Symmetries and dynamics in Particle Physics: the legacy of Yoichiro - - PowerPoint PPT Presentation

Symmetries and dynamics in Particle Physics: the legacy of Yoichiro Nambu

O s a k a C i t y U n i v . 1 3 / 1 2 / 2 0 1 8

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

N a m b u S y m p o s i u m

wA\ Achievements by Y. Nambu

🔶

Spontaneous symmetry breakdown

Chiral symmetry breaking and physics of massless pions /current algebra

🔶

Color degrees of freedom

Quark model to Quantum Chromodynamics

🔶

Strings from dual resonance model

Nambu-Goto action: birth of string theory

🔶

Color (quark) confinement

Work and recollections by colleagues

🔶

“Yoichiro Nambu: remembering an unusual physicist, a mentor, and a friend" Giovanni Jona-Lasinio, Prog. Theor. Exp. Phys. 2016 , 07B102

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“BCS, Nambu-Jona-Lasinio, and Han-Nambu - A sketch of Nambu’s works in 1960-1965 ” Kazuo Fujikawa, arXiv:1602.08193

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“Nambu at Work" Peter G. Freund, arXiv:1511.06955

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“Professor Nambu, String Theory and Moonshine Phenomenon" Tohru Eguchi, arXiv:1608.06036

🔶

“Birth of String Theory" Hiroshi Itoyama arXiv:1604.03701

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“Yoichiro Nambu" Holger B Nielsen, Int. J. Mod. Pays. A (2016)

wA\ “Chiral symmetries and current algebra”

Nambu’s lecture at the “Highlights in Particle Physics”

🔶

Spontaneous symmetry breaking: non gauge theories

Nambu-Goldstone modes (pions), PCAC, soft-pion theorems

🔶

Spontaneous symmetry breaking: gauge theories

Englert-Brout-Higgs mechanism; Weinberg-Salam electroweak theory

🔶

Bilinear quark algebra

🔶

A remark:

Ettore Majorana Erice Summer School, 1972

“ … I am convinced that the strong interactions are also described by some sort of color gauge theory. ”

S u m m e r 1 9 7 2 Need of anomaly cancellation

🔶

Nambu was fully aware that the analogy with superconductivity (with pion physics) was not perfect;

🔶

Nambu played with the idea of giving mass (by the Higgs mechanism) to massive vectors, as

• cfr. Weinberg-Salam

🔶

Nambu was convinced that the strong interactions were described by a gauge theory of color

• cfr. QCD of Fritzch-Gell-Mann-Leutwyler

In retrospect…

3 more Nobel prizes within reach …

J.C. Taylor P . Freund

ρ± ∗ , K∗

Symmetries and dynamics : Nambu’s legacy

① ②

Color and QCD: confinement and XSB Magnetic monopole condensation:

③ ④

Exact solutions in N=2 supersymmetric gauge theories

⑤

Generalized symmetries, mixed ’t Hooft anomalies QCD; Puzzles and solutions Chiral gauge theories NonAbelian vortices and monopoles topological solitons vs gauge dynamics

①

Spontaneous chiral symmetry breaking

Let’s start where Nambu has left out…

XSB:

“The basic principle underlying the model is the idea that field theory may admit, as a result of dynamical instability, extraordinary (nontrivial) solutions that have less symmetries than are built into the Lagrangian. “

• Y. Nambu and G. Jona-Lasinio, Phys. Rev. 1961

Massless Nambu-Goldstone bosons (pions) of broken

Color confinement ? (vs XSB)

SUA(Nf)

Want a deeper understanding of

(&)

(&)

② Confinement = a dual superconductor ?

Nambu, Mandelstam, ’t Hooft, ~‘80

SU(3) → U(1)2 → 1

’t Hooft-Polyakov monopoles (M) ! Dual Abelian gauge theory hMi 6= 0

q ¯ q

Monopole condensation (cfr. Cooper pair condensation) Dual Meissner effect (color chromoelectric fields expelled) Chromoelectric vortex Linearly rising q - q* potential !

Puzzles

• No evidence from lattice GT
• Doubling of the meson spectrum

(*)

• If

XSB

Confinement vs XSB

Accidental SU(NF2 ) : too many NG bosons

• No dynamical mechanism for

hMi 6= 0

Monopoles weak coupled

• Non-Abelian monopole condensation

SU(3) → SU(2) × U(1) → 1

Non-Abelian monopoles

Π

1

( S U ( 2 ) × U ( 1 ) ) = Z

Solution:

(**) (*), (**), (***) OK? (***)

Price:

‘80~’18

k s

• ¯

ψ

L

D S U

A

( N

f

) π

2

( S U ( 3 ) / U ( 1 )

2

) = π

1

( U ( 1 )

2

) = ×

strongly coupled ? no-go theorem ?

NonAbelian magnetic monopoles, gauge dynamics and confinement: theoretical developments ’94 - ‘18

• Exact solutions of N=2

supersymmetric gauge theories

Seiberg-Witten

• Exact quantum mech. monopoles
• N=2 SCFT’s and dualities

A r g y r e s , S e i b e r g , G a i

• t

t

• ,

T a c h i k a w a … …

• NonAbelian vortex
• RG and IRFP and confinement
• NonAbelian monopoles
• Vortices in

High density QCD / cold atoms

• CPN-1 in finite width strip

Pisa, Titech,

Minnesota, Cambridge, Keio

Seiberg-Witten solutions in N=2 supersymmetric theories

🔶

SW curves: all pert/nonpertive effects encoded

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Exact low-energy dual Abelian or nonAbelian Leff

🔶

Analytic demonstration of confinement

🔶

Confinement vs XSB

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Argyres-Douglas vacua

A b e l i a n a n d n

• n

A b e l i a n d u a l M e i s s n e r p i c t u r e

• f

c

• n

fi n e m e n t O K ; B u t N O T s i m i l a r t

• Q

C D

🔶

Argyres-Seiberg S duality, Gaiotto and SCFT

🔶

Confining vacuum near strongly coupled SCFT

M

• s

t i n t e r e s t i n g b u t d i f fi c u l t : s

• m

e a n a l

• g

y t

• Q

C D ? ’94 -

③

’09

’０６

Confinement and RG flow

red curves= deformations by some relevant operators

RG flows

N=2 SCFT

Real-world QCD

N=0 SCFT

aUV = 11NfNc 360 + 31 180(N 2

c − 1)

− aIR = N 2

f − 1

360 .

