Refined geometric transition qq-characters & Hironori Mori - - PowerPoint PPT Presentation

2017/08/08, YITP Workshop “Strings and Fields 2017” @ YITP

HM, Y. Sugimoto (Osaka U.), Phys. Rev. D95 (2017) 026001, arXiv:1608.02849

• T. Kimura (Keio U.), HM, Y. Sugimoto (Osaka U.), arXiv:1705.03467

Hironori Mori (YITP, Kyoto U.)

qq-characters & Refined geometric transition

Quantum fields

SUSY, duality, …

Quantum integrable system

TBA, spin chain, lattice model, …

Quantum geometry

quantum spectral curve, …

qq-character

⇧

Motivation: understand the link of quantum theories

Quantum algebra

DIM algebra, , … Wq,t(g)

Quantum fields

SUSY, duality, …

Quantum integrable system

TBA, spin chain, lattice model, …

Quantum geometry

quantum spectral curve, …

qq-character

⇧

Quantum algebra

DIM algebra, , … Wq,t(g)

qq-character can be derived geometrically from the topological string theory

What we found

Contents

• 1. Y-operator & qq-character
• 2. Refined geometric transition
• 3. qq-character from refined geometric transition
• 4. Summary

ex) SU(N) gauge theory in 4d

Σ = {(x, y) ∈ C × C∗|H(x, y) = 0} H(x, y) = y + 1 y − TN(x) y(x) + 1 y(x) = TN(x)

⟹ Y-operator & qq-character

• Bi

λ = ∂F ∂ai ,

• Ai

λ = ai SW differential : λ = xd(log y)

• SW curve in 4d [Seiberg-Witten 1994]

: a degree-N polynomial TN(x)

y(x) = Y(x) = exp

• ∞
• n=1

On n x−n

• SW curve in 4d [Seiberg-Witten 1994]
• NS limit → quantization [Nekrasov-Shatashvili 2009] [Nekrasov-Pestun-Shatashvili 2013]

(1, 0)

y(x) + 1 y(x − 1) = TN(x; 1)

generating function of chiral ring operators y(x) + 1 y(x) = TN(x) for G = SU(N) SW differential : λ = xd(log y)

• Bi

= W ai = 1Z ⟹ Y-operator q-character “building block”

Y-operator & qq-character

• Key: Ω-deformation (1, 2)

• SW curve in 4d [Seiberg-Witten 1994]

y(x) + 1 y(x) = TN(x) for G = SU(N) ⟹

• generic → “double” quantization [Nekrasov 2015]

(1, 2)

• Y(x) +

1 Y(x − )

• = TN(x; 1, 2)

qq-character

• 5d/6d uplift [Kimura-Pestun 2015, 2016]
• Y(x) +

1 Y(q−1x)

• = TN(x; q1, q2)

q = q1q2, (q1, q2) = (e1, e2)

Y-operator & qq-character

= 1 + 2

• NS limit → quantization [Nekrasov-Shatashvili 2009] [Nekrasov-Pestun-Shatashvili 2013]

(1, 0)

• Key: Ω-deformation (1, 2)

Yk,µ(x) =

Nk

• α=1
• θ1(Qk,α/x)
• (i,j)∈µk,α

θ1(qi

1qj−1 2

Qk,α/x) θ1(qi−1

1

qj−1

2

Qk,α/x) θ1(qi−1

1

qj

2Qk,α/x)

θ1(qi

1qj 2Qk,α/x)

• R4

1,2 × T 2

Elliptic [Kimura-Pestun 2016]

→

k k − 1 k + 1

Y-operator & qq-character

• : building block for qq-character, cf. [Kimura 2016], Kimura’s talk tomorrow, Zhu’s poster

Y(x)

R4

1,2

→ Rational [Nekerasov-Pestun 2012] [Nekrasov 2015]

R4

1,2 × S1 → Trigonometric [Nekerasov-Pestun-Shatashvili 2013] [Kimura-Pestun 2015]

Contents

• 1. Y-operator & qq-character
• 2. Refined geometric transition
• 3. qq-character from refined geometric transition
• 4. Summary

