Differentially algebraic equations in physics Youssef Abdelaziz, - PowerPoint PPT Presentation

Differentially algebraic equations in physics Youssef Abdelaziz, Jean-Marie Maillard (Universit´ e Paris VI) Based on “ Modular forms, Schwarzian conditions, and symmetries of differential equations in physics ”, arXiv 1611.08493 S´ eminaire CALIN Univ. Paris Nord, Villetaneuse, 10/01/2017 1/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Hamiltonian of the Ising model � H = { J v σ j , k σ j +1 , k + J h σ j , k σ j , k +1 } j , k J v , J h : vertical and horizontal coupling constants The spins take the values σ j , k = ± 1. 1 The partition function: exp( − k b T H ) 2/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Nature of power series Algebraic: S ( x ) ∈ Q ( x ) root of a polynomial P ( t , S ( t )) = 0 D-finite: S ( x ) ∈ Q ( x ) satisfying a linear differential equation with polynomial coefficients c r ( t ) S ( t ) ( t ) + s + c 0 ( t ) S ( t ) = 0 Hypergeometric: S ( x ) = � ∞ n =0 s n x n s.t. s n +1 ∈ Q ( n ). E.g., the Gauss s n hypergeometric function: ∞ t n ( a ) n ( b ) n � 2 F 1 ([ a , b ] , [ c ] , x ) = n ! , ( c ) n n =0 3/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Nature of power series Algebraic: S ( x ) ∈ Q ( x ) root of a polynomial P ( t , S ( t )) = 0 D-finite: S ( x ) ∈ Q ( x ) satisfying a linear differential equation with polynomial coefficients c r ( t ) S ( t ) ( t ) + s + c 0 ( t ) S ( t ) = 0 Hypergeometric: S ( x ) = � ∞ n =0 s n x n s.t. s n +1 ∈ Q ( n ). E.g., the Gauss s n hypergeometric function: ∞ t n ( a ) n ( b ) n � 2 F 1 ([ a , b ] , [ c ] , x ) = n ! , ( a ) n := a ( a + 1) · · · ( a + n − 1) ( c ) n n =0 2 F 1 (1 , 1; 2; z ) = − ln(1 − z ) 1 E.g.: 2 F 1 (1 , 1; 1; z ) = 1 − z , z Partition function 2D square Ising model [Viswanathan, 2014] 4 F 3 ([1 , 1 , 3 2 , 3 k = tanh(2 β J ) 2] , [2 , 2 , 2] , 16 k 2 ]) , 2 cosh(2 β J ) 3/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Magnetic susceptibility of 2D Ising model Magnetic susceptibility − → sum ability of a material of two point correlation functions to align itself with ∞ an external imposed � χ (2 n +1) χ := β magnetic field n =0 → 2 n multiple integrals , e.g. χ (3) is given by the double integral: χ (2 n +1) − 4/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Magnetic susceptibility of 2D Ising model Magnetic susceptibility − → sum ability of a material of two point correlation functions to align itself with ∞ an external imposed � χ (2 n +1) χ := β magnetic field n =0 → 2 n multiple integrals , e.g. χ (3) is given by the double integral: χ (2 n +1) − � 2 π � 2 π χ (3) ( s ) = (1 − s ) 1 / 4 1 1 + x 1 x 2 x 3 d φ 1 d φ 2 y 1 y 2 y 3 F 4 π 2 s 1 − x 1 x 2 x 3 0 0 s x j = � 1 + s 2 − s cos φ j + (1 + s 2 − s cos φ j ) 2 − s 2 s y j = (1 + s 2 − s cos φ j ) 2 − s 2 , ( j = 1 , 2 , 3) � φ 1 + φ 2 + φ 3 = 0 � � x i x j f 31 + f 23 and F = f 23 with f ij = (sin φ i − sin φ j ) 2 1 − x i x j 4/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Feynman diagrams are D-finite Feynman diagrams − → first order perturbations of n -fold integral of the operator S (scattering operator) giving the probability of such interactions: n times � �� � n n ∞ � � ι n � � � d 4 x j T S = · · · L ( x j ) n ! n =0 j =1 j =1 L v ( x j ) − → Lagrangian of interaction, T the time ordered product of operators, d 4 x j four-vectors 5/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Multiple integrals of an algebraic object Theorem ( Kashiwara ) n times � �� � � � · · · D-finite function dx 1 · · · dx n → D-finite function (D-finite = solution of linear ODE with polynomial coefficients) 6/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Diagonal of a rational function For a formal power series F given by ∞ ∞ � � F m 1 , ··· , m n z m 1 · · · z m n F ( z 1 , z 2 , · · · , z n ) = · · · n , 1 m 1 =0 m n =0 the diagonal of F is defined as the single variable series: ∞ � F m , ··· , m z m Diag ( F ( z 1 , z 2 , · · · , z n )) := m =0 Example. One of the many diagonals leading to Ap´ ery numbers: n � n � 2 � n + k � 2 1 � � z n Diag = (1 − z 1 − z 2 )(1 − z 3 − z 4 ) − z 1 z 2 z 3 z 4 k k n ≥ 0 k =0 7/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

