♦♦♦ Random variables: Young diagrams λ (1) , . . . , λ ( n )
♦♦♦ Random variables: | λ | colored points z α, i , j In the growth picture z α, i , j = a α + ε ′ ( i − 1) + ε ′′ ( j − 1) 1 ≤ j ≤ λ ( α ) 1 ≤ i ≤ ℓ λ ( α ) , i blue α = 1 , . . . , n
♦♦♦ � λ (1) , . . . , λ ( n ) � Random variables: n partitions Λ = The role of the fugacity: extra weight ℓ λ ( α ) n � � q λ ( α ) q | Λ | = i α =1 i =1
♦♦♦ Boltzmann weights Energy of charges � c ) · ( u ε ′ ,ε ′′ ( z c − z ˜ ε ′ ) ) e − ch ( c ) ch (˜ c ) − u ε ′ ,ε ′′ ( z c − z ˜ c +˜ µ Λ = c ∈ ∂ Λ , ˜ c ∈ ∂ Λ with the charges ch ( c ) = ± 1 as in the pictures
♦♦♦ Boltzmann weights Energy of charges � c ) · ( u ε ′ ,ε ′′ ( z c − z ˜ ε ′ ) ) e − ch ( c ) ch (˜ c ) − u ε ′ ,ε ′′ ( z c − z ˜ c +˜ µ Λ = c ∈ ∂ Λ , ˜ c ∈ ∂ Λ with the charges ch ( c ) = ± 1 as in the pictures ch ( c ) = +1 for new growth points ch ( c ) = − 1 for decay points
The potential � � ∞ � e − tz 1 u ε ′ ,ε ′′ ( z ) = d dt � t t s � (1 − e t ε ′ )(1 − e t ε ′′ ) � Γ( s ) ds 0 s =0 solves u ε ′ ,ε ′′ ( z ) + u ε ′ ,ε ′′ � z − ε ′ − ε ′′ � − u ε ′ ,ε ′′ � z − ε ′ � − u ε ′ ,ε ′′ � z − ε ′′ � = log ( z ) (3) ♦
♦♦♦ Boltzmann weights Energy of charges � c ) · ( u ε ′ ,ε ′′ ( z c − z ˜ ε ′ ) ) e − ch ( c ) ch (˜ c ) − u ε ′ ,ε ′′ ( z c − z ˜ c +˜ µ Λ = c ∈ ∂ Λ , ˜ c ∈ ∂ Λ � A 0 -type model
♦♦♦ Boltzmann weights: generalization Graph γ with the set of vertices Vert γ and the set of edges Edge γ � λ ( α ) � α =1 ... n � λ ( i ,α ) � α =1 ,..., n i n → ( n i ) i ∈ Vert γ , Λ = → (Λ i ) i ∈ Vert γ = i ∈ Vert γ Assignment of masses m e ∈ C to the edges e ∈ Edge γ � � e − ch ( c ) ch (˜ c ) u ε ′ ,ε ′′ ( z c − z ˜ c ) µ λ = × i ∈ Vert γ c ∈ ∂λ i , ˜ c ∈ ∂λ i � � e ch ( c ) ch (˜ c ) u ε ′ ,ε ′′ ( z c − z ˜ c + m e ) × e ∈ Edge γ c ∈ ∂λ s ( e ) , ˜ c ∈ ∂λ t ( e ) This is a γ -quiver model
♦♦♦ When γ is of the � ADE type � � c ) × e − ch ( c ) ch (˜ c ) u ε ′ ,ε ′′ ( z c − z ˜ µ λ = i ∈ Vert γ c ∈ ∂λ i , ˜ c ∈ ∂λ i � � e ch ( c ) ch (˜ c ) u ε ′ ,ε ′′ ( z c − z ˜ c + m e ) × (4) e ∈ Edge γ c ∈ ∂λ s ( e ) , ˜ c ∈ ∂λ t ( e ) The γ ADE -quiver model can be obtained from the � A 0 model by “orbifolding”
♦♦♦ When γ is of the � ADE type � � c ) × e − ch ( c ) ch (˜ c ) u ε ′ ,ε ′′ ( z c − z ˜ µ λ = i ∈ Vert γ c ∈ ∂λ i , ˜ c ∈ ∂λ i � � e ch ( c ) ch (˜ c ) u ε ′ ,ε ′′ ( z c − z ˜ c + m e ) × (5) e ∈ Edge γ c ∈ ∂λ s ( e ) , ˜ c ∈ ∂λ t ( e ) by a subgroup Γ γ of SU (2)
♦♦♦ Proliferation of fugacities q − → ( q i ) i ∈ Vert γ � q | λ | − q | λ i | → i i ∈ Vert γ ♦
♦♦♦ New models by taking limits e.g. q i → 0 for some i ∈ Vert γ and/or a i ,α → ∞ for some ( i , α ) ♦
♦♦♦ New models by taking limits e.g. q i → 0 for some i ∈ Vert γ and/or a i ,α → ∞ for some ( i , α ) For example, A 1 model ♦
♦♦♦ The A 1 model � λ (1) , . . . , λ ( n ) � Random variable, again Λ = ♦
♦♦♦ The A 1 model � λ (1) , . . . , λ ( n ) � Random variable, again Λ = The measure is different � � c ) × e − ch ( c ) ch (˜ c ) · u ε ′ ,ε ′′ ( z c − z ˜ µ Λ = P ( z c ) , c ∈ ∂ Λ , ˜ c ∈ ∂ Λ c ∈ Λ Mass polynomial 2 n � m f ∈ C P ( x ) = ( x − m f ) f =1 ♦
