Non-simply-laced quiver gauge theory from -background Taro Kimura - - PowerPoint PPT Presentation

Non-simply-laced quiver gauge theory from Ω-background

Taro Kimura ♣ 木村太郎

Keio University ♦ 慶應義塾大学

Collaboration with V. Pestun (IH´ ES): [arXiv:1705.04410]

See also [arXiv:1512.08533] [arXiv:1608.04651]

• T. Kimura (Keio U)

August 2017 @ YITP 0 / 15

Quiver gauge theory

Gauge theory with several gauge groups:

• i

Gi

• T. Kimura (Keio U)

August 2017 @ YITP 1 / 15

Quiver gauge theory

Gauge theory with several gauge groups:

• i

Gi Their interaction depicted by quiver diagram

node: vector edge: bifund

• T. Kimura (Keio U)

August 2017 @ YITP 1 / 15

Quiver gauge theory

Gauge theory with several gauge groups:

• i

Gi Their interaction depicted by quiver diagram

node: vector edge: bifund

A3 quiver D4 quiver Analogy with Dynkin diagram

• T. Kimura (Keio U)

August 2017 @ YITP 1 / 15

Quiver gauge theory

Gauge theory with several gauge groups:

• i

Gi Their interaction depicted by quiver diagram

node: vector edge: bifund

• A3 quiver
• D4 quiver

Analogy with Dynkin diagram

• T. Kimura (Keio U)

August 2017 @ YITP 1 / 15

Non-simply-laced quiver

BC2 quiver:

• T. Kimura (Keio U)

August 2017 @ YITP 2 / 15

Non-simply-laced quiver

BC2 quiver: What’s the doubled arrow?

from long to short root

• A1 quiver:

(2 bifunds; no “orientation”)

• T. Kimura (Keio U)

August 2017 @ YITP 2 / 15

Non-simply-laced quiver

BC2 quiver: What’s the doubled arrow?

from long to short root

• A1 quiver:

(2 bifunds; no “orientation”)

What’s the root length in gauge theory?

• T. Kimura (Keio U)

August 2017 @ YITP 2 / 15

Key idea

Quiver W-algebra

Γ-quiver gauge theory W(Γ)-algebra

[TK–Pestun]

AGT: G-gauge theory W(G)-algebra where G = ABCDEFG (finite type)

[Keller–Mekareeya–Song–Tachikawa]

• T. Kimura (Keio U)

August 2017 @ YITP 3 / 15

Key idea

Quiver W-algebra

Γ-quiver gauge theory W(Γ)-algebra

[TK–Pestun]

AGT: G-gauge theory W(G)-algebra where G = ABCDEFG (finite type)

[Keller–Mekareeya–Song–Tachikawa]

Non-simply-laced W-algebra to quiver gauge theory

(q-def of) W(Γ)-algebra for Γ = ADE

[Frenkel–Reshetikhin]

Non-simply-laced Γ-quiver gauge theory

• T. Kimura (Keio U)

August 2017 @ YITP 3 / 15

Γ-quiver gauge theory on R4

ǫ1,2 × S1

Ω-background (equivariant) parameters: (q1, q2) = (eǫ1, eǫ2) q-deformed W-algebra: W(Γ)q1,q2

Langlands dual

W(Γ)q1,q2 = W(LΓ)q2,q1 simply-laced: Γ = LΓ / non-simply-laced: Γ = LΓ (ǫ1, ǫ2) are not equivalent for non-simply-laced quiver

• T. Kimura (Keio U)

August 2017 @ YITP 4 / 15

Non-simply-laced quiver gauge theory: definition

“Root” parameter: di ∈ Z>0 for i ∈ {nodes in Γ}

Γ is simply-laced, if di = d0 ∀i. dij := gcd(di, dj)

Instanton counting (partition function)

Veci: (q1, qdi

2 )

& Hypbf

e:i→j: (q1, qdij 2 )

SO(4) rotation charge depends on the node (eǫ1, ediǫ2) ∈ U(1) × U(1) ⊂ SU(2) × SU(2) = SO(4)

• T. Kimura (Keio U)

August 2017 @ YITP 5 / 15

Partition function

Configuration (instanton mod sp fixed pt): Xi = {xi,α,k}α∈[1,...,ni], k∈[1,...,∞] xi,α,k = qdiλi,α,k

2

qk−1

1

νi,α

Gauge group:

• i∈{node}

U(ni) Partition: (λi,α) = (λi,α,1, λi,α,2, . . .) Ω-background parameter: (q1, q2) = (eǫ1, eǫ2) Coulomb moduli: νi,α = eai,α

• T. Kimura (Keio U)

August 2017 @ YITP 6 / 15

Partition function

Vector multiplet: Zvec

i

=

• (α,k)=(β,k′)
• q1q

di(λi,α,k−λi,β,k′+1) 2

qk−k′

1

ναν−1

β ; qdi 2

• ∞

×

• q

di(λi,α,k−λi,β,k′+1) 2

qk−k′

1

ναν−1

β ; qdi 2

−1

∞

=

• (x,x′)∈X 2

i

• q1qdi

2

x x′ ; qdi

2

• ∞
• qdi

2

x x′ ; qdi

2

−1

∞

= Zvec(Xi; q1, qdi

2 )

