On some Hamiltonian properties of isomonodromic tau functions. - PowerPoint PPT Presentation

On some Hamiltonian properties of isomonodromic tau functions. Andrei Prokhorov Joint work with Alexander Its In the memory of L. D. Faddeev Indiana University-Purdue University Indianapolis Saint-Petersburg State University September 4th, 2018

Painlev´ e equations The Painlev´ e equations appear in several places: conformal field theory, random matrices, statistical mechanics. For example the equation Painlev´ e-IV is given by q tt = q 2 2 q 3 + 4 tq 2 + 2( t 2 − 2 𝜄 ∞ + 1) q − 8 𝜄 2 2 q + 3 t 0 q . (1) The tau function for this equation is given by (see Okamoto, 1980) t 2 (︃ q 2 8 q − q 3 8 − q 2 t 2 − qt 2 2 − 2 𝜄 2 ∫︂ )︃ ln 𝜐 O ( t 1 , t 2 ) = t 0 + ( 𝜄 ∞ − 1) q − 2 𝜄 0 t dt q t 1

Connection problem The Riemann-Hilbert approach provides us with asymptotic for solutions of Painlev´ e equations as t approaches infinity. The asymptotic is parametrised by monodromy data. The natural question is to study the asymptotic of tau function when t 1 and t 2 approach infinity in different directions in complex plane. The connection problem consists in determining such asymptotics. Using the asymptotic for solutions of Painlev´ e equations we can get the asymptotic for tau function up to the term independent of t 1 , t 2 . To find this term is more complicated problem.

Different results Iorgov, Lisovyy, Tykhyy(2013), Its, Lisovyy, Tykhyy(2014), Lisovyy, Nagoya, Roussillon(2018) got the conjectural results for PVI, PIII, PV using the quasiperiodicity of the connection constant and its interpretation as generating function for canonical transformation. Its, P.(2016), Lisovyy, Roussillon (2017), Its, Lisovyy, P.(2018) got the results for PIII, PI, PVI, PII using the extension of JMU form suggested by Bertola based on works by Malgrange. Bothner, Its, P.(2017), Bothner (2018) got the results for PII, PIII, PV using interpretation of extension of JMU in terms of an action. The main result of authors is relation with action for all Painlev´ e equations and Schlesinger equation. In these slides we consider Painlev´ e-IV case.

Lax pair The Lax pair for Painlev´ e-IV case is given by (see Jimbo Miwa, 1981) d Ψ d Ψ dz = A ( z )Ψ( z ) , dt = B ( z )Ψ( z ) A ( z ) = A 1 z + A 0 + A − 1 z , B ( z ) = B 1 z + B 0 (︄ − kq )︄ (︃ t k )︃ A − 1 = 1 − r + 𝜄 0 2 A 0 = , , 2( r − 𝜄 0 − 𝜄 ∞ ) 2 r ( r − 2 𝜄 0 ) − t r − 𝜄 0 z k kq (︃ 1 )︃ (︃ 0 )︃ 0 k A 1 = B 1 = , B 0 = . 2( r − 𝜄 0 − 𝜄 ∞ ) 0 − 1 0 k

The compatibility condition The compatibility condition for the Lax pair has form dA dt − dB dz + [ A , B ] = 0 . It is equivalent to the system ⎧ dq dt = − 4 r + q 2 + 2 tq + 4 𝜄 0 , ⎪ ⎪ ⎪ ⎪ ⎪ (︃ )︃ dt = − 2 dr − q + 4 𝜄 0 ⎨ q r 2 + r + ( 𝜄 0 + 𝜄 ∞ ) q , q ⎪ ⎪ dk ⎪ ⎪ dt = − k ( q + 2 t ) . ⎪ ⎩ The function q ( t ) satisfies Painlev´ e-IV equation (1).

Local behavior of Ψ-function at infinity The first equation of the Lax pair has irregular singularity of Poincar´ e rank 2 at infinity. We have the following formal solution at infinity (︃ z 2 )︃ Ψ ∞ ( z ) = G ∞ ( z ) e Θ ∞ ( z ) , Θ ∞ ( z ) = 𝜏 3 2 + tz − 𝜄 ∞ ln z , (︃ 1 (︃ I + g 1 z + g 2 )︃)︃ G ∞ ( z ) = z 2 + O z → ∞ , z 3 (︃ 1 0 )︃ 𝜏 3 = . 0 − 1

Local behavior of Ψ-function at zero The first equation of the Lax pair has regular singularity at zero. We have the following solution at zero Ψ 0 ( z ) = G 0 ( z ) z 𝜄 0 𝜏 3 , G 0 ( z ) = P 0 ( I + O ( z )) , z → 0 , (︃ − kq 1 − kq )︃ a − σ 3 2 . P 0 = 2 √ kq 𝜄 0 2 r 2 r − 4 𝜄 0 To satisfy the second equation of the Lax pair we need to have dt = 4 𝜄 0 da q a .

