by B_i = 1

2#_ijkF_jk ^r→∞−→ r_i(S· r)

r⁴ , (2.35)

which of course is a well-known radial Dirac monopole field, embedded in the S_i color directions.

The “monopole” mass (the energy of the configuration around its center) can be approximately cal- culated as⁴

H =

%

|r|<R

d³x Tr

& 1

2g²(F_ij)² + 1

g²|Diφ|² '

. (2.36)

Rewriting the Hamiltonian as H = 1

g²

%

|r|<R

d³x Tr & 1

2|Fij − #ijk(Dkφ)|² + ∂_k(#_ijkF_ij φ) '

(2.37) our monopole configuration is seen to satisfy approximately the non-Abelian Bogomol’nyi equations

B_k =Dkφ , (2.38)

3The index a = 1, 2, 3 refers to the SU (2) group utilized to construct the solution; the gauge field Ai and the adjoint scalar φ are both (N + 1) × (N + 1) matrices in the SU (N + 1) color group.

4We take the adjoint scalar field φ to be real here and hence normalize it canonically as a real scalar field.

7

monopole solution [19], embedded in an appropriate corner of SU (N + 1) gauge group. By choosing an SU (2) ⊂ SU(N + 1) group generated by

S₁ = 1 2





0 1

0_{N −1}

1 0



 , S₂ = 1 2





0 −i

0_{N −1}

i 0



 , S₃ = 1 2





1 0

0_{N −1}

0 −1



r χ(r) + v₁





−^{N −1}₂

1_{N −1}

−^{N −1}₂



 , (2.31)

where³

A^a_i(r) = #_ajir^j

r²a(r) , (2.32)

sinh(gv1r) ; χ(r) = coth(gv₁r)− 1

gv1r , (2.33)

a(r) → 1 , χ(r) → sign(v¹) , r → ∞ , (2.34)

(for the antimonopole, χ(r) → −χ(r)). The constant in the φ(r) field is added so that it reduces asymptotically to the vacuum expectation value of Eq. (2.14), in a fixed (here chosen as (0, 0,∞)) direction.

The “magnetic flux” emanated from the magnetic monopole is given by Bi = 1

r⁴ , (2.35)

which of course is a well-known radial Dirac monopole field, embedded in the S_i color directions.

H =

%

|r|<R

d³x Tr

&

1

2g²(F_ij)² + 1

g²|Dⁱφ|² '

. (2.36)

g²

%

|r|<R

d³x Tr & 1

2|F^ij − #^ijk(D^kφ)|² + ∂_k(#_ijkF_ij φ) '

B_k = D^kφ , (2.38)

7 monopole solution [19], embedded in an appropriate corner of SU (N + 1) gauge group. By choosing an SU (2) ⊂ SU(N + 1) group generated by

S1 = 1 2





0 1

0_{N −1}

1 0



 , S2 = 1 2





0 −i

0_{N −1}

i 0



 , S3 = 1 2





1 0

0_{N −1}

0 −1



r χ(r) + v₁





−^{N −1}₂

1_{N −1}

−^{N −1}₂



 , (2.31)

where³

A^a_i(r) = #_ajir^j

r²a(r) , (2.32)

sinh(gv₁r) ; χ(r) = coth(gv1r)− 1

gv₁r , (2.33)

a(r)→ 1 , χ(r)→ sign(v1) , r → ∞ , (2.34)

(for the antimonopole, χ(r)→ −χ(r)). The constant in the φ(r) field is added so that it reduces asymptotically to the vacuum expectation value of Eq. (2.14), in a fixed (here chosen as (0, 0,∞)) direction.

The “magnetic flux” emanated from the magnetic monopole is given by B_i = 1

r⁴ , (2.35)

which of course is a well-known radial Dirac monopole field, embedded in the Si color directions.

H =

%

|r|<R

d³x Tr

&

1

2g²(F_ij)² + 1

g²|Diφ|² '

. (2.36)

g²

%

|r|<R

d³x Tr & 1

2|Fij − #ijk(Dkφ)|² + ∂_k(#_ijkFij φ) '

B_k = Dkφ , (2.38)

7

S

a

∈ su(2)

GUT mono poles ?

SU(5) -> SU(3)xSU(2)xU(1)

Π2(SU(2)/U(1)) = Z

(5)

• Quantum dynamics

• NonAbelian vortices

• NonAbelian monopoles?

• Quantization around the semiclassical monopoles ➪

• Earlier studies and difficulties

• N=2 SQCD :

fully q.m. nonAbelian monopoles in the r-vacua

• Exact Seiberg-Witten solutions in N=2 supersymmetric gauge theories

Many beautiful and subtle features:

• charge fractionalization

• Jackiw-Rebbi effects

• Rubakov effects

• Wrong statistics

Breakthr ough

• Witten effects

’74-’80

SU(3) ➞ SU(2)xU(1)

monopole carrying SU(2) charge?

NO ... well ...

... ... ...