♦♦♦ The A 1 model � λ (1) , . . . , λ ( n ) � Random variable, again Λ = The measure is different � � c ) × e − ch ( c ) ch (˜ c ) · u ε ′ ,ε ′′ ( z c − z ˜ µ Λ = P ( z c ) , c ∈ ∂ Λ , ˜ c ∈ ∂ Λ c ∈ Λ Mass polynomial 2 n � m f ∈ C P ( x ) = ( x − m f ) = P + ( x + ε 1 + ε 2 ) P − ( x ) f =1 n � ( x − m ± P ± ( x ) = f ) f =1 ♦
♦♦♦ The A 1 model is a limit of the � A 2 -model q ± 1 → 0 a ± 1 ,α → m ± α Masses from frozen Coulomb moduli α = 1 , . . . n q 0 → q � � c ) × e − ch ( c ) ch (˜ c ) · u ε ′ ,ε ′′ ( z c − z ˜ q | Λ | µ Λ = ( q P ( z c )) , c ∈ ∂ Λ , ˜ c ∈ ∂ Λ c ∈ Λ ♦
♦♦♦ OBSERVABLES ∞ � 1 Y i ( x ) = x n i exp − k x k TrΦ k i k =1 ↑ four dimensional definition ♦
♦♦♦ Y -operators the statistical model � n i � � ∈ ∂ + λ ( i ,α ) ( x − z c � ) � Y i ( x )[ λ ] = � ∈ ∂ − λ ( i ,α ) ( x − z c � ) α =1 z c � = a α + ε ′ ( i − 1) + ε ′′ ( j − 1) , c � = ( α, � ) , � = ( i , j ) � are the + charges, � are the − charges # { � ∈ ∂ + λ } − # { � ∈ ∂ − λ } = 1 ♦
♦♦♦ Q -operators in the statistical model x − a i ,α n i � � ( − ε ′ ) x − z c − ε ′′ ε ′ Q (1) ( x )[ λ ] = � � i − x − a i ,α x − z c Γ α =1 c ∈ Λ ( i ) ε ′ x − a i ,α n i � � ( − ε ′′ ) x − z c − ε ′ ε ′′ Q (2) ( x )[ λ ] = � � i − x − a i ,α x − z c Γ α =1 c ∈ Λ ( i ) ε ′′ ♦
♦♦♦ Q versus Y − ε ′ � x − a i ,α ni � ε ′ x − z c − ε ′′ Q (1) � � ( x )[ λ ] = i � x − a i ,α � x − z c α =1 Γ − c ∈ Λ ( i ) ε ′ − ε ′′ � x − a i ,α ni � ε ′′ x − z c − ε ′ Q (2) � � ( x )[ λ ] = i � x − a i ,α � x − z c α =1 Γ − c ∈ Λ ( i ) ε ′′ Q (1) Q (2) ( x ) ( x ) i i = = Y i ( x ) Q (1) Q (2) ( x − ε ′ ) ( x − ε ′′ ) i i ♦
♦♦♦ Nonperturbative Dyson-Schwinger equation NN’15 in the A 1 case � � Y ( x + ε ′ + ε ′′ ) + q P ( x ) Y ( x ) has no singularities in x ♦
♦♦♦ Nonperturbative Dyson-Schwinger equation in the A 1 case � � Y ( x + ε ′ + ε ′′ ) + q P ( x ) = T ( x ) Y ( x ) is a degree n polynomial in x ♦
♦♦♦ Back to quantum integrable system Take a limit ε ′ → 0, ε ′′ = � - finite NS = NN-Shatashvili limit’09 Limit shape phenomenon NN, A. Okounkov’03 NN, V. Pestun, S. Shatashvili’14 � Y ( x ) − 1 � = Y ( x ) − 1 , � Y ( x ) � = Y ( x ) Y ( x + � ) + q P ( x ) Y ( x ) = T ( x ) is a degree n polynomial in x ♦
♦♦♦ Back to quantum integrable system Take a limit ε ′ → 0, ε ′′ = � - finite Limit shape phenomenon = ⇒ linear difference equation on Q ( x ) � Q (2) ( x ) − 1 � = Q ( x ) − 1 , � Q (2) ( x ) � = Q ( x ) Q ( x + � ) + q P ( x ) Q ( x − � ) = T ( x ) Q ( x ) The celebrated T − Q equation R. Baxter L. Faddeev, L. Takhtadzhan E. Sklyanin ♦
♦♦♦ Back to quantum integrable system Take a limit ε ′ → 0, ε ′′ = � - finite The celebrated T − Q equation Q ( x + � ) + q P ( x ) Q ( x − � ) = T ( x ) Q ( x ) Has two solutions over quasiconstants ♦
♦♦♦ Back to quantum integrable system Take a limit ε ′ → 0, ε ′′ = � - finite The celebrated T − Q equation Q ( x + � ) + q P ( x ) Q ( x − � ) = T ( x ) Q ( x ) Q ( x + � ) + q P ( x ) � � Q ( x − � ) = T ( x ) � Q ( x ) Has two solutions over quasiconstants Quantum (discrete) Wronskian W ( x ) = Q ( x + � / 2) � Q ( x − � / 2) − Q ( x − � / 2) � Q ( x + � / 2) = quasi-constant up to normalization ♦
