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 N f flavor

5D N=1 SUSY SU(2) gauge theory with N f flavor 5D UV fixed point exists for N f ≦ 7 ’96 Seiberg

5D N=1 SUSY SU(2) gauge theory with N f flavor 5D UV fixed point exists for N f ≦ 7 ’96 Seiberg Global symmetry enhancement at UV fixed point SO (2 N f ) × U (1) ⊂ E N f +1 N f flavors Instanton particle

E E = SU ( 5 ) 8 4 SO ( 14 ) U (1) SO ( 6 ) U (1) E E SU ( 2 ) SU ( 3 ) 7 = × 3 SO ( 12 ) U (1) SO ( 4 ) U (1) E E U ( 1 ) SU ( 2 ) 6 = × 2 SO ( 2 ) U (1) SO ( 10 ) U (1) E = SO ( 10 ) E = SU ( 2 ) 5 1 U (1) SO ( 8 ) 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)

Brane setup for pure SU(2) SYM (1,1) 5-brane (1,-1) 5-brane = 1 D5 + 1 NS5 D5 5 NS5 NS5 6 D5 (1,1) 5-brane (1,-1) 5-brane NS5 0 1 2 3 4 5 D5 0 1 2 3 4 6

pure SU(2) SYM D5 NS5 2 a a : Coulomb moduli parameter 1 g : (Bare) gauge coupling 2 g 2 + 2 a

S-duality for pure SU(2) SYM D5 NS5 S-duality D5 NS5 2 a 1 2 g 2 + 2 a a : Coulomb moduli parameter g : (Bare) gauge coupling

S-duality for pure SU(2) SYM D5 NS5 S-duality 1 D5 2 g 0 2 + 2 a 0 NS5 2 a 2 a 0 1 2 g 2 + 2 a a : Coulomb moduli parameter g : (Bare) gauge coupling

S-duality for pure SU(2) SYM D5 NS5 S-duality 1 D5 2 g 0 2 + 2 a 0 NS5 2 a 2 a 0 1 2 g 2 + 2 a a : Coulomb moduli parameter g : (Bare) gauge coupling g 0 2 = − 1 1 g 2 1 a 0 = a + 4 g 2

S-duality for pure SU(2) SYM D5 NS5 S-duality 1 D5 2 g 0 2 + 2 a 0 NS5 2 a 2 a 0 1 2 g 2 + 2 a a : Coulomb moduli parameter g : (Bare) gauge coupling g 0 2 = − 1 1 Weyl Symmetry for E 1 = SU (2) g 2 ‘97 Aharony,Hanany,Kol 1 a 0 = a + 4 g 2

S-duality for pure SU(2) SYM D5 NS5 S-duality 1 D5 2 g 0 2 + 2 a 0 NS5 2 a 2 a 0 1 2 g 2 + 2 a a : Coulomb moduli parameter g : (Bare) gauge coupling g 0 2 = − 1 1 Weyl Symmetry for E 1 = SU (2) g 2 ‘97 Aharony,Hanany,Kol Coulomb moduli parameter 1 a 0 = a + is also transformed! 4 g 2

Generalization to higher flavor N 0 N 1 N 2 N 3 = = = = f f f f N 5 N 4 N 6 N 7 = = = = f f f f 7-brane 7-brane ’09 Benini-Benvenuti-Tachikawa

(Weyl symmetry of) Transformation induced from + S-duality SO ( 2N f ) × U ( 1 )

(Weyl symmetry of) Transformation induced from + S-duality SO ( 2N f ) × U ( 1 ) Flavors Instanton particle Flavors ↔ Instanton particle (Masses) (Gauge coupling) (Masses ↔ Gauge coupling)

⇒ (Weyl symmetry of) Transformation induced from + S-duality SO ( 2N f ) × U ( 1 ) Flavors Instanton particle Flavors ↔ Instanton particle (Masses) (Gauge coupling) (Masses ↔ Gauge coupling) (Weyl symmetry of) Enhanced symmetry E N f + 1

⇒ (Weyl symmetry of) Transformation induced from + S-duality SO ( 2N f ) × U ( 1 ) Flavors Instanton particle Flavors ↔ Instanton particle (Masses) (Gauge coupling) (Masses ↔ Gauge coupling) (Weyl symmetry of) Enhanced symmetry E N f + 1 Again, Coulomb moduli parameter is also transformed!