’94

’74-’80

‘t Hooft, Mandelstam, Nambu ‘81

• QCD: monopole condensation ➞ quark confinement SU(3) ➞ U(1)

²

(6)

Seiberg-Witten solution in N=2, SU(2) susy gauge theories ’94

• Fields: W = (A

_µ

, λ), Φ = (φ, ψ) Moduli of vacua (degeneracy):

�φ� =

� a 0 0 −a

�

, u = T r �φ

²

�

• L

eff

:

L

_{ef f}

= Im[ �

d

⁴

θ ¯ A ∂F (A)

∂A + �

d

²

θ ∂

²

F (A)

∂A

²

W

^α

W

_α

]

F(A)= prepotential holomorphic

• Duality: L=L

eff

formally inv under SL(2,Z) (ad-bc=1) ⊃ EM duality

� A

_D

A

�

→

� a b c d

� � A

_D

A

� �

δL/δF

_µν⁺

F

_µν⁺

�

→

� a b c d

� � δL/δF

_µν⁺

F

_µν⁺

�

• Assume:

SU(2)/U(1): monopoles

massless monopoles at u= ± Λ

²

A^D= ∂F/∂A

= A

_D ^{τ = F}^��^{(A) =} ^dA_dA^D ⁼ ^θ_2π^{ef f} ⁺ _g^4πi2 ef f

• Which description? Depends on u !

F(A) !

SUR(2)

F_µνF^µν + i ¯λ γ^µD^µλ + iF_µνF˜^µν + ...

(7)

y

²

= (x − u)(x − Λ

²

)(x + Λ

²

)

• solves the theory

dA

_D

du = �

α

dx

y , dA

du = �

β

dx

y ,

^Im

�

α dx

� y β

dx y

> 0

➞ F(A)

•

Perturbative and nonperturbative quantum effects (instantons) fully encoded in ※

• Effective theory near u= Λ

²

Seiberg-Witten curve

L

_{ef f}

(A

_D

, F

_D^µν

, ...) +

�

d

⁴

θ ¯ M e

^V^D

M + (M → ˜ M ) + √ 2 �

d

²

θ M A

_D

M ˜ Magnetic monopole coupled minimally to the dual gauge field

※

M

_n_m_,n_e

= |n

m

A

_D

+ n

_e

A |, A

_D

= �

α

λ, A =

�

β

λ,

(8)

Seiberg-Witten curves for general gauge groups

• SU(N) with N

F

quarks

• SO(N) with N

F

quarks in vector representation

y

²

=

�

N i=1

(x − φ

ⁱ

)

²

− Λ

^2N^−N^F

NF

�

a=1

(x + m

_a

)

y

²

= x

[N/2]

�

i=1

(x − φ

²i

)

²

− 4Λ

^2(N^−N^F^−N^F⁾

x

^2+�

NF

�

a=1

(x + m

²_a

)

• etc.

Co nfinement

(9)

• Abelian dual superconductor (dynamical Abelianization) ?

SU (3) → U(1)

²

→ 1

�M� �= 0

Doubling of the meson spectrum (*)

‘t Hooft, Nambu, Mandelstam ’80

If confinement and XSB both induced by

Accidental SU(N

F2

) : too many NG bosons

SU

_L

(N

_F

) × SU

^R

(N

_F

) → SU

^V

(N

_F

)

☞

• Non-Abelian monopole condensation ?

SU (3) → SU(2) × U(1) → 1

☞ Problems (*) avoided but

Non-Abelian monopoles expected to be strongly coupled (no sign flip of b

0

)

�M

b^a

� = δ

b^a

Λ

Π

₁

(U(1)

²

) = Z × Z

Π

₁

(SU(2) × U(1)) = Z

② Confinement and monopoles

(difficulty for us, not for Nature...)

(10)

Conformal invariance (CFT) and confinement

Often confinement = deformation of an IR f.p. CFT

IR degrees of freedom in CFT

UV CFT Infrared-fixed point CFT

tell us how confinement / XSB work QCD (Quantum Chromodynamics):

quarks and gluons, AF collective behavior of color (confinement, XSB) ?

(11)

Back to p. 5

(12)

L

_{ef f}

(A

_D

, F

_D^µν

, ...) +

�

d

⁴

θ ¯ M e

^V^D

M + (M → ˜ M ) + √ 2 �

d

²

θ M A

_D

M ˜

• Abelian dual superconductor (dynamical Abelianization) ?

N=2 susy SU(2) gauge theory with μ Φ

²

perturbation

YES!

+ �

d

²

θ µ U (A

_D

)

➞ ^A

^D

^{= 0,} ^{�M� =} ^� ^µ _∂A ^∂U

_D

⁼ ^� ^µΛ

(monopole condensation

➞

confinement !! )

Seiberg-Witten ’94

(13)

What do general ! =2 SQCD (softly broken by µΦ ² ) tell us ?

• Abelian dual superconductivity ✔

Seiberg-Witten, ... ... ...