⟺

dictionary Calabi-Yau Fivebrane web Refined topological vertex ⟷ Insertion of Lagrangian brane ⟷ Y-operator

Zclosed Zopen Zinst Y(x)

qq-character ⇒

expect

Refined topological string theory

Nekrasov partition function

NS5

χ(q1, q2)

Lagrangian brane D5 extra D-brane

⟹ We would like to evaluate to construct Y-operator. Zopen

Geometric transition (open/closed duality)

[Gopakumar-Vafa 1998]

Topological vertex

• pen (difficult)

closed (easy) unrefined refined

(1 = −2) (1 = 2)

[Awata-Kanno 2005] [Iqbal-Kozçaz-Vafa 2007]

???

[HM-Sugimoto in progress]

• cf. [Kameyama-Nawata 2017]

Cµνρ(q) Cµνρ(q1, q2) sµ(x) : Schur polynomial

[Aganagic-Klemm-Marinõ-Vafa 2003]

Refined geometric transition

• Geometric transition on the web diagram

⟺

k k + 1 k − 1 k k + 1 k − 1

Lagrangian brane

• µ

Zclosed

µ

Zopen

µ

• µ

˜ Zclosed

µ

Qk−1 Qk Qk+1

tuning Kähler parameters Q

=

Refined geometric transition

⟺

k k + 1 k − 1 k k + 1 k − 1

Lagrangian brane

Qk−1 Qk Qk+1

Q<k = 1 √q1q2 Q>k = √q1q2 ( in the unrefined limit) m, n ∈ Z, q1q2 = 1

• To remove unrelated factors to a Lag. brane attached to the -th line.
• To reproduce the closed string amplitude if no Lag. brane appears.
• To reproduce also the open string contribution in the unrefined limit.

k

• Our proposal [HM-Sugimoto 2016] [Kimura-HM-Sugimoto 2017]

Qk = qm

1 qn 2

√q1q2

Contents

• 1. Y-operator & qq-character
• 2. Refined geometric transition
• 3. qq-character from refined geometric transition
• 4. Summary

Refined topological vertex ⟶ ⟶ Y-operator

Zclosed Zopen Zinst Y(x)

qq-character ⇒

χ(q1, q2)

6d Nekrasov partition function Refined geometric transition D5 NS5

⤳ qq-character from refined geometric transition

[Kimura-HM-Sugimoto 2017]

6d theory on G = U(1) Γ = A1

1

R4

1,2 × T 2

qq-character from refined geometric transition ⟺ ⟺

• How to construct Y-operator

→ Geometric transition twice with specific parameter tuning

Q1 ˜ Q1

Q2 ˜ Q2

• µ

Zclosed

µ

Zopen

µ

Zopen

µ

• µ

ˇ Zclosed

µ

Zopen

µ

⟹ ⟹

Q2 = 1 √q1q2 , ˜ Q2 = q1 √q1q2 Q1 = 1 √q1q2 , ˜ Q1 = q−1

1

√q1q2

• µ

˜ Zclosed

µ

qq-character from refined geometric transition

Q2 = 1 √q1q2 , ˜ Q2 = q1 √q1q2 Q1 = 1 √q1q2 , ˜ Q1 = q−1

1

√q1q2

• µ

Zclosed

µ

Zopen

µ

Zopen

µ

Zopen

µ

=

• (i,j)∈µ

θ1(qi−1

1

qj

2Qx)

θ1(qi

1qj 2Qx)

Zopen

µ

=

• (i,j)∈µ

θ1(qi

1qj−1 2

Qx) θ1(qi−1

1

qj−1

2

Qx) ×θ1(Qx) =

by hand

• Y(x)
• are consistent with Y-operator

Q1 ˜ Q1 Q2 ˜ Q2

• How to construct Y-operator

→ Geometric transition twice with specific parameter tuning the number of difference of θ1(x) q1, q2