SU(N), NF =2N-1

aUV = 7N 2 − N − 5 24

aN=2SCF T = 7N(N − 1) 24

aIR = (2N − 1)2 − 1 48

cUV = 4N 2 − N − 2 12

cN=2SCF T = N(N − 1) 3

cIR = (2N − 1)2 − 1 24

• cUV = 1

20NfNc + N 2

c 1

10 ;

cIR = N 2

f 1

120 ;

Nf < 11 2 Nc

Bolognesi, Giacomelli, KK ‘16 in adjoint repr of GF

③

Summary of

♦

deep insights into quantum behaviors of monopoles in the infrared

♦

and their role in confinement/XSB

♦

not quite a good model for QCD, except for the confinement vacua “near” strongly coupled CFT

♦

Abelian and nonAbelian monopoles = topological solitons ?

NonAbelian vortices and monopoles

🔶

NonAbelian vortex = ANO vortex embedded in CF locked vacuum

🔶

Non-Abelian monopoles are the endpoints of the nonAbelian vortices

non-Abelian monopoles

🔶

Internal orientational zeromodes ~ Fluctuations: CPN-1 sigma model in 2D vortex world sheet

🔶

Notorious problems avoided

Auzzi et. al, Hanany-Tong ’03- Shifman-Yung, Nitta-Ohashi-Sakai …

🔶

Quantum physics of CPN-1 sigma model on finite-width vortex world sheet ; Unbroken isometry SU(N)

Bolognesi, Gudnason, Ohashi, KK ’16-‘18

Standard CPN-1 model at L

• cfr. Gorsky, Pikalov,

Vainshtein’ nov 18

④

NonAbelian monopoles and duality

SU(N + 1)color ⊗ SU(N)flavor

v1

− → (SU(N) × U(1))color ⊗ SU(N)flavor

v2

− → SU(N)C+F Monopole Vortex

C i p r i a n i , D

• r

i g

• n

i , G u d n a s

• n

, K . K . M i c h e l i n i , 2 1 1 C h a t t e r j e e , K . K . 2 1 4

Vortices with fluctuating

• rientational CPN-1 modes

= Origin of the dual SU(N) gauge group

• The monopole ~ N of a new (dual) SU(N) --- isometry group of CPN-1

H a n n a h , T

• n

g , ’ 3 , A u z z i , B

• l
• g

n e s i , E v s l i n , K K , Y u n g ’ 3 S h i f m a n , Y u n g , ’ 3 A u z z i , B

• l
• g

n e s i , E v s l i n , K . K . , ’ 4

Summary of ③, ④

🔶

Quantum behavior of Abelian and nonAbelian monopoles in the IR fairly well (in N=2, N=1, susy gauge theories) understood;

🔶

The topological soliton monopoles and vortices (many applications): origin of the nonAbelian monopoles

🔶

Understanding of confinement /chiral symmetry breaking in the real-world (non supersymmetric) strong-interaction theory however remains a holy grail

🔶

Some new ideas ?

⑤ Generalized symmetries, line and surface operators;

Mixed ’t Hooft anomalies; Symmetry protected topological order (SPT)

🔶

From 0-form symmetries (acting on local operators) to k-form symmetries (acting on line, surface, etc operators) e.g. the center symmetry in SU(N) YM (k=1)

S e i b e r g , K a p u s t i n , A h a r

• n

y , G a i

• t

t

• ,

T a c h i k a w a , W i l l e t , K

• m

a r g

• d

s k i , . . P

• p

p i t z , K i k u c h i , T a n i z a k i , S a k a i , S h i m i z u , Y

• n

e k u r a ,

’05-‘18

ei

H

γ A → ΩN ei

H

γ A ,

ΩN = e2πi/N ∈

N

W i l s

• n

l

• p

P

• l

y a k

• v

l

• p

Criteria for different phases

Phases / symmetry realization of the system not described by the vacuum expectation values

• f a local operator

hφ(x)i , h ¯ ψ(x)ψ(x)i

Nambu-Jona Lasinio L a n d a u

• G

i n z b u r g

: a germ for change of PARADIGM (growing out of Nambu’s teaching…)

H a l d a n e ’ 8 3 W e n , ‘ 8 9

♦

♦

net effect:

🔶

“Gauging” the k-form symmetries

Gauging the (k=1) discrete ∈

N center symmetry

♦

identify configurations related by ∈

N

♦

must eliminate the redundancies

♦

done by coupling the ordinary YM gauge theory to a BF theory

♦

modify the sum over gauge field configurations (path integral)

gets modified to

1 8π2 Z F ^ F = 1 8π2 Z d4x Fµν ˜ F µν 2 1 8π2 Z ( ˜ F B(2)) ^ ( ˜ F B(2)) 2 N , NB(2) = dB(1)

🔶

Consequences

♦

CP (T) inv of SU(N) YM broken at

♦

Periodicity in θ

♦

Can be understood as a mixed discrete CP -

θ ⇠ θ + 2π ! θ ⇠ θ + 2Nπ θ = π

♦

In the infrared, if the system is confined, with mass gap

∈

N

’t Hooft anomaly

• cfr. a standard discrete

’t Hooft anomaly !

’t Hooft anomaly matching requires a double vacuum degeneracy !!

♦

In IR no gauge degrees of freedom: the above result must hold whether or not the “gauging” of

∈

N

Obstruction to gauging a symmetry

♦

A powerful theoretical reasoning!! is actually done

S e i b e r g , K a p u s t i n , G a i

• t

t

• ,

W i l l e t , K

• m

a r g

• d

s k i , . . ‘ 1 7

θ ⇠ θ + 2π ! ⇠ ∆S

Y M

= θ 8π

2

Z F ^ F = π

(Galileo) They must fall at the same velocity

🔶

Generalizations

♦

Adjoint QCD: SU(N) with NF Weyl fermions Nonanomalous discrete

becomes anomalous after 1-form gauging

♦

QCD: SU(N) with NF quarks, NF =N, or gcd(NF , N)= k

⭐ standard center symmetry is absent

λ

i∆S = 2πi Nc

⭐ color-flavor locked

1-form symmetry is present

N.B. k-form symmetries Abelian except k=0

⭐ “gauging”

QCD: SU(N) with NF quarks, arbitrary NF

♦

2NF Nc ;

λ ! e2πi/2NF Ncλ

Anber, Poppits ‘18 Kikuchi, Tanizaki, Misui, Sakai, Shimizu, Yonekura, Kitano,Suyama, Yamada, ’18

Tanizaki ’18

Shimizu, Yonekura ‘18

k ⇢ Nc ⇥ Nf , Nf ⇢ UV (Nf)

k Axial 2NF

broken by mixed anomaly

Nc

Tdec  Tchiral

⭐ flavor Nf 1-form symmetry ⭐ “gauging” ⭐ baryon number

UB(1)