SU(2) with N

F

= 0, 1,2,3

monopole condensation ⇒ confinement & dyn symm. breaking

• Non-Abelian monopole condensation ✔

SU(N) ! =2 SYM : SU(N) ⇒ U(1)

^N-1

SU(N) ⇒ SU(r) x U(1) x U(1) x ....

• Non-Abelian monopoles interacting very strongly !

Argyres,Plesser,Seiberg,’96 Hanany-Oz, ’96

Carlino-Konishi-Murayama ‘00

Beautiful, but don’t look like QCD

r vacua are local, IR free theories

SCFT of highest criticality, EHIY points

Beautiful, interesting and difficult

Argyres,Plesser,Seiberg,Witten, Eguchi-Hori-Ito-Yang, ’96

SU(N), N

F

quarks

r ≤ N

F

/2

Shifman-Yung ’10-’13

Argyres-Seiberg, Gaiotto-Seiberg-Tahcikawa ’07, ’11 Giacomelli-Konishi ’12, ’13

(14)

QMS OF N=2 SU(N) SQCD, Nf quarks

Di Pietr

o, Giacomel li, ’11

w/ bare quark masses mi =m

Argyres,Plesser,Seiberg,’96 Carlino-Konishi-Murayama ‘00

(15)

QMS of N=2, USp(2N) theory with Nf massless quarks

SCFT of highest!

criticality (EHIY point):!

non-Lagrangian!

theory

Our main interest Note

Eguchi-Hori-Ito-Yang

( or SO(N) )

(16)

Effective degrees of freedom in the quantum r vacua of softly broken

N=2 SQCD

• they carry flavor q.n.

• 〈 M

ⁱ^α

〉 = v δ

ⁱα

➯ U(N

f

) ➡ U(r) x U(N

f

-r)

(r ≤ N

f

/ 2 )

Seiberg-Witten ’94

Argyres,Plesser,Seiberg,’96 Hanany-Oz, ’96

Carlino-Konishi-Murayama ‘00

μΦ²perturbation

Shifman-Yung ’10-’13

is added the massless Abelian (M

_k

) and non-Abelian monopoles ( M) all condense, bringing the system to a confinement phase. The form of the eﬀective action describing these light degrees of freedom is dictated by the N = 2 supersymmetry and the gauge and flavor symmetries. The eﬀective superpotential has the form [24, 25, 16]

W

r−vacua

= √

2 Tr( Mφ ˜ M) + √

2 a

D0

Tr( M ˜ M) + √ 2

N

�

−r−1 k=1

a

Dk

M

k

M ˜

k

+

+ µ

� Λ

N

�

−r−1 k=0

c

k

a

Dk

+ 1

2 Trφ

²

�

, (3.1)

where φ and a

_{D 0}

are the adjoint scalar fields in the N = 2 SU(r) × U(1) vector multiplet, a

D k

, k = 1, 2, . . . , N − r − 1 are the adjoint scalars of the Abelian U(1)

^N^−r−1

gauge multiplets.

M

_k

’s are the Abelian monopoles, each carrying one of the magnetic U (1) charges, whereas M (with r color components and in the fundamental representation of the flavor SU (N

_f

) group) are the non-Abelian monopoles. The terms linear in µ is generated by the microscopic N = 1 perturbation µ TrΦ

²

, written in terms of the infrared degrees of freedom a

_Dk

and φ, and c

_k

are some dimensionless constants of order of unity. These quantum r-vacua are known to exist only for r ≤ �

_N

f

2

� .

When small, generic bare quark mass terms

W

_masses

= m

_i

Q

_i

Q ˜

_i

(3.2)

are added in the microscopic theory, the infrared theory gets modified further by the addition

∆W

masses

= m

i

M

ⁱ

M ˜

ⁱ

+

N

�

−r−1 k=1

S

_k^j

m

j

M

k

M ˜

k

, (i, j = 1, 2, . . . , N

f

), (3.3) where S

_k^j

are the j-th quark number carried by the k-th monopole. Supersymmetric vacua are found by minimizing the potential following from Eq. (3.1) with Eq. (3.3), and by vanishing of the D-term potential.

SU (r) U (1)

0

U (1)

1

. . . U (1)

N−r−1

U (1)

B

n

_f

× M r 1 0 . . . 0 0

M

₁

1 0 1 . . . 0 0

... ... ... ... . .. ... ...

M

_N_−r−1

1 0 0 . . . 1 0

Table 1: The massless non-Abelian and Abelian monopoles and their charges at the r vacua at the root of a “non-baryonic” r-th Higgs branch.

5 Understand

this

Giacomelli-Konishi ’12

(17)

Non-Abelian monopoles

H: non-Abelian

2 m· e ∈ Z

“ Monopoles are multiplets of H ^(GNO) ”

cfr.

∼

<Φ> = v = h · T

(Dirac)

The normalization of the generators can be chosen [4] so that the metric of the root vector space is¹⁰

g_ij = !

roots

α_iα_j = δ_ij. (A.4)

The Higgs field vacuum expectation value (VEV) is taken to be of the form

φ₀ = h · T, (A.5)

where h = (h₁, . . . , h_rank_(G)) is a constant vector representing the VEV. The root vectors orthogonal to h belong to the unbroken subgroup H.