⟺

qq-character from refined geometric transition ⟺

Q2 = 1 √q1q2 , ˜ Q2 = q1 √q1q2 Q1 = 1 √q1q2 , ˜ Q1 = q−1

1

√q1q2 Q1 = q−1

1

√q1q2 , ˜ Q1 = 1 √q1q2 Q2 = q1 √q1q2 , ˜ Q2 = 1 √q1q2

⟺

• 1

Y(q−1x)

• Y(x)
• Q1

˜ Q1 Q2 ˜ Q2

• How to construct Y-operator

→ Geometric transition twice with specific parameter tuning

qq-character from refined geometric transition

• 1

Y(q−1x)

• Y(x)
• How to realize qq-character

+qP(x) =

• T(x; q1, q2)
• q

P(x) : gauge coupling : matter contribution

• Γ = A1

⃝ ⟶ Γ = An ⎯ ⃝⎯ ⃝⎯ ⃝⎯

• fundamental rep. of Γ ⟶ higher rank rep. of Γ

• 1. Refined geometric transition
• 2. qq-character

→ propose new prescription for the refined version of geometric transition → provide how to construct Y-operators via refined geometric transition

• Y(x)
• ⟶

Summary

• Relation to supergroup Chern-Simons theory?
• Extension to DE-type quiver, cf. [Hayashi-Ohmori 2017] & Zhu’s poster
• Towards quantum/elliptic integrable models

⟶ brane, ⟶ anti-brane [Vafa 2001] [Mikhaylov-Witten 2014] Outlooks

Auxiliary part

Quiver gauge theory G : gauge group Γ : quiver shape qq-character AGT W(G)-algebra W(Γ)-algebra

⇧

“dual” AGT

BPS/CFT correspondence [Nekrasov 2004]

QFTs with 8 supercharges in 4d/5d/6d CFTs & Integrable systems in 2d

⟺

• Statement
• Gauge/quiver duality

• Why qq-“character”?
• interpreted as a generating current of W-algebra. [Kimura-Pestun 2015, 2016]
• Y(x) +

1 Y(q−1x)

• = TN(x; q1, q2)

G = SU(N)

⟺

Γ = A1

N

BPS/CFT

T(x) =

• n∈Z

Tnx−n ⇒ give the defining commutation relation of ⇒ quantum/elliptic deformed W-algebra. Tn ⟷ character for the fund. rep. of A1 ⟷ weight of A1 y y + 1 y = χ

Y-operator & qq-character

• describes the “double” quantization of the SW geometry.

Y(x) + 1 Y(q−1x) =: T(x) The free field realization of & Y(x)

• cf. q-character for finite dim. rep. of the quantum algebra [Frenkel-Reshetikhin 1998]

• Y(x) +

1 Y(q−1x)

• = T(x; q1, q2)

[Tn, Tm] = −

∞

• k=1

fk (Tn−kTm+k − Tm−kTn+k)−(1 − q1)(1 − q2) 1 − q (qn−q−n)δn+m,0

∞

• k=0

fkxk = exp ∞

• n=1

(1 − qn

1 )(1 − qn 2 )

1 + qn xn n

• ⟷ weight of Γ

Y(x) T(x; q1, q2) ⟷ generating current of W(Γ)-algebra

• ex. 5d SU(2) ⟺ q-Virasoro algebra
• What are Y- and T-operator in an associated Lie algebra?

T(x; q1, q2) =

• n∈Z

Tnx−n

Y-operator & qq-character

k k − 1

k + 1 Qk

Q>k = 1 √q1q2 Q<k = 1 √q1q2 Qk = qm

1 qn 2

√q1q2 Q<k = 1 √q1q2 Qk = qm

1 qn 2

√q1q2 Q>k = √q1q2 Our proposal

⟺ ⟺

[Dimofte-Gukov-Hollands 2010] [Taki 2010]

+ suitable shift of Kähler parameter by hand

The preferred direction does affect the open string sector

qq-character from refined geometric transition

• 1

Y(q−1x)

• Y(x)
• How to realize qq-character

+qP(x) =

• T(x; q1, q2)
• q

P(x) : gauge coupling : matter contribution ↑ poles ↑ poles ↑ No pole! The regularity of qq-character can be checked.