Axial 2NF UB(1) 1 Nf mixed anomaly

Skyrmion needed ! S t e r n p h a s e N

• Hirono, Tanizaki, nov. ’18

Dumitrescu, Cordoba, ‘18

Chatterjee, Nitta, Yasui ’18

🔶

Chiral gauge theories

e.g., SU(N) theory with Weyl fermions standard ’t Hooft anomaly matching constraints for

ψ{ij} , ηB

i ,

B = 1, 2, . . . , N + 4

+ (N + 4) ¯

♦ ⭐

do not tell between

• A. confining vac. with unbroken symm. , w/ massless baryons

B[AB] = ψijηA

i ηB j ,

A, B = 1, 2, . . . , N + 4

⭐

• B. dynamical Higgs

hψijηB

i i = C δjB ,

j, B = 1, 2, . . . N ,

with some massless baryons and 8 N + 1 Nambu-Goldstone bosons

Uψη(1), SU(N + 4),

ψ, η,

♦

A p p e l q u i s t , D u a n , S a n n i n

• ‘

B

• l
• g

n e s i , K K , S h i f m a n ‘ 1 8

SU(N + 4) × Uψη(1) → SU(N)cf × U(1)0

Gauging the color-flavor locked

1-form symmetry

N

Bolognesi, KK, Tanizaki ‘19

Z he Uψη(1), Zψ = ZN+2, t

both suffer mixed anomalies dynamical possibility A excluded; B OK

Y a m a g u c h i , n

• v

‘ 1 8

🔶

Many exciting applications of these ideas being explored

🔶

But understanding of confinement /chiral symmetry breaking in the real-world (non supersymmetric) strong-interaction theory still remains a holy grail

🔶

Should we despair ?

“A truly beautiful idea never really dies”

(Y. Nambu)

THE END

and Thank you organizers,

• Prof. Itoyama, in particular,

and all participants

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

’94

• Fields:

W = (Aµ, λ), Φ = (φ, ψ)

Vacuum moduli (degeneracy): hφi = ✓ a a ◆ , u = Trhφ2i

• Leff :

Leff = Im[ Z d4θ ¯ A∂F(A) ∂A + Z d2θ∂2F(A) ∂A2 W αWα]

F(A)= prepotential holomorphic

• Duality: L=Leff formally inv under SL(2,Z) (ad-bc=1) ⊃ EM duality

✓ AD A ◆ → ✓ a b c d ◆ ✓ AD 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= ± Λ2

AD = ∂ F / ∂ A

= AD

τ = F

00(A) = dAD

dA = θeff 2π + 4πi g2

eff

• Which description? Depends on u !

F(A) !

SUR(2)

FµνF µν + i ¯ λ γµDµλ + iFµν ˜ F µν + ...

y2 = (x − u)(x − Λ2)(x + Λ2)

• solves the theory (see Fig. )

dAD du = I

α

dx y , dA du = I

β

dx y ,

Im H

α dx y

H

β dx y

> 0

➞ F(A)

• Perturbative and nonperturbative quantum effects (instantons) fully encoded in ※
• Effective theory near u= Λ2

Seiberg-Witten curve

Leff(AD, F µν

D , ...) +

Z d4θ ¯ MeVDM + (M → ˜ M) + √ 2 Z d2θ MAD ˜ M

Magnetic monopole coupled minimally to the dual gauge field

※

Mnm,ne = |nmAD + neA|, AD = I

α

λ, A = I

β

λ,

Seiberg-Witten curves for general gauge groups

• SU(N) with NF quarks
• SO(N) with NF quarks in vector representation

y2 =

N

Y

i=1

(x − φi)2 − Λ2N−NF

NF

Y

a=1

(x + ma) y2 = x

[N/2]

Y

i=1

(x − φ2

i )2 − 4Λ2(N−NF −NF )x2+✏ NF

Y

a=1

(x + m2

a)

• etc.

B a c k t

• p

. 1 2

Leff(AD, F µν

D , ...) +

Z d4θ ¯ MeVDM + (M → ˜ M) + √ 2 Z d2θ MAD ˜ M

• Dual superconductor

N=2 susy SU(2) gauge theory with μ Φ2 perturbation

+ Z d2θ µ U(AD)

➞

AD = 0, hMi = r µ ∂U ∂AD = p µΛ

Seiberg-Witten ’94

Monopole condensation ➞ confinement !!

• Dual U(1) Higgs model
• Q: what happens to the “Chromoelectric” vortex ??

monopole condensation

• n=2 vortex? Exotic hadrons? n=1?

Yung-Vainshtein ’01

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

V= λ ( |ϕ|2 - v2 )2

• Fig. 2:

Dϕ ➞ 0; |ϕ|2 ➞ v2

• λ> g2 /2 type I

Abrikosov ’56 Nielsen-Olesen ‘73

• λ< g2 /2 type II
• λ= g2 /2 BPS

∏1(U(1))=Z “ANO” vortex

(Landau-Ginzburg model)

L = 1 4g2

N

(F a

µν)2 +

1 4e2( ˜ Fµν)2 + |Dµq|2 e2 2 | q† q c 1 |2 1 2 g2

N | q† taq |2,

(D1 + iD2) q = 0,

F (0)

12 + e2

2

• c 1N q q†⇥

= 0; F (a)

12 + g2 N

2 q†

i ta qi = 0.

Nonabelian vortex Bogomolnyi equations

(HT, ABEKY ‘03)

Benchmark H= SU(N)xU(1) model with NF = N

NonAbelian Vortices

Broken to U(N-1) by the soliton vortex (“Nambu-Goldstone modes”)

Vortex moduli = SU(N)/ U(N-1) = CPN-1 (= CP1 ~ S2 for U(2) )

• ✏

q = ⇤ ˜ q†⌅ = v2 = v2 ⌥ ↵ ↵ ↵ 1 1 ... 1

• ⌦

2

vortices

Auzzi, Bolognesi,Evslin,Konishi, Yung ’03 Hanany,Tong, Shifman-Yung

q = U ⇧ ⌥ eiφ φ(r) . . . χ(r) . . . . . . χ(r) . . . ... ⌃ ⌦ ⌦ ⌦ ⌦

• U †

U ⇧ ⌥ . . . w . . . ... . . . w ⌃ ⌦ ⌦ ⌦ U †

r=0

Exact SU(N)C+F group

Color-flavor locked vacuum

(orientational zero modes)

S(1+1)

σ

= β

• d2x 1

2 (∂ na)2 , Fluctuations: SO(3) sigma model

B a c k t

• p

. 1 4

classification (’t Hooft)

Phases of SU(N) YM

• (Z(M)

N

, Z(E)

N ) classification (’t Hooft):

If field with x = (a, b) condense, particles X = (A, B) with ⟨x, X⟩ ≡ a B − b A ̸= 0 (mod N) are confined. (e.g. ⟨φ(0,1)⟩ ̸= 0 → Higgs phase.) ⟨χ

0)⟩ ̸= 0 → Confinement: What is χ ?