The Jimbo-Miwa-Ueno form The Jimbo-Miwa-Ueno form is given by )︂ − 1 dG ∞ ( z ) (︃(︂ d Θ ∞ ( z ) )︃ 𝜕 JMU = − res z = ∞ Tr G ∞ ( z ) dt dz dt = − Tr ( g 1 𝜏 3 ) dt [︃ 2 (︃ q + 2 t + 4 𝜄 0 )︃ ]︃ q r 2 − = r + ( 𝜄 0 + 𝜄 ∞ )( r + 2 t ) dt q (︃ q 2 8 q − q 3 8 − q 2 t 2 − qt 2 2 − 2 𝜄 2 )︃ t 0 = + 𝜄 ∞ q + 2 𝜄 ∞ t dt q In general L )︂ − 1 dG 𝜉 ( z ) (︃(︂ d Θ 𝜉 ( z ) )︃ ∑︂ ∑︂ 𝜕 JMU = − res z = a ν Tr G 𝜉 ( z ) dt k dz dt k k =1 a ν

The isomonodromic tau function The isomonodromic tau function is given by t 2 ∫︂ ln 𝜐 JMU ( t 1 , t 2 ) = 𝜕 JMU . t 1 We have the relation t 2 ∫︂ ln 𝜐 JMU ( t 1 , t 2 ) = ln 𝜐 O ( t 1 , t 2 ) + qdt + ( 𝜄 0 + 𝜄 ∞ )( t 2 2 − t 2 1 ) . t 1

Hamiltonian structure We expect 𝜕 JMU ≃ Hdt . Unfortunately if we choose the Hamiltonian in such way, r and q are not Darboux coordinates for Hamiltonian dynamics. [︃ 2 (︃ q + 2 t + 4 𝜄 0 )︃ ]︃ q r 2 − 𝜕 JMU = r + ( 𝜄 0 + 𝜄 ∞ )( r + 2 t ) dt . q dq ⎧ dt = − 4 r + q 2 + 2 tq + 4 𝜄 0 , ⎪ ⎨ dr dt = − 2 (︃ − q + 4 𝜄 0 )︃ q r 2 + r + ( 𝜄 0 + 𝜄 ∞ ) q . ⎪ ⎩ q

Hamiltonian structure Hamiltonian structure for Painlev´ e equations was introduced by Okamoto (1980). It was interpreted in terms of moment map and Hamiltonian reduction in the dual loop algebra * � sl 2 ( R ) in the work by Harnad and Routhier(1995). We want to study the Hamiltonian structure using the extension of Jimbo-Miwa-Ueno form, following works of Bertola (2010), Malgrange(1983), Its, Lisovyy, P.(2018).

Symplectic form Consider the configuration space for Painlev´ e-IV Lax pair consisting of coordinates { q , r , k , a , 𝜄 0 , 𝜄 ∞ } . We denote by 𝜀 the differential in this space. Following the work of Its, Lisovyy, P.(2018) consider the form (︂ A ( z ) 𝜀 G ∞ ( z ) G ∞ ( z ) − 1 )︂ 𝜕 0 = res z = ∞ Tr (︂ A ( z ) 𝜀 G 0 ( z ) G 0 ( z ) − 1 )︂ = Tr ( A − 1 𝜀 G 0 G − 1 + res z =0 Tr 0 − A 1 𝜀 g 2 + A 1 𝜀 g 1 g 1 − A 0 𝜀 g 1 )

Symplectic form In general this form is given by (︂ A ( z ) 𝜀 G 𝜉 ( z ) G 𝜉 ( z ) − 1 )︂ ∑︂ 𝜕 0 = res z = a ν Tr . a ν In all examples considered the symplectic form for Hamiltonian dynamics was given by Ω 0 = 𝜀𝜕 0 . In case of Painlev´ e-IV we have Ω 0 = − 1 q 𝜀 r ∧ 𝜀 q + 1 k 𝜀 k ∧ 𝜀𝜄 ∞ + 1 a 𝜀 a ∧ 𝜀𝜄 0 − 1 q 𝜀 q ∧ 𝜀𝜄 0 .