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

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

Original Nekrasov partition function does not look manifestly E Nf+1 invariant because… 1. Coulomb moduli parameter is transformed.

Original Nekrasov partition function does not look manifestly E Nf+1 invariant because… 1. Coulomb moduli parameter is transformed. β 2. Expanded in terms of instanton factor q = e − 2 g 2

Original Nekrasov partition function does not look manifestly E Nf+1 invariant because… 1. Coulomb moduli parameter is transformed. β 2. Expanded in terms of instanton factor q = e − 2 g 2 2 e − β a 1. Use invariant variable instead of ˜ 8 − Nf e − β a A = q ✓ 1 g 2 → − 1 1 ◆ 4 e β a = e β ( a + 1 8 g 2 ) ˜ 1 a → a + A = q ( N f = 0) g 2 , 4 g 2 ,

Original Nekrasov partition function does not look manifestly E Nf+1 invariant because… 1. Coulomb moduli parameter is transformed. β 2. Expanded in terms of instanton factor q = e − 2 g 2 2 e − β a 1. Use invariant variable instead of ˜ 8 − Nf e − β a A = q ✓ 1 g 2 → − 1 1 ◆ 4 e β a = e β ( a + 1 8 g 2 ) ˜ 1 a → a + A = q ( N f = 0) g 2 , 4 g 2 , 2. Expand in terms of ˜ A

E Nf+1 invariant Nekrasov partition function ∞ X Z k ( a, m i ; ✏ 1 , ✏ 2 ) q k Z ( a, g, m i ; ✏ 1 , ✏ 2 ) = Z pert ( a, m i ; ✏ 1 , ✏ 2 ) k =0 Original form ’12 H-C Kim, S-S.Kim, K.Lee ’14 C.Hwang, J.Kim, S.Kim, J.Park

E Nf+1 invariant Nekrasov partition function ∞ X Z k ( a, m i ; ✏ 1 , ✏ 2 ) q k Z ( a, g, m i ; ✏ 1 , ✏ 2 ) = Z pert ( a, m i ; ✏ 1 , ✏ 2 ) k =0 Original form ’12 H-C Kim, S-S.Kim, K.Lee ’14 C.Hwang, J.Kim, S.Kim, J.Park ∞ Z n ( g, m i ; ✏ 1 , ✏ 2 ) ˜ ˜ X A k = New form n =0 ’14 C.Hwang, J.Kim, S.Kim, J.Park E Nf+1 invariant

E Nf+1 invariant Nekrasov partition function ∞ X Z k ( a, m i ; ✏ 1 , ✏ 2 ) q k Z ( a, g, m i ; ✏ 1 , ✏ 2 ) = Z pert ( a, m i ; ✏ 1 , ✏ 2 ) k =0 Original form ’12 H-C Kim, S-S.Kim, K.Lee ’14 C.Hwang, J.Kim, S.Kim, J.Park ∞ Z n ( g, m i ; ✏ 1 , ✏ 2 ) ˜ ˜ X A k = New form n =0 ’14 C.Hwang, J.Kim, S.Kim, J.Park E Nf+1 invariant " ∞ # ∞ 1 X X F n ( kg, km i ; k ✏ 1 , k ✏ 2 ) ˜ ˜ A nk = exp k n =1 k =0 " ∞ # F n ( g, m i ; ✏ 1 , ✏ 2 ) ˜ ˜ X A n ≡ PE n =1 E Nf+1 invariant

Nekrasov partition function for pure SU (2)  � q + t A 2 + O ( ˜ ˜ (1 − q )(1 − t ) χ E 1 A 4 ) Z = PE 2 2 + q − 1 1 χ E 1 2 , Character of E 1 = SU (2) : = q 2 q = e − �✏ 1 , t = e �✏ 2 1 ˜ 4 e β a , A = q Manifestly E 1 invariant!!

