Fiber-Base duality, Global Symmetry Enhancement and Gopakumar-Vafa - - PowerPoint PPT Presentation

Fiber-Base duality, Global Symmetry Enhancement and Gopakumar-Vafa invariant

Futoshi Yagi (Technion)

Based on arXiv: 1411.2450: V. Mitev, E.Pomoni, M.Taki, FY Work in progress: H.Hayashi, S-S.Kim, K.Lee, M.Taki, FY

5D N=1 SUSY SU(2) gauge theory with Nf flavor

’96 Seiberg

5D UV fixed point exists for Nf ≦ 7 5D N=1 SUSY SU(2) gauge theory with Nf flavor

’96 Seiberg

5D UV fixed point exists for Nf ≦ 7 5D N=1 SUSY SU(2) gauge theory with Nf flavor Global symmetry enhancement at UV fixed point

SO(2Nf) × U(1) ⊂ ENf +1

Instanton particle

Nf flavors

8

E

7

E

6

E ) 10 (

5

SO E = ) 5 (

4

SU E =

) 3 ( ) 2 (

3

SU SU E × =

) 2 ( ) 1 (

2

SU U E × =

) 2 (

1

SU E =

) 14 ( SO ) 12 ( SO ) 10 ( SO ) 8 ( SO ) 6 ( SO ) 4 ( SO ) 2 ( SO

U(1) U(1) U(1) U(1) U(1) U(1) U(1) U(1)

Can we see global symmetry enhancement from brane web?

Can we see global symmetry enhancement from brane web? S-duality (Fiber-base duality in CY language)

(1,-1) 5-brane (1,-1) 5-brane (1,1) 5-brane

Brane setup for pure SU(2) SYM

(1,1) 5-brane = 1 D5 + 1 NS5

0 1 2 3 4 5 0 1 2 3 4 6 NS5 D5

5 6 D5 D5 NS5 NS5

NS5

1 2g2 + 2a

D5

2a

a : Coulomb moduli parameter g : (Bare) gauge coupling

pure SU(2) SYM

NS5 NS5 D5 S-duality

S-duality for pure SU(2) SYM

1 2g2 + 2a

D5

2a

a : Coulomb moduli parameter g : (Bare) gauge coupling

NS5 NS5 D5 S-duality

S-duality for pure SU(2) SYM

1 2g2 + 2a

D5

2a

a : Coulomb moduli parameter g : (Bare) gauge coupling

2a0

1 2g02 + 2a0

NS5 NS5 D5 S-duality

S-duality for pure SU(2) SYM

1 2g2 + 2a

D5

2a

a : Coulomb moduli parameter g : (Bare) gauge coupling

2a0

1 2g02 + 2a0

1 g02 = − 1 g2 a0 = a + 1 4g2

NS5 NS5 D5 S-duality

S-duality for pure SU(2) SYM

1 2g2 + 2a

D5

2a Weyl Symmetry for E1 = SU(2)

‘97 Aharony,Hanany,Kol

a : Coulomb moduli parameter g : (Bare) gauge coupling

2a0

1 2g02 + 2a0

1 g02 = − 1 g2 a0 = a + 1 4g2

NS5 NS5 D5 S-duality

S-duality for pure SU(2) SYM

1 2g2 + 2a

D5

2a Weyl Symmetry for E1 = SU(2) Coulomb moduli parameter is also transformed!

