Some analytical aspects of the Kontsevich matrix model Mattia - PowerPoint PPT Presentation

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité Some analytical aspects of the Kontsevich matrix model Mattia Cafasso Laboratoire Angevin de REcherche en MAthématiques (LAREMA), Angers. Geometry of Integrable Systems. SISSA - International School of Advanced Studies, 04-06-2017.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité Plan of the talk • The Witten’s conjecture and the Witten–Kontsevich tau function. • The Painlevé I hierarchy and the string equation. • Kontsevich’s model. • The convergence of the Kontsevich model to (some) solutions of the Painlevé I hierarchy. Collaboration with M. Bertola, arXiv : 1603.06420 (Comm. in Math. Phys. , 2017).

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité The Deligne-Mumford moduli space of Riemann surfaces � � M g , n := Riemann surfaces with n marked points � ∼ A point in M g , n is a (possibly singular) Riemann surface with n marked points (modulo isomorphisms). M g , n is a complex orbifold of dimension 3 g − 3 + n . We denote with L j the tautological line bundle, their fibers over [ C ] are given by T ⋆ p j C , ψ j will denote the corresponding Chern classes.

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité Intersection numbers Intersection numbers are given by the integrals � � � ψ ℓ 1 τ k 0 0 τ k 1 1 ∧ · · · ∧ ψ ℓ n g , n := 1 . . . n , M g , n where k j = ♯ occurences of j as an exponent. Example : � < τ 2 0 τ 3 ψ 1 ∧ ψ 2 ∧ ψ 3 ∧ ψ 2 4 ∧ ψ 0 5 ∧ ψ 0 1 τ 2 > = 6 . The numbers k i satisfy ∞ � 3 g − 3 + n = jk j , k = 0 � n = k j . Let’s define k j � T � � τ k 0 0 τ k 1 1 . . . τ k ℓ j F ( T 0 , T 1 , . . . , ) := ℓ . . . � k j !

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité Witten’ conjecture (Kontsevich theorem) : Let ˜ R n [ U ] be the Lenard polynomials defined by the recursion � ∂ U ∂ ˜ ∂ 3 R n + 1 1 + 2 U ∂ + 1 � ˜ ˜ R 0 [ U ] = U , = R n . 2 n + 1 ∂ T 3 ∂ T 0 ∂ T 0 ∂ T 0 4 0 Theorem : The formal series k j � T j � � τ k 0 0 τ k 1 1 . . . τ k ℓ F ( T 0 , T 1 , . . . , ) := ℓ . . . � k j ! is uniquely determined by the following conditions : 1) U := ∂ 2 F is a solution of the Korteweg de-Vries hierarchy ∂ T 2 0 ∂ U ∂ ˜ = R i [ U ] , i ≥ 0 . ∂ T i ∂ T 0 2) F satisfies the string equation + T 2 ∂ F ∂ F � 0 = 2 . T i + 1 ∂ T 0 ∂ T i i ≥ 0 In other words, e F = τ is a tau function for the KdV hierarchy, uniquely determined by the Virasoro constraints or -equivalently- by its initial value ln τ = T 3 6 . 0

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité String equation and the Painlevé I hierarchy ∂ 2 F ∂ 2 F  � = + T 0 T i + 1   ∂ T 2 ∂ T i ∂ T 0   0  i ≥ 0  � T i + 1 ˜ = ⇒ U − T 0 = R i + 1 [ U ] . � �  ∂ 2 F ∂ 2 F i ≥ 0  = ˜   R i  ∂ T 2  ∂ T 0 ∂ T i 0 Putting T i = 0 for all i � = 0 , N we get the collection of equations T N ˜ R N [ U ] = U − T 0 , N ≥ 1 known as Painlevé I hierarchy. Remark : The same equations can be written as L := ∂ 2 � � = 1 , − U . L , M ∂ T 2 0 (Douglas, “String in less than one dimension”).

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité The Kontsevich matrix model � � i M 3 3 − YM 2 + ixM � Tr dM e H n Z n ( x ; Y ) := , ( Matrix Airy function ) � dM e − Tr ( YM 2 ) H n � M = M † ∈ Mat n × n ( C ) � H n := Y := diag ( y 1 , . . . , y n ) ( x is added for later convenience). Theorem (Kontsevich, 1992) : When n → ∞ , the following formal identity holds F ( � τ WK ( � T ) = lim n →∞ ln Z n ( 0 ; Y ) , i.e. T ) = lim n →∞ Z n ( 0 ; Y ) , under the identification (Miwa’s variables) n 1 � T j = T j ( Y ) := − ( 2 j − 1 )!! , y 2 j + 1 ℓ = 1 ℓ for | Y | → ∞ .

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité A natural question How do we choose Y in such a way that Z n ( 0 ; Y ) converges to a solution of the PI hierarchy ? What are the properties of such solutions ?