The monopole solutions are constructed from various SU (2) subgroups of G that do not commute with H,

S₁ = 1

√2α²(E_α+ E_−α); S₂ = − i

√2α²(E_α− E−α); S₃ = α^∗ · T, (A.6) where α is a root vector associated with a pair of broken generators E_±α. α^∗ is a dual root vector defined by

α^∗ ≡ α

α · α. (A.7)

The symmetry breaking (A.1) induces the Higgs mechanism in such an SU (2) subgroup, SU (2) → U(1). By embedding the known ’t Hooft-Polyakov monopole [34, 27] lying in this subgroup and adding a constant term to φ so that it behaves correctly asymptotically, one easily constructs a solution of the equation of motion [6, 19]:

A_i(r) = A^a_i(r, h · α) Sa; φ(r) = χ^a(r, h· α) Sa+ [ h− (h · α) α^∗]· T, (A.8) where

A^a_i(r) = %_aijr^j

r²A(r); χ^a(r) = r^a

r χ(r), χ(∞) = h · α (A.9) is the standard ’t Hooft-Polyakov-BPS solution. Note that φ(r = (0, 0,∞)) = φ0.

10In the Cartan basis the Lie algebra of the group G takes the form

[Hi, Hk] = 0, (i, k = 1, 2, . . . , r); [Hi, Eα] = αiEα; [Eα, E_−α] = αⁱHi; (A.2) [Eα, Eβ] = NαβEα+β (α + β &= 0). (A.3) αi = (α1, α2, . . .) are the root vectors.

43

significantly relaxed in cases in which the unbroken group is smaller. In this way one finds that the only real restriction is that the number of flavors be at least equal to 2r if the monopole transforms in the fundamental representation of SU (r). (See e.g., Eq.(3.2).)

3 Quantum Nonabelian Monopoles

The above example of the SU (N + 1) model nicely illustrates the fact that a semiclassical treatment alone is not enough to ensure that the set of apparently degenerate monopoles associated with the symmetry breaking G

^!φ"#=0

−→ H are truly nonabelian. The reason is that the “unbroken” gauge group H may well dynamically break down to an abelian subgroup. If this occurs, one has only an approximately degenerate set of monopoles whose masses differ by e.g., O(

_!φ"^Λ²

). For this reason, the very concept of nonabelian monopoles is never really semi-classical, in sharp contrast to the case of abelian monopoles. Only if the “unbroken” gauge group H is not further broken dynamically do the unconfined (topologically stable) nonabelian monopoles and dual gauge bosons appear in the quantum theory.

Another subtlety is that it is in general not justified to study the system G

^!φ"#=0

−→ H with a nonabelian subgroup H as a limiting situation of a maximal breaking, -

G

^!φ"#=0

−→ U(1)

^R

, where R is the rank of the group G, by letting some of the eigen-

values of #φ$ to coincide, as is sometimes done in the literature. To do so would introduce fictitious degrees of freedom corresponding to massless, infinitely extended

“solitons”. In this limit all fields tend to constant values and so in fact these are not solitons at all. Indeed, in the case G = SU (N ), such “massless monopoles” do not represent any topological invariant as the fundamental group of any restored SU (N ) is trivial.

²

It is hardly possible to overemphasize the importance of the fact [4, 6, 19] that nonabelian monopoles, if they exist quantum mechanically, transform as irreducible multiplets of the dual group ˜ H, not under H itself. Monopoles transforming under

2

This is analogous to what would happen to the ’t Hooft - Polyakov monopole of the spontaneously broken SU (2) −→U(1) theory, if one were to apply the semi-classical formulae na¨ıvely in the limit

^v

v → 0. We believe that this fact, together with the fact that the magnetic monopoles are multiplets of the dual of H (see the following paragraph), are responsible for some difficulties found in such an approach [10].

8 The normalization of the generators can be chosen [4] so that the metric of the root vector space is

¹⁰

g

_ij

= !

roots

α

_i

α

_j

= δ

_ij

. (A.4)

The Higgs field vacuum expectation value (VEV) is taken to be of the form

φ

₀

= h · T, (A.5)

where h = (h

₁

, . . . , h

_rank_(G)

) is a constant vector representing the VEV. The root vectors orthogonal to h belong to the unbroken subgroup H.