(’t Hooft) (1981)

Monopole condensation, confinement and Susy breaking

W = (Aµ, λ), Φ = (φ, ψ)

N=2 susy SU(2) gauge theory with Physics depends on

1 2 3 4 5 6

• 1.0
• 0.5

0.5 1.0

Konishi ’96 Evans, Hsu, Schwetz ’96

B a c k t

• p

. 1 2

Similar behavior of the standard QCD vacua from Leff with nonzero mu , md

Di Vecchia, Veneziano ’80

What do general 𝒪=2 SQCD (softly broken by µΦ2 ) tell us ?

• Abelian dual superconductivity ✔

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

SU(2) with NF = 0, 1,2,3 monopole condensation ⇒ confinement & dyn symm. breaking

• Non-Abelian monopole condensation ✔

SU(N) 𝒪 =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 !

A r g y r e s , P l e s s e r , S e i b e r g , ’ 9 6 H a n a n y

• O

z , ’ 9 6 Carlino-Konishi-Murayama ‘00

Beautiful, but don’t look like QCD

Beautiful, but don’t look like QCD

r vacua are local, IR free theories SCFT of highest criticality, EHIY points

Beautiful and interesting

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

SU(N), NF quarks

r ≤ NF /2

Shifman-Yung ’10-’13

Argyres-Seiberg, Gaiotto-Seiberg-Tahcikawa ’07, ’11 Giacomelli, Bolognesi, -Konishi ’12, - 17

Dynamical Abelianization

CONFINEMENT 13

r=1 r = nf /2

• - -

Non Abelian monopoles Abelian monopoles (Non-baryonic) Higgs Branches Baryonic Higgs Branch Coulomb Branch Dual Quarks

QMS of N=2 SQCD (SU(n) with nf quarks)

r=0 <Q> 0

< > 0

N=1 Confining vacua (with 2 perturbation) N=1 vacua (with 2 perturbation) in free magnetic phas

SCFT

Φ Φ

Φ ≠

m = mcr next slide

Di Pietro, Giacomelli ’11

CONFINEMENT 14

Non Abelian monopoles Higgs Branches Special Higgs Branch Coulomb Branch Dual Quarks

QMS of N=2 USp(2n) Theory with nf Quarks

<Q> 0 < > 0 N=1 Confining vacua (with 2 perturbation) N=1 vacua (with 2 perturbation) in free magnetic phas SCFT

SCFT of highest criticality EHIY point non-Lagrangian

Carlino-Konishi-Murayama ‘00

Φ

Φ Φ

m ≠ 0

previous slide (m = 0) (Universality)

10

≠

≠

Quantum space of vacua in N=2 SQCD with critical mass

Effective low-energy degrees of freedom in the quantum r vacua of softly broken N=2 SQCD

• they carry flavor q.n.
• 〈M i α〉= v δi α ➯ U(Nf) ➡ U(r) x U(Nf -r)

(r ≤ Nf / 2 )

S e i b e r g

• W

i t t e n ’ 9 4 A r g y r e s , P l e s s e r , S e i b e r g , ’ 9 6 H a n a n y

• O

z , ’ 9 6 Carlino-Konishi-Murayama ‘00

μΦ2 perturbation

S h i f m a n

• Y

u n g ’ 1

• ’

1 3

SU(r) U(1)0 U(1)1 . . . U(1)N−r−1 U(1)B nf × M r 1 . . . M1 1 1 . . . . . . . . . . . . . . . ... . . . . . . MN−r−1 1 . . . 1

NonAbelian monopoles

Giacomelli-Konishi ’12

b a c k t

• p

. 1 1

Non-Abelian monopoles

H: non-Abelian

2 m· e ∈ Z

“Monopoles are multiplets of H (GNO)”

cfr.

∼

<Φ> = v = h · T

(Dirac)

S1 = 1 √ 2α2(Eα + E−α); S2 = − i √ 2α2(Eα − E−α); S3 = α∗ · T,

g G

φ̸=0

− → H

Ai(r) = Aa

i (r, h · α) Sa;

φ(r) = χa(r, h · α) Sa + [ h − (h · α) α∗] · T,

α∗ ≡ α α · α.

H generated by

∼

H H

U(N) U(N)

SU(N) SU(N)/Z SO(2N) SO(2N) SO(2N+1) USp(2N) ∼

N

Fij = ⌅ijk rk r3(⇥ · T), 2 ⇥ · ∈ Z

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

∼

root vectors

GNO

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:

No multiplets of H The real issue: how do they transform under H ? GNO

∼

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 Dorey... ’96

Coleman, ... ’81

How do they do that ???

A b

• u

e l s a a d e t . a l . ‘ 8 3

But NonAbelian monopoles are ubiquitous in N=2 theories!!!

Back to p.9

C

• l

e m a n , N e l s

• n

, ‘ 8 4

Back to p.14

Study:

• High-energy system (v2 ≈ 0) has regular monopoles if π2 (G/H) ≠ 1
• Low-energy system (v1 ≈ ∞) has vortices if π1 (H) ≠ 1

HC+F ⌅ Hcolor ⇥ GF

G

hφi=v1

• ! H

hqi=v2

• ! 1,

v1 v2

No, if π1 (G)=1 No, as π2 (G)=1

SU(N + 1)color ⌦ SU(N)flavor

v1

• ! (SU(N) ⇥ U(1))color ⌦ SU(N)flavor
• r

v2

• ! SU(N)C+F

e.g.,

Idea

🔶

The full theory cannot have vortices

🔶 🔶

Vortex ~ CPN-1 effective action The monopole from the HE breaking absorbs / generates the CPN-1 modes: nonAbelian monopole

b(z,t)

endow the monopole with fluctuating CPN-1 modes

b(t)

Exact sequence of homotopy groups

· · · → π2(G) → π2(G/H) → π1(H) → π1(G) → · · ·

e.g. G=SU(N), USp(2N): π1 = 1 ⇒ No Dirac monopoles (Wu-Yang)

G=SO(N) π1 =Z2 , Z2 monopoles; G=SU(N)/ZN : ZN monopoles;

• π2 (G) = 1 ⇒ No regular monopoles (i. are confined by vortices)
• If π1(G) = 1 ⇒

Vortices end at regular monopoles

‘t Hooft SO(3)/U(1)

• If π1(G) = Z2 ⇒ k=2 vortices end at regular monopoles!

Apply to:

G

v1

• ⌅ H

v2

• ⌅ 1,

‘t Hooft, Nucl. Phys. B79 (’74) 276

• Monopoles and vortices are related

Ricapitulating:

topology, stability and symmetry via

• Transformation properties of the monopoles under HC+F

from those of nonAbelian vortices

N.B.