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité The main idea It’s easy to prove that Z n ( x ; Y ) can be written as a “wronskian” - type determinant � � 1 Ai ( j − 1 ) ( y 2 � n k + x ) j = 1 ( y j ) det 2 n 2 3 Tr Y 3 + x Tr Y k , j ≤ n Z n ( x ; Y ) = 2 n π 2 e Re y j > 0 , � j < k ( y j − y k ) and this suggest a link with Darboux transformations... Let’s consider the system � � − i  0 ∂ λ Ψ 0 ( x ; λ ) = Ψ 0 ( x ; λ ) ,  i ( λ + x ) 0    � � − i 0   ∂ x Ψ 0 ( x ; λ ) = Ψ 0 ( x ; λ ) ,   i ( λ + x ) 0 and let’s add poles on the points { λ 1 , . . . , λ n } , λ k = y 2 k to get the new system

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité The main idea II n  λ + x A j � 2 − ∂ x a ( n ) � � ∂ λ Ψ n = A Ψ n , A = i σ + − i σ − − λ − λ j ,      j = 1      (1) � λ − 2 ∂ x a ( n ) � ∂ x Ψ n = U Ψ n , U = i σ + − i σ − ,        A k  ∂ λ k Ψ n = − λ − λ k Ψ n , k = 1 , . . . , n .   The isomonodromic (Jimbo-Miwa-Ueno) tau function associated to the system above is defined by the equations ∂ x ln τ n = a ( n ) ∂ λ k ln τ n = res λ k Tr A 2 d λ, and we will prove that x 3 12 Z n ( x , Y ) . τ n ( x , { λ k } ) = e Once this equality is established, one can study the large n limit of the Riemann–Hilbert problem associated to the system (1)...

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité An extension of the Kontsevich matrix model I : Remark : 3 Ai ( λ ) = e − 2 3 λ 2 � 1 + O ( λ − 3 � 2 ) , λ → ∞ , 2 √ πλ 1 4 1  2 √ π √ y ( 1 + O ( y − 3 )) for Re y > 0 ,     3 y 3 + xy Ai ( y 2 + x ) =  2 = ⇒ e 3 y 3 + 2 xy 4 e   2 √ π √ y ( 1 + O ( y − 3 )) for Re y < 0 ,    � � 1 Ai ( j − 1 ) ( y 2 � n k + x ) j = 1 ( y j ) 3 Tr Y 3 + x Tr Y det 2 n 2 k , j ≤ n ⇒ Z n ( x ; Y ) = 2 n π 2 e = , � j < k ( y j − y k ) this expression admits a “regular” expansion if Re ( y i ) > 0 for all i .

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité An extension of the Kontsevich matrix model II : → Y ( 0 ) ⊔ Y ( 1 ) ⊔ Y ( 2 ) ; Y �− � Ai ( k − 1 ) �  ( y 2  j + x ) yj ∈ Y ( 0 ) 0 1 ≤ k ≤ n   3 Tr Y 3 + x Tr Y 2  � �  e Ai ( k − 1 ) ( y 2 j + x ) Z n ( x ; Y ( 0 ) , Y ( 1 ) , Y ( 2 ) ) = C n   j < k ( y j − y k ) det yj ∈ Y ( 1 ) . 1   �  1 ≤ k ≤ n    � Ai ( k − 1 ) � ( y 2 j + x )   yj ∈ Y ( 2 ) 2 1 ≤ k ≤ n 2 i π Ai s ( λ ) := Ai ( ω s λ ) , ω := e 3 . This determinant have a regular expansion if Y ( a ) ∋ y j → ∞ , y j ∈ S a . S 0 : ; S 1 : S 2 : . For what follows let’s introduce the parameters { λ i , µ j } such that y i = √ λ i if Re ( y i ) > 0 et y j = −√ µ j if Re ( y j ) ≤ 0 . √ √ √ µ j + � λ λ j − λ d ( � � � µ ) := √ √ λ, � . √ µ j − � λ λ j + λ j = 1 j = 1

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité A Riemann-Hilbert problem for Z n Theorem (M. Bertola, M.C.) : x 3 12 τ n ( x ,� Z n ( x ; Y ( 0 ) , Y ( 1 ) , Y ( 2 ) ) = e λ, � µ ) , τ n tau function of the Riemann–Hilbert problem with asymptotics � � 1 + a ( n ) ( x ; � 4 1 + i σ 1 µ ) Γ n ( λ ) ∼ λ − σ 3 λ, � σ 3 + O ( λ − 1 ) √ √ , 2 λ � d − 1 ( � λ, � 1 µ ) e 4 3 λ 3 2 + 2 x √ λ 0 � 1 √ � 3 λ 4 − 2 x � − λ 2 d ( � λ, � µ ) e 3 � � 1 0 1 1 0 − 1 0 � 0 1 √ λ 1 x 3 2 + 2 4 λ µ ) e 3 � λ, � d − 1 ( �

The Witten’s conjecture and the PI hierarchy Isomonodromic tau functions Convergence Universalité The matrix √ 3 µ ) := Γ n e ( − 2 Ψ n ( λ ; x ,� 2 − x λ ) σ 3 D − 1 ( λ ) 3 λ λ, � n 2 n 1  √ √  ( √ µ j − � � � ( λ j + λ ) λ ) 0   j = 1 j = 1   D ( λ ) :=  n 2 n 1  √ √ ( √ µ j +   � � � ( λ j − λ ) λ ) 0   j = 1 j = 1 is a solution of the isomonodromic system ∂ ∂λ Ψ n ( λ ; x ,� µ ) = A ( λ ; x ,� µ )Ψ n ( λ ; x ,� λ, � λ, � λ, � µ ) ∂ ∂ x Ψ n ( λ ; x ,� µ ) = U ( λ ; x ,� µ )Ψ n ( λ ; x ,� λ, � λ, � λ, � µ ) µ ) = − A k ( x ,� ∂ λ, � µ ) ∂λ k Ψ n ( λ ; x ,� Ψ n ( λ ; x ,� λ, � λ, � µ ) , λ − λ k µ ) = − B k ( x ,� ∂ λ, � µ ) ∂µ k Ψ n ( λ ; x ,� Ψ n ( λ ; x ,� λ, � λ, � µ ) . λ − µ k