Darboux coordinates We can choose Darboux coordinates as p 1 = − r q , q 1 = q , t ∫︂ p 2 = ln k = − ( q + 2 t ) dt , q 2 = 𝜄 ∞ c 1 t 4 𝜄 0 ∫︂ p 3 = ln a − ln q = q dt − ln q , q 3 = 𝜄 0 . c 2

Hamiltonian Jimbo-Miwa-Ueno form in these coordinates take form 2 p 2 1 q 1 + p 1 ( q 2 (︁ )︁ 𝜕 JMU = 1 + 2 q 1 t + 4 q 3 ) + ( q 1 + 2 t )( q 3 + q 2 ) dt The deformation equations take form ⎧ dq 1 dp 3 ⎪ dt = 4 p 1 q 1 + q 2 1 + 2 q 1 t + 4 q 3 , dt = − 4 p 1 − q 1 − 2 t , ⎪ ⎪ ⎨ dp 1 dp 2 ⎪ dt = − 2 p 2 ⎪ 1 − 2 p 1 q 1 − 2 p 1 t − q 3 − q 2 , dt = − ( q 1 + 2 t ) . ⎪ ⎩ These equations become Hamiltonian system with Hamiltonian given by the equation 𝜕 JMU = Hdt

Counterexample We can choose Darboux coordinates in different way p 1 = − r ˜ q + f ( t ) , q 1 = q , t ∫︂ p 2 = ln k = − ( q + 2 t ) dt , q 2 = 𝜄 ∞ c 1 t 4 𝜄 0 ∫︂ p 3 = ln a − ln q = q dt − ln q , q 3 = 𝜄 0 . c 2

Counterexample Jimbo-Miwa-Ueno form in these coordinates take form p 1 − f ) 2 q 1 + ( ˜ p 1 − f )( q 2 (︁ 𝜕 JMU = 2( ˜ 1 + 2 q 1 t + 4 q 3 ) +( q 1 + 2 t )( q 3 + q 2 )) The deformation equations take form ⎧ dq 1 1 + 2 q 1 t + 4 q 3 , dp 3 p 1 q 1 − 4 fq 1 + q 2 ⎪ dt = 4 ˜ dt = − 4 ˜ p 1 + 4 f − q 1 − 2 t , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dp 1 p 1 − f ) 2 − 2( ˜ dt = − 2( ˜ p 1 − f )( q 1 + t ) − q 3 − q 2 + f ′ , ⎪ ⎪ ⎪ ⎪ dp 2 ⎪ ⎪ dt = − ( q 1 + 2 t ) . ⎪ ⎩ These equations become Hamiltonian system with Hamiltonian given by the equation 𝜕 JMU = ( ˜ H − q 1 f ′ ) dt

Hamiltonian We can ask, what Hamiltonian induce isomonodromic deformation with respect to described symplectic structure. Consider the form in the configuration space (︃ 𝜖 A ( z ) )︃ ∑︂ 𝜀 G 𝜉 ( z ) G 𝜉 ( z ) − 1 𝛽 = res z = a ν Tr 𝜖 t a ν (︃ d ( 𝜀 Θ 𝜉 ( z )) G 𝜉 ( z ) − 1 𝜖 G 𝜉 ( z ) )︃ ∑︂ − res z = a ν Tr dz 𝜖 t a ν Conjecture The form 𝛽 is exact and the Hamiltonian is given by 𝛽 = 𝜀 H .

Extension of Jimbo-Miwa-Ueno form We consider the extended configuration space. For Painlev´ e-IV it has coordinates { t , q 1 , p 1 , q 2 , p 2 , q 3 , p 3 } We denote by ” d ” the differential in this space. Following Its, Lisovyy, P.(2018) we consider the form (︂ A ( z ) dG ∞ ( z ) G ∞ ( z ) − 1 )︂ 𝜕 = res z = ∞ Tr (︂ A ( z ) dG 0 ( z ) G 0 ( z ) − 1 )︂ = Tr ( A − 1 dG 0 G − 1 + res z =0 Tr 0 − A 1 dg 2 + A 1 dg 1 g 1 − A 0 dg 1 ) .

Extension of Jimbo-Miwa-Ueno form In general this form is given by (︂ A ( z ) dG 𝜉 ( z ) G 𝜉 ( z ) − 1 )︂ ∑︂ 𝜕 = res z = a ν Tr . a ν Using the first choice of Darboux coordinates and Hamiltonian we can rewrite it for Painlev´ e-IV case as 𝜕 = p 1 dq 1 + p 2 dq 2 + p 3 dq 3 − Hdt − p 2 q 2 − p 3 q 3 + q 2 2 − q 2 (︃ Ht 2 − p 1 q 1 2 − q 3 2 + q 2 )︃ 3 2 + d (2) 2 2