The monopole solutions are constructed from various SU (2) subgroups of G that do not commute with H,

S

₁

= 1

√ 2α

²

(E

_α

+ E

_−α

); S

₂

= − i

√ 2α

²

(E

_α

− E

−α

); S

₃

= α

^∗

· T, (A.6) where α is a root vector associated with a pair of broken generators E

_±α

. α

^∗

is a dual root vector defined by

α

^∗

≡ α

α · α . (A.7)

The symmetry breaking (A.1) induces the Higgs mechanism in such an SU (2) subgroup, SU (2) → U(1). By embedding the known ’t Hooft-Polyakov monopole [34, 27] lying in this subgroup and adding a constant term to φ so that it behaves correctly asymptotically, one easily constructs a solution of the equation of motion [6, 19]:

A

_i

(r) = A

^a_i

(r, h · α) S

^a

; φ(r) = χ

^a

(r, h · α) S

^a

+ [ h − (h · α) α

^∗

] · T, (A.8) where

A

^a_i

(r) = %

_aij

r

^j

r

²

A(r); χ

^a

(r) = r

^a

r χ(r), χ( ∞) = h · α (A.9) is the standard ’t Hooft-Polyakov-BPS solution. Note that φ(r = (0, 0, ∞)) = φ

⁰

.

[H_i, H_k] = 0, (i, k = 1, 2, . . . , r); [H_i, E_α] = α_iE_α; [E_α, E_−α] = αⁱH_i; (A.2) [E_α, E_β] = N_αβE_α+β (α + β &= 0). (A.3) αi = (α1, α2, . . .) are the root vectors.

43 The normalization of the generators can be chosen [?] so that the metric of the root vector space is

¹⁰

g

_ij

= !

roots

α

_i

α

_j

= δ

_ij

. (A.4)

The Higgs field vacuum expectation value (VEV) is taken to be of the form

φ

₀

= h · T, (A.5)

where h = (h

₁

, . . . , h _rank

_(G)

) is a constant vector representing the VEV. The root vectors orthogonal to h belong to the unbroken subgroup H.

The monopole solutions are constructed from various SU (2) subgroups of G that do not commute with H,

S

₁

= 1

√ 2α

²

(E

_α

+ E

_−α

); S

₂

= − i

√ 2α

²

(E

_α

− E

−α

); S

₃

= α

^∗

· T, (A.6) where α is a root vector associated with a pair of broken generators E

_±α

. α

^∗

is a dual root vector defined by

α

^∗

≡ α

α · α . (A.7)

The symmetry breaking (??) induces the Higgs mechanism in such an SU (2) subgroup, SU (2) → U(1). By embedding the known ’t Hooft-Polyakov monopole [?, ?]

lying in this subgroup and adding a constant term to φ so that it behaves correctly asymptotically, one easily constructs a solution of the equation of motion [?, ?]:

A

_i

(r) = A

^a_i

(r, h · α) S

^a

; φ(r) = χ

^a

(r, h · α) S

^a

+ [ h − (h · α) α

^∗

] · T, (A.8) where

A

^a_i

(r) = %

_aij

r

^j

r

²

A(r); χ

^a

(r) = r

^a

r χ(r), χ( ∞) = h · α (A.9) is the standard ’t Hooft-Polyakov-BPS solution. Note that φ(r = (0, 0, ∞)) = φ

⁰

.

The mass of a BPS monopole is then given by M =

"

dS · Tr φ B, B = r

_i

(S · r)

r

⁴

. (A.10)

[Hi, Hk] = 0, (i, k = 1, 2, . . . , r); [Hi, Eα] = αi Eα; [Eα, E_−α] = αⁱ Hi; (A.2) [Eα, Eβ] = Nαβ Eα+β (α + β &= 0). (A.3) α_i = (α₁, α₂, . . .) are the root vectors.

43 H generated by

∼ H H

U(N) U(N)

SU(N) SU(N)/Z

SO(2N) SO(2N)

SO(2N+1) USp(2N)

∼

N

For the cases SO(N + 2) → SO(N) × U(1) and USp(2N + 2) → USp(2N) × U (1), where TrH

_i

H

_j

= C δ

_ij

, one finds

M = 4π C h · α

^∗

g = 4 π v

g , (A.14)

while for SO(2N ) → SU(N) × U(1), SO(2N + 1) → SU(N) × U(1), and U Sp(2N ) → SU(N) × U(1), the mass is

M = 8π C h · α

^∗

g = 8 π v

g . (A.15)

In order to get the U (1) magnetic charge

¹¹

(the last column of Table 3), we first divide by an appropriate normalization factor in the mass formula Eq.(A.10)

F

_m

=

�

dS · Tr φ B N

_φ

=

�

dS · B

⁽⁰⁾

, B = r

_i

(S · r)

r

⁴

, (A.16)

as was done in Eq.(2.14). The result, which is equal to 4πg

_m

by definition, gives the magnetic charge. The latter must then be expressed as a function of the minimum U (1) electric charge present in the given theory, which can be easily found from the normalized (such that Tr T

^(a)

T

^(a)

=

¹₂

) form of the relevant U (1) generator.

For example, in the case of the symmetry breaking, SO(2N ) → U(N), the adjoint VEV is of the form, φ = √

4N v T

⁽⁰⁾

, where T

⁽⁰⁾

is a 2N × 2N block- diagonal matrix with N nonzero submatrices

√ⁱ

4N

� 0 1

−1 0

�

. Dividing the mass (A.15) by √

N v and identifying the flux with 4πg

_m

one gets g

_m

=

√²

N g

. Finally, in terms of the minimum electric charge of the theory e

₀

=

√^g

4N

( which follows from the normalized form of T

⁽⁰⁾

above) one finds

g

_m

= 2

√ N g = 2

N · 1

2 e

₀

. (A.17)

The calculation is similar in other cases.