♦♦♦ Back to quantum integrable system Take a limit ε ′ → 0, ε ′′ = � - finite The celebrated T − Q equation Q ( x + � ) + q P ( x ) Q ( x − � ) = T ( x ) Q ( x ) Q ( x + � ) + q P ( x ) � � Q ( x − � ) = T ( x ) � Q ( x ) Both solutions are used in the functional Bethe ansatz E. Sklyanin Where is � Q ( x ) in gauge theory? ♦
♦♦♦ Nonperturbative Dyson-Schwinger equation in the A 1 case � � Y ( x + ε ′ + ε ′′ ) + q P ( x ) = T ( x ) Y ( x ) has no singularities in x can be shown directly (residue matching) ♦
♦♦♦ Nonperturbative Dyson-Schwinger equation in the A 1 case � � Y ( x + ε ′ + ε ′′ ) + q P ( x ) = T ( x ) Y ( x ) has no singularities in x can be shown conceptually ♦
♦♦♦ Nonperturbative Dyson-Schwinger equation in the A 1 case X 1 , x = Y ( x + ε ′ + ε ′′ ) + q P ( x ) Y ( x ) A 1 fundamental qq -character ♦
♦♦♦ THE qq -CHARACTERS X w ,ν = PARTITION FUNCTION OF A POINT-LIKE DEFECT D w ,ν ♦
♦♦♦ THE qq -CHARACTERS X w ,ν = PARTITION FUNCTION OF A POINT-LIKE DEFECT D w ,ν which can be engineered using intersecting branes ♦
♦♦♦ Brane-world scenarios ♦
♦♦♦ Local model: ∪ a < b C 2 ab ⊂ C 4 For example, when 1 ≤ a , b ≤ 3 ♦
♦♦♦ Local IIB string model: � � × R 1 , 1 ⊂ R 1 , 9 ∪ a < b C 2 D 5’s and D 5’s spanning ab ⇒ (0 , 2) susy in R 1 , 1 When 1 ≤ a , b ≤ 4 = ♦
♦♦♦ Chan-Paton spaces N ab for the stack of branes spanning C 2 ab with complex coordinates z a , z b ♦
♦♦♦ Useful pictures n 12 = dim N 12 branes along C 2 12 ♦
♦♦♦ Useful pictures n 12 = dim N 12 branes along C 2 12 and n 23 = dim N 23 branes alond C 2 23 ♦
♦♦♦ n 12 = dim N 12 branes along C 2 n 23 = dim N 23 branes along C 2 12 , 23 , n 13 = dim N 13 branes along C 2 n 24 = dim N 23 branes along C 2 13 , 24 , n 14 = dim N 14 branes along C 2 n 34 = dim N 34 branes along C 2 14 , 34
♦♦♦ = ♦
♦♦♦ Integrate out the degrees of freedom on all but one of the stacks To produce observables on the remaining stack of branes ♦
♦♦♦ How one integrates out the degrees of freedom? Using unbroken supersymmetry and localisation ♦
♦♦♦ Degrees of freedom associated with k instantons Chan-Paton spaces K = C k =# instanton charge , N ab = C n ab ♦
♦♦♦ Degrees of freedom associated with instantons: Rectangular complex I ab , J ab matrices, 1 ≤ a < b ≤ 4 ♦
♦♦♦ Degrees of freedom associated with instantons Square complex matrices B a , a = 1 , . . . , 4 ♦
♦♦♦ Local ADHM data ♦
♦♦♦ Partition function of gauge origami ♦
♦♦♦ Partition Z -function of gauge origami Equivariant integral over the space of solutions of generalized ADHM equations ♦
♦♦♦ Generalized ADHM equations µ ab + ε abcd µ † cd = 0 µ ab = [ B a , B b ] + I ab J ab ♦
♦♦♦ Generalized ADHM equations I. µ ab + ε abcd µ † cd = 0 µ ab = [ B a , B b ] + I ab J ab � � I ab I † ab − J † [ B a , B † a ] + ab J ab = ζ · 1 K a a < b 7 hermitian k × k matrix equations ♦
♦♦♦ Generalized ADHM equations I. µ ab + ε abcd µ † cd = 0 , for all 1 ≤ a < b ≤ 4, where µ ab = [ B a , B b ] + I ab J ab and � � [ B a , B † I ab I † ab − J † µ ≡ a ] + ab J ab = ζ · 1 K a a < b 7 hermitian k × k matrix equations Divide by the U ( k ) action ♦
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