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: G i i T. Kimura (Keio U) August 2017 @ YITP 1 / 15

Quiver gauge theory � Gauge theory with several gauge groups: G i i 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: G i i Their interaction depicted by quiver diagram node: vector edge: bifund A 3 quiver D 4 quiver Analogy with Dynkin diagram T. Kimura (Keio U) August 2017 @ YITP 1 / 15

Quiver gauge theory � Gauge theory with several gauge groups: G i i Their interaction depicted by quiver diagram node: vector edge: bifund � � A 3 quiver D 4 quiver Analogy with Dynkin diagram T. Kimura (Keio U) August 2017 @ YITP 1 / 15

Non-simply-laced quiver BC 2 quiver: T. Kimura (Keio U) August 2017 @ YITP 2 / 15

Non-simply-laced quiver BC 2 quiver: What’s the doubled arrow ? from long to short root � A 1 quiver: (2 bifunds; no “orientation”) T. Kimura (Keio U) August 2017 @ YITP 2 / 15

Non-simply-laced quiver BC 2 quiver: What’s the doubled arrow ? from long to short root � A 1 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 R 4 ǫ 1 , 2 × S 1 Ω -background (equivariant) parameters: ( q 1 , q 2 ) = ( e ǫ 1 , e ǫ 2 ) q -deformed W-algebra: W (Γ) q 1 ,q 2 Langlands dual W (Γ) q 1 ,q 2 = W ( L Γ) q 2 ,q 1 simply-laced: Γ = L Γ / non-simply-laced: Γ � = L Γ ( ǫ 1 , ǫ 2 ) 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: d i ∈ Z > 0 for i ∈ { nodes in Γ } Γ is simply-laced, if d i = d 0 ∀ i . d ij := gcd( d i , d j ) Instanton counting (partition function) e : i → j : ( q 1 , q d ij Vec i : ( q 1 , q d i Hyp bf 2 ) & 2 ) SO(4) rotation charge depends on the node ( e ǫ 1 , e d i ǫ 2 ) ∈ 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): x i,α,k = q d i λ i,α,k q k − 1 X i = { x i,α,k } α ∈ [1 ,...,n i ] , k ∈ [1 ,..., ∞ ] ν i,α 2 1 � Gauge group: U( n i ) i ∈{ node } Partition: ( λ i,α ) = ( λ i,α, 1 , λ i,α, 2 , . . . ) Ω -background parameter: ( q 1 , q 2 ) = ( e ǫ 1 , e ǫ 2 ) Coulomb moduli: ν i,α = e a i,α T. Kimura (Keio U) August 2017 @ YITP 6 / 15

Partition function Vector multiplet: � � � d i ( λ i,α,k − λ i,β,k ′ +1) q k − k ′ β ; q d i Z vec ν α ν − 1 = q 1 q i 2 1 2 ∞ ( α,k ) � =( β,k ′ ) � � − 1 d i ( λ i,α,k − λ i,β,k ′ +1) q k − k ′ β ; q d i ν α ν − 1 × q 2 1 2 ∞ � � � � − 1 � x x q 1 q d i x ′ ; q d i q d i x ′ ; q d i = 2 2 2 2 ∞ ∞ ( x,x ′ ) ∈X 2 i = Z vec ( X i ; q 1 , q d i 2 ) ( q 1 , q d i Replacement : ( q 1 , q 2 ) 2 ) for i ∈ { nodes } T. Kimura (Keio U) August 2017 @ YITP 7 / 15

Partition function Bifund hyper: � � − 1 � � � x x e q 1 q d ij x ′ ; q d ij e q d ij x ′ ; q d ij Z bf µ − 1 µ − 1 e : i → j = 2 2 2 2 ∞ ∞ ( x,x ′ ) ∈X i ×X j d i /d ij − 1 � � − 1 � � � � x x e q − rd ij e q − rd ij q 1 q d i x ′ ; q d i q d i x ′ ; q d i µ − 1 µ − 1 = 2 2 2 2 2 2 ∞ ∞ r =0 ( x,x ′ ) ∈X i ×X j d i /d ij − 1 � Z bf ( µ e q rd ij ; X i , X j ; q 1 , q d i = 2 ) 2 r =0 Multiplication { µ e , µ e q d ij 2 , µ e q 2 d ij , . . . , µ e q d i − d ij Bifund mass: { µ e } } 2 2 � � { m e } { m e , m e + ǫ 2 d ij , . . . , m e + ǫ 2 ( d i − d ij ) } T. Kimura (Keio U) August 2017 @ YITP 8 / 15

Example BC 2 quiver: Root parameter: ( d 1 , d 2 ) = (2 , 1) with d 12 = 1 Vector multiplet Z vec = Z vec ( X 1 ; q 1 , q 2 2 ) & Z vec = Z vec ( X 2 ; q 1 , q 2 ) 1 2 Bifund hypermultiplet Z bf 1 → 2 = Z bf ( µ ; X 1 , X 2 ; q 1 , q 2 2 ) Z bf ( µq 2 ; X 1 , X 2 ; q 1 , q 2 2 ) Z bf 2 → 1 = Z bf ( µ − 1 q 1 q 2 ; X 2 , X 1 ; q 1 , q 2 ) cf. B-type quiver in 3d [Dey–Hanany–Koroteev–Mekareeya] T. Kimura (Keio U) August 2017 @ YITP 9 / 15

Example A (2) quiver: 1 Root parameter: ( d 1 , d 2 ) = (4 , 1) with d 12 = 1 Vector multiplet Z vec = Z vec ( X 1 ; q 1 , q 4 2 ) & Z vec = Z vec ( X 2 ; q 1 , q 2 ) 1 2 Bifund hypermultiplet 4 � Z bf ( µq r − 1 Z bf ; X 1 , X 2 ; q 1 , q 4 1 → 2 = 2 ) 2 r =1 Z bf 2 → 1 = Z bf ( µ − 1 q 1 q 2 ; X 2 , X 1 ; q 1 , q 2 ) T. Kimura (Keio U) August 2017 @ YITP 10 / 15

Non-perturbative aspects: Seiberg–Witten geometry Coulomb branch of 4d N = 2 theory: � � ∂ F a i = λ and = λ ∂a i A i B i Contour integral on Seiberg–Witten curve λ = xdy Σ = { ( x, y ) ∈ C × C ∗ | H ( x, y ) = 0 } with y U( n ) SYM theory: H ( x, y ) = y + y − 1 − ( x n + · · · ) � � y + y − 1 = x n + · · · =: T n ( x ) Σ = Prepotential: F = lim ǫ 1 , 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 d i ∈ Z > 0 for i ∈ { nodes } Instanton counting with ( q 1 , q d i 2 ) Multiplication of bifund hyper (doubled arrow) Non-perturbative tests: (quantum) Seiberg–Witten geometry T. Kimura (Keio U) August 2017 @ YITP 15 / 15