The asymptotic gauge field can be written as F

_ij

= �

_ijk

r

_k

r

³

(β · T), 2 β · α ∈ Z (A.18)

11

In this calculation it is necessary to use the generators normalized as Tr T

^(a)

T

^(b)

=

¹₂

δ

ab

, such that B = B

⁽⁰⁾

T

⁽⁰⁾

+ . . . .

44

Goddard-Nuyts-Olive, E.Weinberg, Lee,Yi, Bais, Schroer, .... ‘77-80

∼

③

root vecto rs

(18)

Difficulties

➀ Topological obstructions

e.g., SU(3) ➝ SU(2)×U(1)

∄ monopoles ∼ (2, 1 ) ∗

“No colored dyons exist”

② Non-normalizable gauge zero modes:

Monopoles not multiplets of H

The real issue:

how do they transform under H ? ∼

cfr.

Jackiw-Rebbi Flavor Q.N. of monopoles

via

fermion zeromodes

N.B. : H and H relatively nonlocal

∼

Φ = diag(v,v,-2v)

↵

Weinberg, ’82,’96 Coleman, Nelson, ‘84 Dorey... ’96

Coleman, ... ’81

How do they do that ???

Abouelsaad et.al. ‘83

But N=2 theories do generate nonAbelian monopoles !!!

(19)

Study :

• Flavor symmetry:

H

C+F

necessary not to break the degeneracy;

• High-energy system (v

2

≈ 0) has regular monopoles if π

2

(G/H) ≠ 1

• Low-energy system (v

1

≈ ∞) has vortices if π

1

(H) ≠ 1

• v

2

~ an infrared cut-off

[14]. Actually, the latter can be interpreted as the GNOW monopoles becoming light due to the dynamics, at least in SU (N ) theories [15]. For SO(N ) or in U Sp(2N ) theories the relation between Seiberg’s duals and GNOW monopoles are less clear [15]. For instructive discussions on the relation between Seiberg’s duals and semiclassical monopoles in a class of N = 1, SO(N) models with matter fields in vector and spinor representations, see Strassler [16].

Dynamics of the system is thus a crucial ingredient: if the dual group were in Higgs phase, the multiplet structure among the monopoles would get lost, generally. Therefore one must study the dual ( ˜ H) system in confinement phase.

²

But then, according to the standard electromagnetic duality argument, one must analyse the electric system in Higgs phase. The monopoles will appear confined by the confining strings which are nothing but the vortices in the H system in Higgs phase.

We are thus led to study the system with a hierarchical symmetry breaking,

G

^�φ

−→ H

¹^��=0 ^�φ

−→ ∅,

²^��=0

(1.7)

where

|�φ

¹

�| � |�φ

²

�|, (1.8)

instead of the original system (1.1). The smaller VEV breaks H completely. Also, in order for the degeneracy among the monopoles not to be broken by the breaking at the scale |�φ

²

�|, we assume that some global color-flavor diagnonal group

H

_C+F

⊂ H

^color

⊗ G

^F

(1.9)

remains unbroken.

It is hardly possible to emphasize the importance of the role of the massless flavors too much.

This manifests in several diﬀerent aspects.

(i) In order that H must be non-asymptotically free, there must be suﬃcient number of massless flavors: otherwise, H interactions would become strong at low energies and H group can break itself dynamically;

(ii) The physics of the r vacua [9, 11] indeed shows that the non-abelian dual group SU (r) appear only for r ≤

^N₂^f

. This limit can be understood from the renormalization group: in order for a nontrivial r vacuum to exist, there must be at least 2 r massless flavors in the fundamental theory;

(iii) Non-abelian vortices [17, 18], which as we shall see are closely related to the concept of nonabelian monopoles, require a flavor group. The non-abelian flux moduli arise as a result of an exact, unbroken color-flavor diagonal symmetry of the system, broken by individual soliton vortex.

The idea that the dual group transformations among the monopoles at the end of the vortices follow from those among the vortices (monopole-vortex flux matching, etc.), has been discussed

2

The non-abelian monopoles in the Coulomb phase suﬀer from the diﬃculties already discussed.

3 G ^�φ�=v −→ H

¹

^�q�=v −→ 1,

²

v ₁ � v ²

No, if π

1

(G)=1

No, as π

2

(G)=1

(20)

Figure 1: A pictorial representation of the exact homotopy sequence, (3.1), with the leftmost figure corresponding to π2(G/H).

taken into account, having mass large but not infinite (Fig. 2). The low-energy vortices become unstable also through heavy monopole pair productions which break the vortices in the middle (albeit with small, tunneling rates [40]), which is really the same thing. Note that, even if the eﬀect of such string breaking is neglected, a monopole-vortex-antimonopole configuration is not topologically stable anyway: its energy would become smaller if the string becomes shorter (so such a composite, generally, will get shorter and shorter and eventually disappear).