2 F 1 , modular forms, and physics The Gauss hypergeometric function 2 F 1 → PHYSICS!, e.g. the differential operator of χ 2 n +1 factorizes into operators that annihilate 2 F 1 functions. [ A. Bostan, S. Boukraa, S. Hassani, M. van Hoeij, J.-M. Maillard, J.-A. Weil, N. Zenine, The Ising model: from elliptic curves to modular forms and Calabi-Yau equations , 2011] [ M. Assis, S. Boukraa, S. Hassani, M. van Hoeij, J.-M. Maillard, B. M. McCoy, Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations , 2012] ∞ q n � n 3 E 4 ( q ) = 1 + 240 1 − q n n =0 � � 4 [ 1 12 , 5 12] , [1] , 1728 = 2 F 1 j ( τ ) q = exp(2 i πτ ), j ( τ ) → j -invariant 8/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Modular forms as pullbacked 2 F 1 functions Emergence of modular forms in physics through 2 F 1 functions Modular forms emerge through covariance properties of 2 F 1 : � � � � 2 F 1 [ α, β ] , [ γ ] , p 1 ( x ) = A ( x ) 2 F 1 [ α, β ] , [ γ ] , p 2 ( x ) A ( x ), p 1 ( x ) and p 2 ( x ) are rational functions. p 1 ( x ) and p 2 ( x ) are called pullbacks, the 2 F 1 is thus called pullbacked. For instance: [ 1 12 , 5 1728 x � � 2 F 1 12] , [1] , = (5 + 10 x + x 2 ) 3 5 + 10 x + x 2 1728 x 5 [ 1 12 , 5 � 1 / 4 � � � 12] , [1] , F 1 3125 + 250 x + x 2 (3125 + 250 x + x 2 ) 3 . 2 9/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Modular forms as pullbacked 2 F 1 functions Emergence of modular forms in physics through 2 F 1 functions Modular forms emerge through covariance properties of 2 F 1 : � � � � 2 F 1 [ α, β ] , [ γ ] , p 1 ( x ) = A ( x ) 2 F 1 [ α, β ] , [ γ ] , p 2 ( x ) A ( x ), p 1 ( x ) and p 2 ( x ) are rational functions. p 1 ( x ) and p 2 ( x ) are called pullbacks, the 2 F 1 is thus called pullbacked. For instance: [ 1 12 , 5 1728 x � � 2 F 1 12] , [1] , = (5 + 10 x + x 2 ) 3 5 + 10 x + x 2 1728 x 5 [ 1 12 , 5 � 1 / 4 � � � 12] , [1] , F 1 3125 + 250 x + x 2 (3125 + 250 x + x 2 ) 3 . 2 � � � � w.l.o.g we have: A ( x ) 2 F 1 [ α, β ] , [ γ ] , y ( x ) = 2 F 1 [ α, β ] , [ γ ] , x A ( x ) and y ( x ) algebraic functions. 9/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Modular forms as pullbacked 2 F 1 functions Emergence of modular forms in physics through 2 F 1 functions Modular forms emerge through covariance properties of 2 F 1 : � � � � 2 F 1 [ α, β ] , [ γ ] , p 1 ( x ) = A ( x ) 2 F 1 [ α, β ] , [ γ ] , p 2 ( x ) A ( x ), p 1 ( x ) and p 2 ( x ) are rational functions. p 1 ( x ) and p 2 ( x ) are called pullbacks, the 2 F 1 is thus called pullbacked. For instance: [ 1 12 , 5 1728 x � � 2 F 1 12] , [1] , = (5 + 10 x + x 2 ) 3 5 + 10 x + x 2 1728 x 5 [ 1 12 , 5 � 1 / 4 � � � 12] , [1] , F 1 3125 + 250 x + x 2 (3125 + 250 x + x 2 ) 3 . 2 � � � � w.l.o.g we have: A ( x ) 2 F 1 [ α, β ] , [ γ ] , y ( x ) = 2 F 1 [ α, β ] , [ γ ] , x A ( x ) and y ( x ) algebraic functions. Modular equation M ( x , y ( x )) = 0: 1953125 x 3 y 3 − 187500 x 2 y 2 ( x + y ) + 375 xy (16 x 2 − 4027 xy + 16 y 2 ) − 64( x + y )( x 2 + 1487 xy + y 2 ) + 110592 xy = 0 9/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Schwarzian condition Theorem ( Abdelaziz–Maillard, 2016 ) If we have a pullback given by: � � � � A ( x ) 2 F 1 [ α, β ] , [ γ ] , x = 2 F 1 [ α, β ] , [ γ ] , y ( x ) then we have the following “Schwarzian condition”: W ( x ) − W ( y ( x )) y ′ ( x ) 2 + { y ( x ) , x } = 0 W ( x ) := p ′ ( x ) + p ( x ) 2 where − 2 q ( x ) 2 p ( x ) = ( α + β + 1) x − γ αβ with q ( x ) = x ( x − 1) x ( x − 1) NB: The Schwarzian derivative is defined by � y ′′ ( x ) � 2 { y ( x ) , x } := y ′′′ ( x ) y ′ ( x ) − 3 2 y ′ ( x ) 10/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics

Schwarzian condition Theorem ( Abdelaziz–Maillard, 2016 ) If we have a pullback given by: � � � � A ( x ) 2 F 1 [ α, β ] , [ γ ] , x = 2 F 1 [ α, β ] , [ γ ] , y ( x ) then we have the following “Schwarzian condition”: W ( x ) − W ( y ( x )) y ′ ( x ) 2 + { y ( x ) , x } = 0 W ( x ) := p ′ ( x ) + p ( x ) 2 where − 2 q ( x ) 2 p ( x ) = ( α + β + 1) x − γ αβ with q ( x ) = x ( x − 1) x ( x − 1) NB: The Legendre derivative is defined by � y ′′ ( x ) � 2 { y ( x ) , x } := y ′′′ ( x ) y ′ ( x ) − 3 2 y ′ ( x ) 10/27 Youssef Abdelaziz, Jean-Marie Maillard Differentially algebraic equations in physics