SO(3)

v1

− → U(1)

v2

− → ∅

• describes both ’t Hooft-Polyakov-Georgi-Glashow

and SU(2) Seiberg-Witten

• mathematics similar, but physics different

Monopole condensation, confinement and Susy breaking

W = (Aµ, λ), Φ = (φ, ψ)

N=2 susy SU(2) gauge theory with Physics depends on

• 1.0
• 0.5

Konishi ’96 Evans, Hsu, Schwetz ’96

↔ ↔ −

2 4 6 8 10 0.2 0.4 0.6 0.8 1 2 4 6 8 10 0.2 0.4 0.6 0.8 1

qkA(x) =

• ei n ϕφ1(r)

ei k ϕφ2(r)

• ,

A3

i (x) = −εij

xj r2 ((n − k) − f3(r)) , A8

i (x) = −

√ 3 εij xj r2 ((n + k) − f8(r))

Low-energy effective action

SU(N), SO(2N) USp(2N) models

Gudnason,Jiang,Konishi ’10

X = 1 + B†B , Y = 1N−1 + BB†

Reducing matrix

plex unit N-com nc = X− 1

2

BX− 1

2

! = √

• ve can be put

= 2D CPN-1 action !

A u z z i , B

• l
• g

n e s i , E v s l i n , K

• n

i s h i , Y u n g ’ 3 H a n a n y , T

• n

g , S h i f m a n

• Y

u n g ‘ 3

B =      b1 b2 . . . bN−1     

CPN-1 inhomogeneous coord

• Global flavor SU(N)CF symmetry unbroken (no Nambu-Goldstone bosons in 4D)
• Soliton monopole-vortex complex breaks it to SU(N-1)xU(1)

➯ orientational zeromodes (can fluctuate)

b(z,t)

〈Φ〉 ∿ m SU(N+1) SU(N) x U(1)

〈Q〉 ∿ √ m μ

1

• Gauge symmetry completely (hierarchically) broken

endow the monopole with fluctuating CPN-1 modes

b(t)

= Origin of the dual gauge group

• The monopole ~ N of a new (dual) SU(N) --- isometry group of CPN-1
• N.B. M-V-M as a whole is neutral (a singlet) : monopoles are confined. (A dual model of quark confinement)

So:

CPN vs CPN-1

SU(N+1)C (SU(N)xU(1))C

CPN

SU(N)F x

{

SU(N)C+F

hφi ⇠ v1

SU(N-1)xU(1)

CPN-1

⊃ SU(2)C ⊃ U(1)

monopole m

• n
• p
• l

e

• v
• r

t e x c

• m

p l e x

SU(N)F x

hqi ⇠ v2 1N

SU(N)

naive “nonAbelian monopole”

Monopole Moduli Vortex Moduli ~ CPN-1 SU(N) 1(H) 2(G/H)

Chatterjee, Konishi ’14

Cipriani, Gudnason, ... ’11

CPN-1 Origins model on a finite-width world sheet

Bolognesi,Konishi,Ohashi, ’16 Betti, Bolognesi,Konishi,Ohashi, ’18 Bolognesi,Gudnason, Konishi, Ohashi ‘18

t h e n

• n
• n
• r

m a l i z a b l e 3 D g a u g e z e r

• m
• d

e s

• f

t h e m

• n
• p
• l

e , w h e n d r e s s e d b y fl a v

• r

c h a r g e s , t u r n i n t

• n
• r

m a l i z a b l e 2 D m

• d

e s

• n

t h e v

• r

t e x w

• r

l d s h e e t .

🔶

No spontaneous breaking of SU(N) (isometry group of CPN-1 )

CPN-1 model on a finite-width world sheet

Boundary conditions:

Separating and integrating over ni , (i=2,3,.. N) (Large N approximation)

n1 = σ

Seff = Z d2x

• (N − 1) tr log(−DµDµ + λ) + (Dµσ)∗Dµσ − λ(|σ|2 − r)
• .

Generalized gap equations

E = N X

n

ωn + Z L

• (∂xσ)2 + λ(σ2 r)
• dx ,
• ∂2

x + λ(x)

• fn(x) = ω2

n fn(x) ,

∂2

xσ(x) λ(x)σ(x) = 0 .

N 2 X

n

fn(x)2 ωn + σ(x)2 r = 0 .

(*)

D-D : n1

• L

2

• = n1

L

2

• = pr ,

ni

• L

2

• = ni

L

2

• = 0 ,

i > 1 . N-N : @xni

• − L

2

• = @xni

L

2

• = 0 ,

∀i.

x ∈ [−L 2 , L 2 ]

Bolognesi, KK.,Ohashi, ’16 Betti, Bolognesi, KK.,Ohashi, ‘17 Bolognesi, Gudnason, KK ,Ohashi, ’18

λ x 5 10 15 20 25 −6 −4 −2 2 4 6 σ2 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −6 −4 −2 2 4 6

Figure 1: The functions λ(x) (left) and σ2(x) (right) which are solutions to the gap equa- tion, Eq. (2.6), for various values of L ranging L = 1 ∼ 12. Λ = 1 in this figure. The innermost (outermost) curve corresponds to L = 1 (L = 12).

A new random-walk method Back to p.14

✓ n1 L

2

• n2

L

2

• ◆

= ✓ 1 ◆ √r✏ ; ✓ n1

• − L

2

• n2
• − L

2

• ◆

= ✓ ei cos α ei sin α −e−i sin α e−i cos α ◆ ✓ √r✏ ◆ ∼ ✓ cos α sin α ◆ √r✏ ;

The most general Dirichlet conditions

N 2 X

n

fn(x)2 ωn e−✏!n + σ1(x)2 + σ2(x)2 − r✏ = 0 , ∂2

xσ1(x) − λ(x)σ1(x) = 0 ,

∂2

xσ2(x) − λ(x)σ2(x) = 0 ,

The gap equations

λ x Λ2 = 1 5 10 15 20 25 −2 −1.5 −1 −0.5 0.5 1 1.5 2 α = 0 α = π λ x Λ2 = 1 0.8 1 1.2 1.4 1.6 1.8 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2

(∂E / ∂α) / N α

• Eq. (4.23)
• Eq. (4.22)
• Eq. (4.12)

−0.002 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 π/8 π/4 π/2 3π/4 π 3π/8 5π/8 7π/8

∂E ∂α = +2 sin α W > 0 ,

Bolognesi,Gudnason, Konishi, Ohashi ‘18

Does N=2 SQCD teach us anything useful?