Nekrasov partition function for pure SU (2)  � q + t A 2 + O ( ˜ ˜ (1 − q )(1 − t ) χ E 1 A 4 ) Z = PE 2 1 2 + q − 1 χ E 1 2 , Character of E 1 = SU (2) : = q 2 q = e − �✏ 1 , t = e �✏ 2 1 ˜ 4 e β a , A = q Manifestly E 1 invariant!! W boson F1 D1 Instanton

Nekrasov partition function for N f =1 " 1 1 2 t q 2 ⇣ ⌘ χ SU (2) − 3 4 ˜ 7 + u 2 Z = PE ( u 1 ) u 2 A 7 − 2 (1 − q )(1 − t ) # q + t 1 A 2 + O ( ˜ (1 − q )(1 − t ) χ SU (2) 2 ˜ A 3 ) + ( u 1 ) u 7 2 − 1 1 2 e − 1 χ 2 ( u 1 ) = u 1 + u 1 4 β m SU (2) : u 1 = q E 2 = SU (2) × U (1) u 2 = q − 1 2 e − 7 4 β m U (1) : 2 ˜ 7 e − β a A = q

Nekrasov partition function for N f =1 " 1 1 2 t q 2 ⇣ ⌘ χ SU (2) − 3 4 ˜ 7 + u 2 Z = PE ( u 1 ) u 2 A 7 − 2 (1 − q )(1 − t ) # q + t 1 A 2 + O ( ˜ (1 − q )(1 − t ) χ SU (2) 2 ˜ A 3 ) + ( u 1 ) u 7 2 − 1 1 2 e − 1 χ 2 ( u 1 ) = u 1 + u 1 4 β m SU (2) : u 1 = q E 2 = SU (2) × U (1) u 2 = q − 1 2 e − 7 4 β m U (1) : 2 ˜ 7 e − β a A = q Hypermultiplet Vector multiplet Hypermultiplet

The vector multiplet and the hypermultiplet are included in the fundamental representation of E Nf+1 corresponding to the following nodes E E = SU ( 5 ) 8 (conjectured) 4 248 3875 10 5 E E SU ( 2 ) SU ( 3 ) 7 = × 3 2 56 133 3 3 E U (1) E U ( 1 ) SU ( 2 ) 6 = × 2 -3/7, 1/7 27 27 2 E = SO ( 10 ) 5 E = SU ( 2 ) 1 2 16 10

Gopakumar-Vafa’s expansion 2 3 j L j R M ( j L ,j R ) t k L + k R q k L − k R X X X X Q C C Z = PE 4 5 ( t − t − 1 )( q − q − 1 ) j L ,j R k L = − j L k R = − j R C ∈ H 2 ( X, Z ) X : Calabi-Yau manifold Q C = e − R C ω , ω : K¨ ahler form ( Q C = e − 2 β a , e − β ( a − m ) , q k e − 2 β a , · · · ) M ( j L ,j R ) : Refined Gopakumar-Vafa invariant C Gopakumar-Vafa ‘98 Iqbal, Kozcaz, Vafa ‘07 M ( j L ,j R ) Nekrasov partition function Set of integers C Non-Negative integer (After the convention change ˜ A → − ˜ A ) Consistent with the result from topological B-model [Huang, Klemm, Poretschkin ‘13]

Summary SO (2 N f ) × U (1) + S-duality = E N f +1 Nekrasov partition function is invariant Refined Gopakumar-Vafa invariants from Nekrasov partition function agrees with topological B-model computation