Replacement: (q1, q2) (q1, qdi

2 ) for i ∈ {nodes}

• T. Kimura (Keio U)

August 2017 @ YITP 7 / 15

Partition function

Bifund hyper: Zbf

e:i→j =

• (x,x′)∈Xi×Xj
• µ−1

e q1qdij 2

x x′ ; qdij

2

−1

∞

• µ−1

e qdij 2

x x′ ; qdij

2

• ∞

=

• (x,x′)∈Xi×Xj

di/dij−1

• r=0
• µ−1

e q−rdij 2

q1qdi

2

x x′ ; qdi

2

−1

∞

• µ−1

e q−rdij 2

qdi

2

x x′ ; qdi

2

• ∞

=

di/dij−1

• r=0

Zbf(µeqrdij

2

; Xi, Xj; q1, qdi

2 )

Multiplication

Bifund mass: {µe} {µe, µeqdij

2 , µeq2dij 2

, . . . , µeqdi−dij

2

}

• {me}

{me, me + ǫ2dij, . . . , me + ǫ2(di − dij)}

• T. Kimura (Keio U)

August 2017 @ YITP 8 / 15

Example

BC2 quiver:

Root parameter: (d1, d2) = (2, 1) with d12 = 1

Vector multiplet

Zvec

1

= Zvec(X1; q1, q2

2) & Zvec 2

= Zvec(X2; q1, q2)

Bifund hypermultiplet

Zbf

1→2 = Zbf(µ; X1, X2; q1, q2 2)Zbf(µq2; X1, X2; q1, q2 2)

Zbf

2→1 = Zbf(µ−1q1q2; X2, X1; q1, q2)

• cf. B-type quiver in 3d

[Dey–Hanany–Koroteev–Mekareeya]

• T. Kimura (Keio U)

August 2017 @ YITP 9 / 15

Example

A(2)

1

quiver:

Root parameter: (d1, d2) = (4, 1) with d12 = 1

Vector multiplet

Zvec

1

= Zvec(X1; q1, q4

2) & Zvec 2

= Zvec(X2; q1, q2)

Bifund hypermultiplet

Zbf

1→2 = 4

• r=1

Zbf(µqr−1

2

; X1, X2; q1, q4

2)

Zbf

2→1 = Zbf(µ−1q1q2; X2, X1; q1, q2)

• T. Kimura (Keio U)

August 2017 @ YITP 10 / 15

Non-perturbative aspects: Seiberg–Witten geometry

Coulomb branch of 4d N = 2 theory: ai =

• Ai

λ and ∂F ∂ai =

• Bi

λ

Contour integral on Seiberg–Witten curve Σ = {(x, y) ∈ C × C∗ | H(x, y) = 0} with λ = xdy y U(n) SYM theory: H(x, y) = y + y−1 − (xn + · · · ) Σ =

• y + y−1 = xn + · · · =: Tn(x)
• Prepotential: F = lim

ǫ1,2→0 log Z

• T. Kimura (Keio U)

August 2017 @ YITP 11 / 15

Seiberg–Witten geometry for ADE-quiver gauge theory

determined by the fundamental characters of ADE-group

[Nekrasov–Pestun]

• T. Kimura (Keio U)

August 2017 @ YITP 12 / 15

Seiberg–Witten geometry for ADE-quiver gauge theory

determined by the fundamental characters of ADE-group

[Nekrasov–Pestun]

Seiberg–Witten geometry for Γ-quiver gauge theory

determined by the fundamental characters of Γ-group

[TK–Pestun]

• T. Kimura (Keio U)

August 2017 @ YITP 12 / 15

Quantum Seiberg–Witten geometry for ADE-quiver

determined by the fundamental q-characters of ADE-group

[Nekrasov–Pestun–Shatashvili]

Quantum Seiberg–Witten geometry for Γ-quiver

determined by the fundamental q-characters of Γ-group

[TK–Pestun]

• T. Kimura (Keio U)

August 2017 @ YITP 13 / 15

Doubly quantum Seiberg–Witten geometry for ADE-quiver

determined by the fundamental qq-characters of ADE-group

[Nekrasov] [TK–Pestun]

Doubly quantum Seiberg–Witten geometry for Γ-quiver

determined by the fundamental qq-characters of Γ-group

[TK–Pestun]

• T. Kimura (Keio U)

August 2017 @ YITP 14 / 15

Summary

Non-simply-laced quiver gauge theory

“Root” parameter di ∈ Z>0 for i ∈ {nodes} Instanton counting with (q1, qdi

2 )

Multiplication of bifund hyper (doubled arrow) Non-perturbative tests: (quantum) Seiberg–Witten geometry

• T. Kimura (Keio U)

August 2017 @ YITP 15 / 15