However, this does not mean that such a monopole-vortex-antimonopole configuration cannot be dynamically stabilized, or that they are not relevant as physical configurations. A rotation can stabilize easily such a monopole-vortex-antimonopole configuration dynamically. After all, we believe that the real-world mesons are quark-string-antiquark bound states of this sort, the endpoints rotating almost with a speed of light! An excited meson can and indeed do decay through a quark pair production into states made of two lighter mesons. Only the lightest mesons are truly stable. The same occurs with our monopole-vortex-antimonopole configurations. The lightest such systems, after the rotation modes are appropriately quantized, are truly stable bound states of solitons, even though they cannot be simply described as static, semiclassical configurations.

Our model is thus a reasonably faithful (dual) model of the quark confinement in QCD.

It is crucial in our argument that the monopoles of high-energy theory and the vortices of low-energy theory are both BPS only approximately; in other words, they are almost BPS but not exactly.⁶ They are unstable in the full theory. But the fact that there exists a limit (of a large ratio of the mass scales, ^v_v¹

2 → ∞) in which these solitons become exactly BPS and stable, means that the magnetic flux through the surface of a small sphere surrounding the monopole and the vortex magnetic flux through a plane perpendicular to the vortex axis, must match exactly. These questions (the flux matching) have been discussed extensively already in [19].

Our argument, applied to the simplest case, G = SO(3), and H = U (1), is precisely the one adopted by ’t Hooft [1] in his pioneering paper, to argue that there must be a regular monopole of charge two (with respect to the Dirac’s minimum unit): as the vortex of winding number k = 2 must be trivial in the full theory (with π₁(SO(3)) = Z2), such a vortex must end at a regular monopole. What is new here, as compared to the case discussed by ’t Hooft [1] is that now the unbroken group H is non-abelian and that the low-energy vortices carry continuous,

6The importance of non-BPS soliton configurations have also been emphasized by Strassler [16].

10

“Homotopy exact sequence”

The gauge field equations take a slightly more complicated form than in the U (N ) model (2.1):

∂

_z

(Ω

⁻¹

∂

_z_¯

Ω) = − g

_N²

2 Tr ( t

^a

Ω

⁻¹

q q

^†

) t

^a

− e

²

4N Tr ( Ω

⁻¹

q q

^†

− 1), Ω = S S

^†

.(2.33) The last equation reduces to the master equation Eq. (2.10) in the U (N ) limit, g

_N

= e.

The advantage of the moduli matrix formalism is that all the moduli parameters appear in the holomorphic, moduli matrix H

₀

(z). Especially, the transformation property of the vortices under the color-flavor diagonal group can be studied by studying the behavior of the moduli matrix.

3 Topological stability, vortex-monopole complex and confinement

The fact that there must be a continuous set of monopoles, which transform under the color-flavor SU (N ) group follows from the following exact homotopy sequence

· · · → π

²

(G) → π

²

(G/H) → π

¹

(H) → π

¹

(G) → · · · (3.1) where π

₂

(G) = π

₁

(G) = ∅, in the system under consideration, G = SU(N + 1), H =

SU(N)×U(1)

ZN

∼ U(N). (Fig. 1). The nontrivial configuration of the scalar field can be interpreted as representing π

₂

(G/H) while the gauge field consfiguration can be classified according to π

₁

(H) [42]. It follows that

π

₂

� SU (N + 1) U (N )

�

= π

₂

(CP

^N

) ∼ π

¹

(U (N )) = Z : (3.2) each nontrivial element of π

₁

(U (N )) is associated with a nontrivial element of π

₂

(

^SU(N+1)_U(N)

).

Recalling that the latter represents the topological classification of gauge and scalar fields, this result is consistent as the theory does not admit Dirac monopoles: all monopoles are regular ’t Hooft-Polyakov monopoles.

However, there is something of a puzzle: when the small VEV’s are taken into account, which break the “unbroken” gauge group completely, these monopoles must disappear somehow. A related puzzle is this: the low-energy theory develops vortices since H is completely broken. The vortex flux is quantized by (in our case, with H =

^SU(N)×U(1)_Z

N

)

π

₁

(H) = Z. (3.3)

Again, when the massive monopoles associated with the breaking G → H are taken into account, ı.e., in the full theory, the vortices visible and stable in the low-energy approximation must disappear, as π

₁

(G) = ∅.

Actually, these two apparent puzzles are the two faces of a medal. The solution is that the massive monopoles are confined by the vortices and disappear from the spectrum; on the other hand, the vortices of the low-energy theory end at the heavy monopoles once the latter are

9 e.g. G=SU(N), USp(2N): π

1

= 1 No Dirac monopoles

^(Wu-Yang)

G=SO(N) π

1

=Z

2

, Z

2

monopoles; G=SU(N)/Z

N

: Z

N

monopoles;

- π

2

(G) = 1 No regular monopoles (i. are confined by vortices)

- If π

1

(G) = 1 Vortices end at regular monopoles _{‘t Hoo} ^ft

SO(3) /U(1)

- If π

1

(G) = Z

2

k=2 vortices end at regular monopoles!