🔶 A recent observation: the most singular (“Argyres-Douglas”) SCFT,

in N=2 SU(2) theory with NF=1,2,3, and SU(N) theory with NF flavors, under an N=1 perturbation, flows down (RG) towards an infrared-fixed-point theory described by massless mesons M in the adjoint representation of GF

🔶 Further relevant deformations (shift of bare mass parameters)

µΦ2|F = µ ψψ + . . .

G i a c

• m

e l l i , ’ 1 5 , B

• l
• g

n e s i , G i a c

• m

e l l i , K K ‘ 1 5

☞ Confinement and flavor symmetry breaking: M (or a part of it) are now the massless Nambu-Goldstone particles

Strongly-coupled nonAbelian monopoles in action

Blown-up RG flow

Why remarkable:

🔶 N=2 SCFT is a complicated, nonlocal theory of strongly interacting

massless monopoles, dyons and quarks *

🔶 N=1 SCFT is a theory of weak-coupled local theory of mesons M 🔶 In the nearby N=1 confining vacuum, M ~ NG bosons of symmetry

breaking **

🔶 Analogy with the real-world QCD? * **

Massless mesons in the adjoint representation of GF in the IR

Giacomelli, ’15, Bolognesi,Giacomelli, KK ‘15

Tools:

🔶 Seiberg-Witten curves for N=2 gauge theories

☞ SCFT by appropriate tuning of Gc , Gf , Ui (VEVs), g’s and m’s

🔶 Trace anomalies (any theory)

hT µ

µ i =

1 16π2  c (R2

µνρσ 2R2 µν + R2

3 ) a (R2

µνρσ 4R2 µν + R2)

• (Weyl)^2

Euler density

🔶 For any N=1 susy theory

a = 3 32(3Tr R3 Tr R) ; c = 1 32(9Tr R3 5Tr R) ,

Tr= sum over Weyl fermions

( R= UR (1) charge )

Anselmi, Freedman, Grisaru, Johansen (’98)

🔶 For N=2 susy fields

N

N

Tr R3

N =2 = Tr RN =2 = 48(a c) ;

Tr RN =2IaIb = δab(4a 2c) .

SUR(2) × UR(1); RN =2 ≡ UR(1) charge, I3 ⊂ SUR(2)

Any N=2 theory has global R symmetries

Tachikawa: Lecture Notes In Physics ‘15

Gaiotto, ‘09

Chacaltana, Distler, ‘10

a and c

Argyres-Plesser-Seiberg-Witten ‘96 Eguchi-Hori-Ito-Yan ‘96 Argyres-Douglas ‘95 Argyres-Seiberg ‘07 Shapere-Tachikawa ‘08

Most singular (“Argyres-Douglas”) SCFT in SQCD

🔶

Flowing down from N=2 SCFT to N=1 SCFT

µΦ2|F = µ ψψ + . . .

Bonelli, Giacomelli, Maruyoshi, Tanzini (’13)

🔶 ☞ relations

Dijkgraaf, Vafa ’03 Cachazo-Seiberg-Witten ‘03

e.g., for SU(2), NF =1,

RN =1 = 5 6RN =2 + 1 3I3

🔶 Known of N=2 SCFT ⟹

for SU(N), NF =2N-1,

RN =1 = 2 3RN =2 + 2 3I3

{RN =2 , I3} ↔ RN =1 {RN =2 , I3} RN =1

🔶 Known Tr R3 , Tr R of the IR theory - > a, c of the IR theory 🔶 N=1 curves (N=2 SW curves + factorization condition)

e.g., for SU(2), NF =1,

a = 1 48 , c = 1 24 ,

for SU(2), NF =3,

a = 1 6 , c = 1 3 ,

This is 32 -1= 8 massless meson chiral fields !! Weff = µ T r M 3 + . . .

Giacomelli, ‘15

🔶 In all cases,

a c = 1 2

• cfr. 1

2 ≤ a c ≤ 3 2

“Conformal collider bounds”

H

• f

m a n , M a l d a c e n a , ‘ 8 H

• f

m a n , L i , M e l t z e r , R e j

• n
• B

a r r e r a , R ‘ 1 6

end up with the curve 8 > > > < > > > : v02 = z03 ; w2 = µ02z0 ; z0w = µ0v0 , Ω = dv0dwdz0 . is simply the SW curve of the AD theory Imposing now the constraint [8] D R-charges:

(note)

SU(N), NF =2N-1 AD vacuum:

y2 = PN(x)2 − 4Λ Y

i

✓ x + mi − Λ N ◆ . y2 = x2N−1; λSW = xdy y . setting mi = m∗ = Λ/N

⟹

− D(Uk) = 2k − 1 2 ; k = 2, . . . , N ,

δm = 1 2N − 1 X

i

mi − m∗, m∗ ≡ Λ N .

µΦ2|F = µ ψψ + . . .

N=1 SCFT

8 < : y2 = x2N1(x 4Λ) ; w2 = µ2x(x 4Λ) . variables

         z2 + x2N1 = 0 ; w02 + µ02x = 0 ; µ0z = w0xN1 , Ω = dw0dxdz z .

change of variables

RIR = 2 3(RN=2 + I3)

🔶 ’t Hooft anomaly matching conditions Tr R3 , Tr R , Tr R(GF )2

☞

F r e e m a s s l e s s m e s

• n

s i n t h e a d j

• i

n t r e p r e s e n t a t i

• n
• f

S U ( N

F

) ! ! !

W(Ψ) = 1 2 X

i

˜ miΨi + δm Tr Ψ2 + µ0Tr Ψ3,

N=2 SCFT

Ψ

Ψ2 = 0

deformation, ˜ mi = δm = 0,

deformation, δm 6= 0

r vacua

a,c at IR

（＊）

chiral multiplets

RG flows

N=2 SCFT

Real-world QCD

N=0 SCFT

aUV = 11NfNc 360 + 31 180(N 2

c − 1)

− aIR = N 2

f − 1

360 .

SU(N), NF =2N-1

aUV = 7N 2 − N − 5 24

aN=2SCF T = 7N(N − 1) 24

aIR = (2N − 1)2 − 1 48

cUV = 4N 2 − N − 2 12

cN=2SCF T = N(N − 1) 3

cIR = (2N − 1)2 − 1 24

• cUV = 1

20NfNc + N 2

c 1

10 ;

cIR = N 2

f 1

120 ;

Nf < 11 2 Nc

⑥ Confinement and XSB in “chiral” QCD

ψ{ij} , χ[ij] , ηA

i ,

(A = 1, 2, . . . , 8) ,

model:

• YM with left-handed

fermions

⇣ ⌘

¯

¯

Gf = SU(8) × U1(1) × U2(1) ×

N∗ .

U1(1) : ψ ! ei

α N+2ψ ;

η ! e−i α

8 η ;

U2(1) : ψ ! ei

β N+2ψ ;

χ ! e−i

β N−2χ

N∗ ,

N ∗ = GCD{N + 2, N 2, 8}

global symmetry

🔶 Asymptotically free 🔶 No gauge invariant bifermion condensates (four-fermion condensates?)

