Refined geometric transition qq-characters & Hironori Mori (YITP, Kyoto U.) 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 2017/08/08, YITP Workshop “Strings and Fields 2017” @ YITP
Motivation: understand the link of quantum theories Quantum integrable system TBA, spin chain, lattice model, … Quantum fields SUSY, duality, … Quantum geometry Quantum algebra quantum spectral curve, … DIM algebra, , … W q,t ( g ) ⇧ qq-character
What we found Quantum integrable system TBA, spin chain, lattice model, … qq-character can be derived geometrically Quantum fields from the topological string theory SUSY, duality, … Quantum geometry Quantum algebra quantum spectral curve, … DIM algebra, , … W q,t ( g ) ⇧ qq-character
Contents 1. Y-operator & qq-character 2. Refined geometric transition 3. qq-character from refined geometric transition 4. Summary
Y-operator & qq-character • SW curve in 4d [Seiberg-Witten 1994] Σ = { ( x, y ) ∈ C × C ∗ | H ( x, y ) = 0 } λ = ∂ F � � SW di ff erential : λ = x d(log y ) λ = a i , ∂ a i B i A i ex) SU(N) gauge theory in 4d H ( x, y ) = y + 1 1 ⟹ y ( x ) + y ( x ) = T N ( x ) y − T N ( x ) : a degree-N polynomial T N ( x )
Y-operator & qq-character 1 • SW curve in 4d [Seiberg-Witten 1994] y ( x ) = T N ( x ) for G = SU(N) y ( x ) + • Key: Ω -deformation ( � 1 , � 2 ) - NS limit → quantization [Nekrasov-Shatashvili 2009] [Nekrasov-Pestun-Shatashvili 2013] ( � 1 , 0) � = � W � SW di ff erential : λ = x d(log y ) = � 1 Z � a i B i 1 y ( x ) + y ( x − � 1 ) = T N ( x ; � 1 ) ⟹ q-character � � ∞ generating function of � O n � � y ( x ) = � Y ( x ) � = exp � x − n chiral ring operators n n =1 Y-operator “building block”
Y-operator & qq-character 1 • SW curve in 4d [Seiberg-Witten 1994] y ( x ) = T N ( x ) for G = SU(N) y ( x ) + • Key: Ω -deformation ( � 1 , � 2 ) - NS limit → quantization [Nekrasov-Shatashvili 2009] [Nekrasov-Pestun-Shatashvili 2013] ( � 1 , 0) - generic → “double” quantization [Nekrasov 2015] ( � 1 , � 2 ) � = � 1 + � 2 � � 1 Y ( x ) + = T N ( x ; � 1 , � 2 ) ⟹ Y ( x − � ) qq-character • 5d/6d uplift [Kimura-Pestun 2015, 2016] q = q 1 q 2 , ( q 1 , q 2 ) = ( e � 1 , e � 2 ) � � 1 Y ( x ) + = T N ( x ; q 1 , q 2 ) Y ( q − 1 x )
Y-operator & qq-character • : building block for qq-character, cf. [Kimura 2016], Kimura’s talk tomorrow, Zhu’s poster Y ( x ) → Rational [Nekerasov-Pestun 2012] [Nekrasov 2015] R 4 � 1 , � 2 � 1 , � 2 × S 1 → Trigonometric [Nekerasov-Pestun-Shatashvili 2013] [Kimura-Pestun 2015] R 4 → R 4 � 1 , � 2 × T 2 Elliptic [Kimura-Pestun 2016] � � N k 1 q j − 1 q j θ 1 ( q i − 1 θ 1 ( q i Q k, α /x ) 2 Q k, α /x ) � � 2 1 Y k,µ ( x ) = � θ 1 ( Q k, α /x ) � q j − 1 θ 1 ( q i − 1 1 q j θ 1 ( q i Q k, α /x ) 2 Q k, α /x ) 1 2 α =1 ( i,j ) ∈ µ k, α k + 1 k − 1 k
