Kontsevich-Penner model and open intersection numbers Geometry of - PowerPoint PPT Presentation

Kontsevich-Penner model and open intersection numbers Geometry of Integrable Systems SISSA, Trieste, 7-9 June 2017 Giulio Ruzza, SISSA Joint work with Marco Bertola, SISSA/Concordia University

Moduli spaces of Riemann surfaces M g , n = { stable compact Riemann surfaces of genus g with n marked points } / isomorphism (Deligne and Mumford, 1969). M g , n is a compact smooth complex orbifold. • • dim C M g , n = 3 g − 3 + n . Stability condition: 2 g − 2 + n > 0.

Moduli spaces of Riemann surfaces M g , n = { stable compact Riemann surfaces of genus g with n marked points } / isomorphism (Deligne and Mumford, 1969). M g , n is a compact smooth complex orbifold. • • dim C M g , n = 3 g − 3 + n . Stability condition: 2 g − 2 + n > 0.

Free energy and Witten-Kontsevich theorem Psi classes ψ i := c 1 ( L i ) ∈ H 2 � � M g , n , Q , L i tautological line bdle of cotangent spaces at the i − th marked point. Intersection numbers � � τ d 1 ··· τ dn � = � τ r 0 0 τ r 1 M g , n ψ d 1 1 ∧···∧ ψ dn 1 ··· � := ( r j = ♯ { i : d i = j } ) n Generating function � � tr 0 0 tr 1 exp � = � t 3 t 2 t 2 1 ··· 0 t 3 � τ r 0 0 τ r 1 6 + t 1 24 + t 0 t 2 F ( t 0 , t 1 ,... ):= 1 ··· � r 0 ! r 1 ! ··· = 0 24 + 24 + 1 48 + ··· t j τ j r ∗ j ≥ 0 Theorem (E. Witten - M. Kontsevich, 1991) exp F is a KdV tau function. In particular U := ∂ t 0 F satisfies ∂ 3 U ∂ U = U ∂ U + 1 ∂ t 1 ∂ t 0 12 ∂ t 3 0

Free energy and Witten-Kontsevich theorem Psi classes ψ i := c 1 ( L i ) ∈ H 2 � � M g , n , Q , L i tautological line bdle of cotangent spaces at the i − th marked point. Intersection numbers � � τ d 1 ··· τ dn � = � τ r 0 0 τ r 1 M g , n ψ d 1 1 ∧···∧ ψ dn 1 ··· � := ( r j = ♯ { i : d i = j } ) n Generating function � � tr 0 0 tr 1 exp � = � t 3 t 2 t 2 1 ··· 0 t 3 � τ r 0 0 τ r 1 6 + t 1 24 + t 0 t 2 F ( t 0 , t 1 ,... ):= 1 ··· � r 0 ! r 1 ! ··· = 0 24 + 24 + 1 48 + ··· t j τ j r ∗ j ≥ 0 Theorem (E. Witten - M. Kontsevich, 1991) exp F is a KdV tau function. In particular U := ∂ t 0 F satisfies ∂ 3 U ∂ U = U ∂ U + 1 ∂ t 1 ∂ t 0 12 ∂ t 3 0

Free energy and Witten-Kontsevich theorem Psi classes ψ i := c 1 ( L i ) ∈ H 2 � � M g , n , Q , L i tautological line bdle of cotangent spaces at the i − th marked point. Intersection numbers � � τ d 1 ··· τ dn � = � τ r 0 0 τ r 1 M g , n ψ d 1 1 ∧···∧ ψ dn 1 ··· � := ( r j = ♯ { i : d i = j } ) n Generating function � � tr 0 0 tr 1 exp � = � t 3 t 2 t 2 1 ··· 0 t 3 � τ r 0 0 τ r 1 6 + t 1 24 + t 0 t 2 F ( t 0 , t 1 ,... ):= 1 ··· � r 0 ! r 1 ! ··· = 0 24 + 24 + 1 48 + ··· t j τ j r ∗ j ≥ 0 Theorem (E. Witten - M. Kontsevich, 1991) exp F is a KdV tau function. In particular U := ∂ t 0 F satisfies ∂ 3 U ∂ U = U ∂ U + 1 ∂ t 1 ∂ t 0 12 ∂ t 3 0