Bolognesi, Konishi, Shifman ‘18

🔶 Planar equivalence to SYM at large N ?

Armoni, Shifman ‘12

🔶 How is Gf realized at low energies?

Studies of “chiral varieties” of QCD

] Stuart Raby, Savas Dimopoulos, and Leonard Susskind, Tumbling Gauge Theo- ries, Nucl. Phys. B169, 373 (1980). ] T. Appelquist, A. G. Cohen, M. Schmaltz and R. Shrock, New constraints on chiral gauge theories, Phys. Lett. B 459, 235 (1999) [hep-th/9904172]; T. Ap- pelquist, Z. y. Duan and F. Sannino, Phases of chiral gauge theories, Phys.

• Rev. D 61, 125009 (2000) [hep-ph/0001043]; Y. L. Shi and R. Shrock, Ak ¯

F chi- ral gauge theories, Phys. Rev. D 92,105032 (2015) [arXiv:1510.07663 [hep-th]];

• Y. L. Shi and R. Shrock, Renormalization-Group Evolution and Nonperturba-

tive Behavior of Chiral Gauge Theories with Fermions in Higher-Dimensional Representations, Phys. Rev. D 92, 125009 (2015) [arXiv:1509.08501 [hep-th]]. 4] E. Poppitz and Y. Shang, Chiral Lattice Gauge Theories Via Mirror-Fermion Decoupling: A Mission (im)Possible?, Int. J. Mod. Phys. A 25, 2761 (2010) [arXiv:1003.5896 [hep-lat]].

Assume that bi-fermion condensates play the key role

hφiAi ⇠ hψijηA

j i ,

A = 1, 2, . . . , 8 , h˜ φi

ji = hψikχkji ,

(bifundamental in color and in SU(8))

(adjoint of color SU(N))

Two possible dynamical scenarios

🔶 Partial color-flavor locking and dynamical Abelianization

hφiAi = Λ3 B B @ c18 0N−8,8 1 C C A , h˜ φi

ji = Λ3

B B B B B @ a 18 d1 ... dN−12 b 14 1 C C C C C A ,

SU(N)c ⇥ SU(8)f ⇥ U(1)2 ! SU(8)cf ⇥ U(1)N−11 ⇥ SU(4)c .

B a r y

• n

s

• m condensates and the η fields,

B{AB} = ψ{ij}ηA

j ηB i

• A,B symm ⇠ hφiAiηB

i

• A,B symm

contribute 8+4=12 contribute N-12

˜ BA

j = ψikχkjηA i ⇠ h˜

φi

jiηA i

The sum = N = SU(8)3 |UV

⇥ U(1)2 ⇥ SU(8)f

broken by the adjoint condensate

🔶 Split color-flavor locked SU(8)

h

ji

SU(N)c ⇥ SU(8)f ⇥ U(1)2 ! Y

i

SU(Ai)cf ⇥ Y U(1)cf ⇥ U(1)N−11 ⇥ SU(4)c .

’t Hooft OK

🔶 Full Abelianization

SU(N)c ⇥ SU(8)f ⇥ U(1)2 ! U(1)N−1 ⇥ SU(8)f .

weakly coupled and saturate ’t Hooft trivially

model:

Partial color-flavor locking and partial dynamical Abelianization

• r

Full dynamical Abelianization

🔶 No planar equivalence to SYM 🔶 Dynamical Abelianization (ubiquitous in N=2 theories) 🔶 MAC criterion supports

hφiAi ⇠ hψijηA

j i ,

A = 1, 2, . . . , 8 ,

h˜ φi

ji = hψikχkji ,

🔶 Four-fermion condensates less likely

• Confinement mechanism (probably very) different from the naive dual

superconductor picture;

• Gauge systems generate dynamically richer IR effective systems than is

naïvely expected, e.g. with a larger degrees of freedom;

• Ordinary Q C D:

U(1)2 or SU(2) x U(1) #d.f < #d.f

Abelian monopoles strongly-coupled nonAbelian monopoles

Conclusion / Lessons

• Confinement vacuum ~ IRFP conformal vacuum ?

Strongly-coupled nonAbelian monopoles and IR fixed point

The END

Topology (Mapping: Space → G)

• Charged particle ψ(x) in a monopole field

exp ige

• ∂Ω

Ai dxi → exp ige

• S2 dS · H = exp 4πige gm

(H = ∇gm r ). 2 ge gm = n, n ∈ Z, Π1(U(1)) = Z

• U(1) in a nonabelian theory G (Wu, Yang): monopole ∼ Π1(G) ̸= ∅.
• ’t Hooft-Polyakov monopoles: Π2(SU(2)/U(1)) = Π1(U(1)) = Z;
• G

⟨φ⟩̸=0

− → H : similar → monopoles with nonabelian charges if Π2(G/H) ̸= ∅

Dirac ~1930 Regular monopoles F i b e r b u n d l e

B a c k t

• p

. 4

Back to p.18

Argyres-Douglas vacuum (SCFT)

SU(N) SW curve (pure YM) y2 = PN(x)2 − Λ2N = x2N − X

k

Uk xN−k − Λ2N λ = 1 2πi ∂P(x) ∂x xdx y SW differential aDi = I

αi

λ; ai = I

βi

λ; SU(3) pure YM: a (1,0) monopole, a (1,1) dyon and the (0,1) quark Mnm,i,ne,i = √ 2 |nm,iaDi + ne,iai|

Argyrers, Douglas ’95

massless simultaneously at

u = Tr Φ2 ' 0, v = Tr Φ3 ' 3Λ3,

M1 M2 M3 M4 A1 A3 A4 A5 A2 A6 M6 M5

Figure 3: Zero loci of the discriminant of the curve of N = 2, SU(3), nf = 4 theory at small m.

U = Tr Φ2 ' 3m2 ; V = Tr Φ3 ' 2m3

Auzzi, Grena, K.K. ‘13

AD vac in SU(3), Nf=4

Matrix Charge M1, M4 (±1, 1, 0, 0)4 A2, A5 (±1, −1, ∓1, 0)4 M2, M5 (±2, 2, ∓1, 0) A3, A6 (±2, −2, ±1, 0) M3, M6 (0, 2, ±1, 0) A1, A4 (±4, −2, ∓1, 0)

back to p.11

Argyres-Seiberg’s S duality

• SU(3) with NF = 6 hypermultiplets ( ’ s ) at infinite coupling

Qi, ˜ Qi

SU(2) w/ (2 · 2 ⊕ SCFTE6) SU(3) w/ (6 · 3 ⊕ ¯ 3)

=

SU(2) × SU(6) ⊂ E6

Flavor symmetry ~ SU(6) x U(1)

g = ∞ g = 0

Minahan-Nemeschansky ’96

Jackiw-Rebbi

H ψ = [ iγ0γiDi − gΦ ]ψ = 0,

[L + s + T, H] = 0 ψ = ψ(0) = δi

α f(r)

ˆ ψ = ψ(0)b0 +

• i=0

ψ(i)bi + . . .