Apply to:

Abstract

It is argued that the dual transformation of non-Abelian monopoles occurring in a system with gauge symmetry breaking G −→ H is to be defined by setting the low-energy H system in Higgs phase, so that the dual system is in confinement phase. The transformation law of the monopoles follows from that of monopole-vortex mixed configurations in the system (with a large hierarchy of energy scales, v

₁

� v

²

)

G −→ H

^v¹

−→ 1,

^v²

under an unbroken, exact color-flavor diagonal symmetry H

_C+F

∼ ˜ H.

The transformation property among the regular monopoles characterized by π

₂

(G/H), follows from that among the non-Abelian vortices with flux quantized according to π

₁

(H), via the isomorphism π

₁

(G) ∼ π

₁

(H)/π

₂

(G/H). Our idea is tested against the concrete models – softly-broken N = 2 supersymmetric SU(N), SO(N) and USp(2N) theories, with appropriate number of flavors. The results obtained in the semiclassical regime (at v

₁

� v

²

� Λ) of these models are consistent with those inferred from the fully quantum-mechanical low-energy eﬀective action of the systems (at v

₁

, v

₂

∼ Λ).

(21)

‘t Ho oft, N

ucl. Phys. B79 (’74)

276

(22)

Monopole Moduli Vortex Moduli

~ CP

^N-1

SU(N)

1 (H)

2 (G/H)

Figure 2: The non-trivial vortex moduli implies a corresponding moduli of monopoles.

adopted by ’t Hooft in his pioneering paper [1] to argue that there must be a regular monopole of charge two (with respect to the Dirac’s minimum unit): as the vortex of winding number k = 2 must be trivial in the full theory (π ₁ (SO(3)) = Z ² ), such a vortex must end at a regular monopole. What is new here, as compared to the case discussed by ’t Hooft [1] is that now the unbroken group H is non-Abelian and that the low-energy vortices carry continuous, non-Abelian flux moduli. The monopoles appearing as the endpoints of such vortices must carry the same continuous moduli (Fig. 2).

The fact that the vortices of the low-energy theory are BPS saturated (which allows us to analyze their moduli and transformation properties elegantly, as discussed in the next section), while in the full theory there are corrections which make them non BPS (and unstable), could cause some concern. Actually, the rigor of our argument is not aﬀected by those terms which can be treated as perturbation. The attributes characterized by integers such as the transformation property of certain configurations as a multiplet of a non-Abelian group which is an exact symmetry group of the full theory, cannot receive renormalization. This is similar to the current algebra relations of Gell-Mann which are not renormalized. Conserved vector current (CVC) of Feynman and Gell-Mann [39] also hinges upon an analogous situation. ⁷ The results obtained in the BPS limit (in the limit v ₂ /v ₁ → 0) are thus valid at any finite values of v ² /v ₁ .

4 Dual gauge transformation among the monopoles

7 The absence of “colored dyons” [5] mentioned earlier can also be interpreted in this manner.

11

(23)

• Monopoles and vortices are related

④ NonAbelian Vortices

topology, stability and symmetry via

• Transformation properties under H

C+F

from those of nonabelian vortices

(24)

ANO vortex (Abelian Higgs - U(1) - model)

V= λ ( |ϕ| ² - v ² ) ²

Fig. 2:

Given the points f, p and the spaceM, the vortex solution is still not unique. Any exact symmetry of the system (internal symmetry ˜G^{q} as well as spacetime symmetries such as Poincar´e invariance) broken by an individual vortex solution gives rise to vortex zero modes (moduli), V.

The vortex-center position moduli V ∼ C, for instance, arise as a result of the breaking of the translation invariance in R². The breaking of the internal symmetry ˜G^{q} (Eq. (3.5)) by the individual vortex solution gives rise to orientational zeromodes in the U (N ) models extensively studied in last several years. See [32, 33, 34] for more recent results on this issue.

Our main interest here, however, is the vortex moduli which arises from the non-trivial vacuum moduliM itself. Due to the BPS nature of our vortices, the gauge field equation (see Eq. (A.3))⁶

F₁₂Î = g_I²!q^†TÎq− ξÎ"

, (3.9)

reduces, in the strong-coupling limit (or in any case, sufficiently far from the vortex center), to the vacuum equation defining M . This means that a vortex configuration can be approximately seen as a non-linear σ-model (NLσM) lump with target space M (for non-trivial element of π2(M)). Various distinct maps

S² #→ M , (3.10)

of the same homotopy class correspond to physically inequivalent solutions; each of these corre- sponds to a vortex with the equal tension

Tmin = −ξ^I

#

d²x F₁₂^I > 0 , (3.11)

6The index I denotes generally all the generators of the gauge group considered. A non-vanishing (FI) param- eter ξ is assumed only for U (1) factor(s).