Free energy and Witten-Kontsevich theorem Psi classes ψ i := c 1 ( L i ) ∈ H 2 � � M g , n , Q , L i tautological line bdle of cotangent spaces at the i − th marked point. Intersection numbers � � τ d 1 ··· τ dn � = � τ r 0 0 τ r 1 M g , n ψ d 1 1 ∧···∧ ψ dn 1 ··· � := ( r j = ♯ { i : d i = j } ) n Generating function � � tr 0 0 tr 1 exp � = � t 3 t 2 t 2 1 ··· 0 t 3 � τ r 0 0 τ r 1 6 + t 1 24 + t 0 t 2 F ( t 0 , t 1 ,... ):= 1 ··· � r 0 ! r 1 ! ··· = 0 24 + 24 + 1 48 + ··· t j τ j r ∗ j ≥ 0 Theorem (E. Witten - M. Kontsevich, 1991) exp F is a KdV tau function. In particular U := ∂ t 0 F satisfies ∂ 3 U ∂ U = U ∂ U + 1 ∂ t 1 ∂ t 0 12 ∂ t 3 0

Kontsevich matrix integral � � � i M 3 3 − M 2 Y H n d M exp Tr � Z n ( Y ) := H n d M exp Tr ( − M 2 Y ) H n = R n 2 = n × n hermitian matrices, Y = diag ( y 1 , ..., y n ) . • Z n ( Y ) is a KdV tau function in Miwa variables 2 − 2 k + 1 3 ( 2 k + 1 )!! Tr Y − ( 2 k + 1 ) T k ( Y ) := − • Feynman diagramatic expansion as n → ∞ for large Y of log Z n ( Y ) is F ( t 0 ( Y ) , t 1 ( Y ) , ... ) where t k ( Y ) := − 2 − 2 k + 1 3 ( 2 k − 1 )!! Tr Y − ( 2 k + 1 )

Kontsevich matrix integral � � � i M 3 3 − M 2 Y H n d M exp Tr � Z n ( Y ) := H n d M exp Tr ( − M 2 Y ) H n = R n 2 = n × n hermitian matrices, Y = diag ( y 1 , ..., y n ) . • Z n ( Y ) is a KdV tau function in Miwa variables 2 − 2 k + 1 3 ( 2 k + 1 )!! Tr Y − ( 2 k + 1 ) T k ( Y ) := − • Feynman diagramatic expansion as n → ∞ for large Y of log Z n ( Y ) is F ( t 0 ( Y ) , t 1 ( Y ) , ... ) where t k ( Y ) := − 2 − 2 k + 1 3 ( 2 k − 1 )!! Tr Y − ( 2 k + 1 )

Kontsevich matrix integral � � � i M 3 3 − M 2 Y H n d M exp Tr � Z n ( Y ) := H n d M exp Tr ( − M 2 Y ) H n = R n 2 = n × n hermitian matrices, Y = diag ( y 1 , ..., y n ) . • Z n ( Y ) is a KdV tau function in Miwa variables 2 − 2 k + 1 3 ( 2 k + 1 )!! Tr Y − ( 2 k + 1 ) T k ( Y ) := − • Feynman diagramatic expansion as n → ∞ for large Y of log Z n ( Y ) is F ( t 0 ( Y ) , t 1 ( Y ) , ... ) where t k ( Y ) := − 2 − 2 k + 1 3 ( 2 k − 1 )!! Tr Y − ( 2 k + 1 )

The Riemann-Hilbert problem Question Z n is genuinely analytic for Re y k > 0. Does F represent an asymptotic expansion? Answer: consider RHP in the λ -plane � M 1 Γ ( n ) + = Γ ( n ) − M j Γ ( n ) ( λ ) ∼ λ − σ 3 2 ( 1 + O ( λ − 1 4 1 + i σ 1 2 )) λ → ∞ √ M 3 M 0 M j := D − 1 − e − θ − S j e θ + D + � 0 1 � M 2 S 0 := [ 1 1 0 1 ] S 1 := [ 1 0 1 1 ] S 2 := S 3 := [ 1 0 1 1 ] − 1 0 � √ � √ n � θ := 2 λ j + λ 3 0 2 σ 3 √ 3 λ D := √ λ j − λ 0 j = 1

The Riemann-Hilbert problem Question Z n is genuinely analytic for Re y k > 0. Does F represent an asymptotic expansion? Answer: consider RHP in the λ -plane � M 1 Γ ( n ) + = Γ ( n ) − M j Γ ( n ) ( λ ) ∼ λ − σ 3 2 ( 1 + O ( λ − 1 4 1 + i σ 1 2 )) λ → ∞ √ M 3 M 0 M j := D − 1 − e − θ − S j e θ + D + � 0 1 � M 2 S 0 := [ 1 1 0 1 ] S 1 := [ 1 0 1 1 ] S 2 := S 3 := [ 1 0 1 1 ] − 1 0 � √ � √ n � θ := 2 λ j + λ 3 0 2 σ 3 √ 3 λ D := √ λ j − λ 0 j = 1