Contents 1. Y-operator & qq-character 2. Refined geometric transition 3. qq-character from refined geometric transition 4. Summary
⇒ Refined topological string theory NS5 dictionary ⟺ D5 Lagrangian brane extra D-brane Calabi-Yau Fivebrane web Refined topological vertex ⟷ Nekrasov partition function Z closed Z inst expect Insertion of Lagrangian brane ⟷ Y-operator qq-character Y ( x ) χ � ( q 1 , q 2 ) Z open ⟹ We would like to evaluate to construct Y-operator. Z open
Topological vertex closed (easy) open (di ffi cult) unrefined C µ νρ ( q ) s µ ( x ) : Schur polynomial ( � 1 = − � 2 ) [Aganagic-Klemm-Marinõ-Vafa 2003] ??? refined C µ νρ ( q 1 , q 2 ) ( � 1 � = � 2 ) [Awata-Kanno 2005] [HM-Sugimoto in progress] [Iqbal-Kozçaz-Vafa 2007] cf. [Kameyama-Nawata 2017] Geometric transition (open/closed duality) [Gopakumar-Vafa 1998]
Refined geometric transition • Geometric transition on the web diagram k + 1 k + 1 k k k − 1 k − 1 ⟺ Q k +1 Q k Lagrangian brane Q k − 1 = � � Z closed ˜ Z closed Z open µ µ µ µ µ tuning Kähler parameters Q �
Refined geometric transition • Our proposal [HM-Sugimoto 2016] [Kimura-HM-Sugimoto 2017] k + 1 k + 1 k k k − 1 k − 1 ⟺ Q k +1 Q k Lagrangian brane Q k − 1 Q k = q m 1 q n 1 2 Q � <k = Q � >k = √ q 1 q 2 √ q 1 q 2 √ q 1 q 2 ( in the unrefined limit) m, n ∈ Z , q 1 q 2 = 1 - To remove unrelated factors to a Lag. brane attached to the -th line. k - To reproduce the closed string amplitude if no Lag. brane appears. - To reproduce also the open string contribution in the unrefined limit.
Contents 1. Y-operator & qq-character 2. Refined geometric transition 3. qq-character from refined geometric transition 4. Summary
⇒ qq-character from refined geometric transition NS5 6d theory on R 4 � 1 , � 2 × T 2 ⤳ G = U(1) 1 D5 Γ = A 1 Refined topological vertex ⟶ 6d Nekrasov partition function Z closed Z inst [Kimura-HM-Sugimoto 2017] Refined geometric transition ⟶ Y-operator qq-character Y ( x ) χ � ( q 1 , q 2 ) Z open
qq-character from refined geometric transition • How to construct Y-operator → Geometric transition twice with specific parameter tuning ⟺ ⟺ ˜ Q 2 Q 2 ˜ Q 1 Q 1 q − 1 q 1 1 1 ˜ ˜ 1 Q 2 = , Q 2 = Q 1 = , Q 1 = √ q 1 q 2 √ q 1 q 2 √ q 1 q 2 √ q 1 q 2 ⟹ ⟹ � � � ˜ Z closed Z closed ˇ Z open Z closed Z open Z open µ µ µ µ µ µ µ µ µ
qq-character from refined geometric transition • How to construct Y-operator → Geometric transition twice with specific parameter tuning q 1 1 ˜ Q 2 = , Q 2 = √ q 1 q 2 √ q 1 q 2 ⟺ q − 1 1 ˜ 1 Q 1 = , Q 1 = √ q 1 q 2 √ q 1 q 2 ˜ Q 2 Q 2 � � � Z closed Z open Z open Y ( x ) × θ 1 ( Q x ) = µ µ µ by hand ˜ µ Q 1 Q 1 θ 1 ( q i − 1 q j 2 Q x ) � Z open 1 = the number of µ θ 1 ( x ) 1 q j θ 1 ( q i 2 Q x ) ( i,j ) ∈ µ di ff erence of q 1 , q 2 1 q j − 1 θ 1 ( q i Q x ) are consistent with Y-operator � Z open 2 = µ q j − 1 θ 1 ( q i − 1 Q x ) 1 2 ( i,j ) ∈ µ