ˆ ψ†

A|M⇥ ,

A = 1, 2, . . . Nf

• d3r |ψ(0)|2 < ∞

monopole bckgrd gauge su(2) spin 3D normalizable zero mode

➱ Flavor multiplets of monopoles

⏎

a and c coefficients for free particles

c = 1 120(NS + 6 NF + 12 NV ) ; a = 1 360(NS + 11 NF + 62 NV ) N a = 1 360(4 + 11) = 1 24 ; c = 1 120(4 + 6) = 1 12 . a = 1 360(2 + 11 + 62) = 5 24 ; c = 1 120(2 + 6 + 12) = 1 6 .

☞

For N=2 hypermultiplet (pair of chiral multiplets Q and Q bar):

☞

For N=2 vector multiplet:

Back to p.37 ☞

− cUV = 1 20NfNc + N 2

c − 1

10 ; aUV = 11NfNc 360 + 31 180(N 2

c − 1) ,

− cIR = N 2

f − 1

120 ; aIR = N 2

f − 1

360 .

For real-world QCD:

🔶 The result has been checked by following alternative RG paths

UV IR N = 2 IR N = 1 UV N = 1 µ ⌧ Λ2 µ Λ1 IR

Figure 1: RG flow for various values of µ.

N=1 Leff of SQCD

(M, B, q, …)

Seiberg, ‘94

Carlino, K.K. Murayama ‘00 Di Pietro, Giacomelli ’12

B

• l
• g

n e s i , G i a c

• m

e l l i , K K ‘ 1 5

“Left RG route” ( )

µ ⌧ Λ2

discussed above; “Right RG route” ( ):

µ Λ2

the system first flows down to N=1 SQCD with

Weff = m TrM − 1 2µ ✓ Tr M 2 − 1 N (Tr M)2 ◆ + 1 Λ

3N−Nf Nf −N

1

(det M)

1 Nf −N ,

η∗ = m∗ p 2 = ωk 2N Nf N 2Nf/(4N−2Nf)Λ ,

M 2

ij − 2↵Mij + ↵2ij = 0 ;

↵ ≡ Nµm∗ 4N − 2Nf = Tr M Nf .

• r

f M 2 = 0 , f M ≡ M − ↵I . （＊＊）

A bonus: our mesons M ~ Meson M in the Seiberg’s N=1 duality SU(N), NF Q’s < —- > SU(NF - N), NF q’s, M; Result valid for any NF

N.B.

（＊） = （＊＊）

SU(3) with NF = 6

y2 = (x3 − ux − v)2 − f(τ)x6, s to one in the infinite coupling limit. In or

− p − − p − − As f → 1 the curve clearly degenerates to a genus one curve. Th ferential develops a pair of poles at infinity

: setting u = 0 th

With the redefinition x → −ix v ; y → 2y v; v → 2iv, we finally get y2 = x3 − v4; ∂λ ∂v = dx y ,

E6 Minahan-Nemeschansky theory

f → 1 τ → 0

means

g → ∞

• r

SCFT at u=v=0

v=0 ➯

But this is SCFT SU(2), NF = 4 theory!

y2 = [(1 - √f) x3 - u x - v ] [(1 + √f) x3 - u x - v ] y2 = - v [ 2 x3 - v ] , ∂λ/∂v = dx /y

Back to p. 38

Vortex 2D dynamics in Higgs phase (U(2))

S(1+1)

σ

= β

• d2x 1

2 (∂ na)2 ,

+ fermionic terms

N=(2,2) CP1 sigma model : 2 vacua ➞ kinks = (Abelian) monopole!

• (Seiberg-Witten)

Dynamical Abelianization

• Tong, Shifman-Yung

Global SU(2) unbroken (Coleman) ≡ Gauge dynamics in 4D in Coulomb phase

• 2D - 4D duality

Dorey

dt dz

V a f a , H

• r

i Shifman et. al.

• Two types of Chebyshev* vacua ( ϕ1 = ϕ2 = ... = 0; ϕn 2 = ± Λ 2 ; ϕm det’d by Cheb. polynom. )

xy2 ∼

• xn(x − φ2

n)

⇥2 − 4Λ4x2n = x2n(x − φ2

n − 2Λ2)(x − φ2 n + 2Λ2).

y2 ∿ x2n singular SCFT (EHIY point); strongly interacting, relatively non-local monopoles and dyons

• A strategy: resolve the vacuum by adding small mi =m ≠ 0 (i.e.,

Vacua in confinement phase surviving N=1, μ Φ2 perturbation)

GST duals and confinement

⟹

✓Nf ◆ + ✓Nf 2 ◆ + . . . ✓Nf Nf ◆ = 2Nf−1

even r vacua, from one of the Chebyshev vacua

• dd r vacua, from the other Chebyshev

vacua

Carlino-Konishi-Murayama ‘00

✓Nf 1 ◆ + ✓Nf 3 ◆ + . . . ✓ Nf Nf 1 ◆ = 2Nf−1 .

USp(2N) theory w/ NF = 2n

Figs.

• 40
• 20

20 40

• 15
• 10
• 5

5 10 15

Jackiw-Rebbi flavor dressing color fields color flavor fields correlated UNCORRELATED

↩

Argyres-Seiberg’s S duality

• SU(3) with NF = 6 hypermultiplets ( ’ s ) at infinite coupling

Qi, ˜ Qi

SU(2) w/ (2 · 2 ⊕ SCFTE6) SU(3) w/ (6 · 3 ⊕ ¯ 3)

=

SU(2) × SU(6) ⊂ E6

Flavor symmetry ~ SU(6) x U(1)

g = ∞ g = 0

Minahan-Nemeschansky ’96

Axial / chiral anomalies

• No conservation in all channels
• Observable (calculable) effects
• Gauge vertices:

Gauge symmetry destroyed (inconsistency)

• External “gauge” fields:

Abelian / nonAbelian anomalies

♦ ♦ Anomaly cancellation needed

OK in

(NOT an inconsistency)

♦

• bservable effects
• ’t Hooft’s anomaly matching conditions

Back to p.4

(Nambu’s Erice lecture!)

• etc. Wess-Zumino-Witten (5D) action
• Deep roots

Fukuda, Steinberger, Schwinger, Adler, Bell, Jackiw, Bardeen, …