‘97 Aharony,Hanany,Kol

a : Coulomb moduli parameter g : (Bare) gauge coupling

2a0

1 2g02 + 2a0

1 g02 = − 1 g2 a0 = a + 1 4g2

=

f

N 1 =

f

N 2 =

f

N 3 =

f

N 4 =

f

N

5 =

f

N

’09 Benini-Benvenuti-Tachikawa

6 =

f

N

7-brane

7 =

f

N

7-brane

Generalization to higher flavor

(Weyl symmetry of) Transformation induced from

S-duality

+

SO(2Nf) × U(1)

(Weyl symmetry of) Transformation induced from

S-duality

+

Flavors ↔ Instanton particle (Masses ↔ Gauge coupling)

SO(2Nf) × U(1)

Flavors (Masses) Instanton particle (Gauge coupling)

(Weyl symmetry of) Transformation induced from

S-duality

+

(Weyl symmetry of)

Enhanced symmetry

⇒

Flavors ↔ Instanton particle (Masses ↔ Gauge coupling)

SO(2Nf) × U(1)

Flavors (Masses) Instanton particle (Gauge coupling)

ENf +1

(Weyl symmetry of) Transformation induced from

S-duality

+

(Weyl symmetry of)

Enhanced symmetry

⇒

Flavors ↔ Instanton particle (Masses ↔ Gauge coupling)

SO(2Nf) × U(1)

Flavors (Masses) Instanton particle (Gauge coupling)

ENf +1

Again, Coulomb moduli parameter is also transformed!

Can we write Nekrasov partition function in manifestly ENf+1 invariant way ?

Original Nekrasov partition function does not look manifestly ENf+1 invariant because…

Original Nekrasov partition function does not look manifestly ENf+1 invariant because…

• 1. Coulomb moduli parameter is transformed.

Original Nekrasov partition function does not look manifestly ENf+1 invariant because…

• 1. Coulomb moduli parameter is transformed.
• 2. Expanded in terms of instanton factor q = e−

β 2g2

Original Nekrasov partition function does not look manifestly ENf+1 invariant because…

• 1. Coulomb moduli parameter is transformed.
• 2. Expanded in terms of instanton factor
• 1. Use invariant variable

˜ A = q

2 8−Nf e−βa

e−βa

instead of

q = e−

β 2g2

✓ 1 g2 → − 1 g2 , a → a + 1 4g2 , ˜ A = q

1 4 eβa = eβ(a+ 1 8g2 )

(Nf = 0) ◆

Original Nekrasov partition function does not look manifestly ENf+1 invariant because…

• 1. Coulomb moduli parameter is transformed.
• 2. Expanded in terms of instanton factor
• 1. Use invariant variable

˜ A = q

2 8−Nf e−βa

e−βa

instead of

• 2. Expand in terms of ˜

A

q = e−

β 2g2

✓ 1 g2 → − 1 g2 , a → a + 1 4g2 , ˜ A = q

1 4 eβa = eβ(a+ 1 8g2 )

(Nf = 0) ◆

Original form

’12 H-C Kim, S-S.Kim, K.Lee ’14 C.Hwang, J.Kim, S.Kim, J.Park

ENf+1 invariant Nekrasov partition function

Z(a, g, mi; ✏1, ✏2) = Zpert(a, mi; ✏1, ✏2)

∞

X

k=0

Zk(a, mi; ✏1, ✏2)qk

Original form

’12 H-C Kim, S-S.Kim, K.Lee ’14 C.Hwang, J.Kim, S.Kim, J.Park

ENf+1 invariant Nekrasov partition function

Z(a, g, mi; ✏1, ✏2) = Zpert(a, mi; ✏1, ✏2)

∞

X

k=0

Zk(a, mi; ✏1, ✏2)qk =

∞

X

n=0

˜ Zn(g, mi; ✏1, ✏2) ˜ Ak

New form

’14 C.Hwang, J.Kim, S.Kim, J.Park

ENf+1 invariant

Original form

’12 H-C Kim, S-S.Kim, K.Lee ’14 C.Hwang, J.Kim, S.Kim, J.Park

ENf+1 invariant Nekrasov partition function

Z(a, g, mi; ✏1, ✏2) = Zpert(a, mi; ✏1, ✏2)

∞

X

k=0

Zk(a, mi; ✏1, ✏2)qk =

∞

X

n=0

˜ Zn(g, mi; ✏1, ✏2) ˜ Ak

New form

’14 C.Hwang, J.Kim, S.Kim, J.Park

ENf+1 invariant ENf+1 invariant

= exp " ∞ X

k=0

1 k

∞

X

n=1

˜ Fn(kg, kmi; k✏1, k✏2) ˜ Ank # ≡ PE " ∞ X

n=1

˜ Fn(g, mi; ✏1, ✏2) ˜ An #

Manifestly E1 invariant!!