qq-character from refined geometric transition • How to construct Y-operator → Geometric transition twice with specific parameter tuning q 1 1 ˜ Q 2 = , Q 2 = √ q 1 q 2 √ q 1 q 2 ⟺ q − 1 1 ˜ 1 Q 1 = , Q 1 = √ q 1 q 2 √ q 1 q 2 � � Y ( x ) ˜ Q 2 Q 2 ˜ Q 1 q 1 1 ˜ Q 2 = , Q 2 = Q 1 √ q 1 q 2 √ q 1 q 2 ⟺ q − 1 1 ˜ 1 Q 1 = , Q 1 = √ q 1 q 2 √ q 1 q 2 � � 1 Y ( q − 1 x )
qq-character from refined geometric transition • How to realize qq-character : gauge coupling q : matter contribution P ( x ) � � 1 � � � � Y ( x ) + q P ( x ) = T ( x ; q 1 , q 2 ) Y ( q − 1 x ) • Γ = A 1 ⃝ ⟶ Γ = A n ⎯ ⃝ ⎯ ⃝ ⎯ ⃝ ⎯ • fundamental rep. of Γ ⟶ higher rank rep. of Γ
Summary 1. Refined geometric transition → propose new prescription for the refined version of geometric transition 2. qq-character → provide how to construct Y-operators via refined geometric transition � � Y ( x ) ⟶ Outlooks • Relation to supergroup Chern-Simons theory? ⟶ brane, ⟶ anti-brane [Vafa 2001] [Mikhaylov-Witten 2014] • Extension to DE-type quiver, cf. [Hayashi-Ohmori 2017] & Zhu’s poster • Towards quantum/elliptic integrable models
Auxiliary part
BPS/CFT correspondence [Nekrasov 2004] • Statement QFTs with 8 supercharges CFTs & Integrable systems ⟺ in 4d/5d/6d in 2d • Gauge/quiver duality Quiver gauge theory G : gauge group Γ : quiver shape AGT “dual” AGT W(G)-algebra W( Γ )-algebra ⇧ qq-character
Y-operator & qq-character � � 1 • Why qq-“character”? Y ( x ) + = T N ( x ; q 1 , q 2 ) Y ( q − 1 x ) BPS/CFT ⟷ weight of A 1 y ⟺ N y + 1 ⟷ character for the fund. rep. of A 1 y = χ � G = SU(N) Γ = A 1 cf. q-character for finite dim. rep. of the quantum algebra [Frenkel-Reshetikhin 1998] • describes the “double” quantization of the SW geometry. • interpreted as a generating current of W-algebra. [Kimura-Pestun 2015, 2016] 1 The free field realization of & Y ( x ) Y ( x ) + Y ( q − 1 x ) =: T ( x ) � T n x − n ⇒ give the defining commutation relation of T ( x ) = T n n ∈ Z ⇒ quantum/elliptic deformed W-algebra.
Y-operator & qq-character • What are Y- and T-operator in an associated Lie algebra? � � 1 Y ( x ) + = � T ( x ; q 1 , q 2 ) � Y ( q − 1 x ) Y ( x ) ⟷ weight of Γ T ( x ; q 1 , q 2 ) ⟷ generating current of W( Γ )-algebra � ex. 5d SU(2) ⟺ q-Virasoro algebra T n x − n T ( x ; q 1 , q 2 ) = n ∈ Z ∞ f k ( T n − k T m + k − T m − k T n + k ) − (1 − q 1 )(1 − q 2 ) � ( q n − q − n ) δ n + m, 0 [ T n , T m ] = − 1 − q k =1 � ∞ � ∞ (1 − q n 1 )(1 − q n x n 2 ) f k x k = exp � � 1 + q n n n =1 k =0
k + 1 k k − 1 ⟺ Our proposal Q k Q k = q m 1 q n 1 2 Q � <k = Q � >k = √ q 1 q 2 √ q 1 q 2 √ q 1 q 2 + suitable shift of Kähler parameter by hand ⟺ [Dimofte-Gukov-Hollands 2010] [Taki 2010] Q k = q m 1 q n 1 1 2 Q � <k = Q � >k = √ q 1 q 2 √ q 1 q 2 √ q 1 q 2 The preferred direction does a ff ect the open string sector
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