Kontsevich matrix integral as isomonodromic tau function The jumps of Ψ n := Γ n e − θ D − 1 do not depend on λ, λ 1 , ..., λ n ⇒ isomonodromy equations � ∂ ∂λ Ψ n ( λ ; λ 1 , ..., λ n ) = A n ( λ ; λ 1 , ..., λ n )Ψ n ( λ ; λ 1 , ..., λ n ) ∂ ∂λ j Ψ n ( λ ; λ 1 , ..., λ n ) = U n , j ( λ ; λ 1 , ..., λ n )Ψ n ( λ ; λ 1 , ..., λ n ) ⇒ isomonodromic tau function τ n ( λ 1 , ..., λ n ) (M. Jimbo, T. Miwa and K. Ueno, 1981) ∂ log τ n ( λ 1 , ..., λ n ) = res λ = λ j d λ Tr A 2 n ( λ ; λ 1 , ..., λ n ) ∂λ j Theorem (M. Bertola - M. Cafasso, 2016) τ n ( λ 1 , ..., λ n ) = Z n ( Y ) , λ j = y 2 j .

Kontsevich matrix integral as isomonodromic tau function The jumps of Ψ n := Γ n e − θ D − 1 do not depend on λ, λ 1 , ..., λ n ⇒ isomonodromy equations � ∂ ∂λ Ψ n ( λ ; λ 1 , ..., λ n ) = A n ( λ ; λ 1 , ..., λ n )Ψ n ( λ ; λ 1 , ..., λ n ) ∂ ∂λ j Ψ n ( λ ; λ 1 , ..., λ n ) = U n , j ( λ ; λ 1 , ..., λ n )Ψ n ( λ ; λ 1 , ..., λ n ) ⇒ isomonodromic tau function τ n ( λ 1 , ..., λ n ) (M. Jimbo, T. Miwa and K. Ueno, 1981) ∂ log τ n ( λ 1 , ..., λ n ) = res λ = λ j d λ Tr A 2 n ( λ ; λ 1 , ..., λ n ) ∂λ j Theorem (M. Bertola - M. Cafasso, 2016) τ n ( λ 1 , ..., λ n ) = Z n ( Y ) , λ j = y 2 j .

Formulae for closed intersection numbers Theorem (M. Bertola - B. Dubrovin - D. Yang, 2015) Let � − � ( 6 g − 5 )!! ( 6 g − 1 )!! 24 g − 1 ( g − 1 )! λ − 6 g + 4 24 g g ! λ − 6 g  − 1  2   g ≥ 1 g ≥ 0   Θ( λ ) := � �     6 g + 1 ( 6 g − 1 )!! ( 6 g − 5 )!! 24 g g ! λ − 6 g + 2 1 24 g − 1 ( g − 1 )! λ − 6 g + 4     6 g − 1 2   g ≥ 0 g ≥ 1 � � � ∞ � n ( 2 kj + 1 )!! F n ( λ 1 ,...,λ n ):= τ kj j = 1 2 kj + 1 k 1 ,..., kn = 0 λ j Then � ∞ ( 6 g − 3 )!! 24 g g ! λ − 6 g + 2 F 1 ( λ )= g = 1 Tr ( Θ ( λσ ( 1 ) ) ··· Θ ( λσ ( n ) )) � λ 2 1 + λ 2 F n ( λ 1 ,...,λ n )= − 1 � − δ n , 2 2 n ≥ 2 n � 2 ( λ 2 2 ) λ 2 σ ( j ) − λ 2 1 − λ 2 σ ∈ Sn � j ∈ Z / n Z σ ( j + 1 )

Moduli spaces of open Riemann surfaces M g , k , l =moduli spaces of open (i.e. with boundary ) Riemann surfaces ( g = doubled genus, k = ♯ bdry markings, l = ♯ int. markings). Rigorous study initiated by Pandharipande, Solomon and Tessler, 2015. • • • • Main challenges: M g , k , l is a real orbifold with real boundary, possibly nonorientable ⇒ difficulties in the definition of intersection numbers. dim R M g , k , l = 3 g − 3 + k + 2 l . Stability condition : 2 g − 2 + k + 2 l > 0.

Moduli spaces of open Riemann surfaces M g , k , l =moduli spaces of open (i.e. with boundary ) Riemann surfaces ( g = doubled genus, k = ♯ bdry markings, l = ♯ int. markings). Rigorous study initiated by Pandharipande, Solomon and Tessler, 2015. • • • • Main challenges: M g , k , l is a real orbifold with real boundary, possibly nonorientable ⇒ difficulties in the definition of intersection numbers. dim R M g , k , l = 3 g − 3 + k + 2 l . Stability condition : 2 g − 2 + k + 2 l > 0.