Nekrasov partition function for pure SU(2)

q = e−✏1, t = e✏2

Z = PE  q + t (1 − q)(1 − t)χE1

2

˜ A2 + O( ˜ A4)

• ˜

A = q

1 4 eβa,

Character of E1 = SU(2) : χE1

2

= q

1 2 + q− 1 2 ,

Manifestly E1 invariant!!

Nekrasov partition function for pure SU(2)

q = e−✏1, t = e✏2

Z = PE  q + t (1 − q)(1 − t)χE1

2

˜ A2 + O( ˜ A4)

• F1

D1 W boson Instanton

˜ A = q

1 4 eβa,

Character of E1 = SU(2) : χE1

2

= q

1 2 + q− 1 2 ,

Nekrasov partition function for Nf=1

E2 = SU(2) × U(1) SU(2) : u1 = q

1 2 e− 1 4 βm

U(1) : u2 = q− 1

2 e− 7 4 βm

Z = PE " − q

1 2 t 1 2

(1 − q)(1 − t) ⇣ χSU(2)

2

(u1)u2

− 3

7 + u2 4 7

⌘ ˜ A + q + t (1 − q)(1 − t)χSU(2)

2

(u1)u

1 7

2 ˜

A2 + O( ˜ A3) #

χ2(u1) = u1 + u1

−1

˜ A = q

2 7 e−βa

Nekrasov partition function for Nf=1

E2 = SU(2) × U(1) SU(2) : u1 = q

1 2 e− 1 4 βm

U(1) : u2 = q− 1

2 e− 7 4 βm

Z = PE " − q

1 2 t 1 2

(1 − q)(1 − t) ⇣ χSU(2)

2

(u1)u2

− 3

7 + u2 4 7

⌘ ˜ A + q + t (1 − q)(1 − t)χSU(2)

2

(u1)u

1 7

2 ˜

A2 + O( ˜ A3) #

χ2(u1) = u1 + u1

−1

˜ A = q

2 7 e−βa

Vector multiplet Hypermultiplet Hypermultiplet

8

E

7

E

6

E ) 10 (

5

SO E =

) 5 (

4

SU E =

) 3 ( ) 2 (

3

SU SU E × =

) 2 ( ) 1 (

2

SU U E × =

) 2 (

1

SU E =

The vector multiplet and the hypermultiplet are included in the fundamental representation of ENf+1 corresponding to the following nodes 3875 248

(conjectured)

56 133 27 27 16 10 10 5 2 2 3 3 2

• 3/7, 1/7

U(1)

Z = PE 2 4 X

C∈H2(X,Z)

X

jL,jR jL

X

kL=−jL jR

X

kR=−jR

M (jL,jR)

C

tkL+kRqkL−kR (t − t−1)(q − q−1) QC 3 5

Nekrasov partition function Set of integers

Gopakumar-Vafa’s expansion

Gopakumar-Vafa ‘98 Iqbal, Kozcaz, Vafa ‘07

M (jL,jR)

C

X : Calabi-Yau manifold QC = e−

R

C ω,

ω : K¨ ahler form (QC = e−2βa, e−β(a−m), qke−2βa, · · · ) M (jL,jR)

C

: Refined Gopakumar-Vafa invariant

(After the convention change ˜ A → − ˜ A)

Non-Negative integer Consistent with the result from topological B-model

[Huang, Klemm, Poretschkin ‘13]

Nekrasov partition function is invariant

Summary SO(2Nf) × U(1) + S-duality = ENf +1

Refined Gopakumar-Vafa invariants from Nekrasov partition function agrees with topological B-model computation