q-garnier system and its autonomous limit

Transcription

1 q-garnier system and its autonomous limit Yasuhiko Yamada (Kobe Univ.) Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics ESI, 24.Mar

2 Introduction Garnier system [Garnier (1912)] is a 2N variable extension of Painlevé VI equation (N = 1). q-garnier system [Sakai (2005)] is its q-difference analog. In this talk, I will study the q-garnier system and its autonomous limit from geometric/physical point of view. Ref: [Nagao-Y, (arxiv )] 2

3 Autonomous limit of discrete Painlevé equations are integrable system (QRT system). Example : q-painlevé VI [Jimbo-Sakai (1996)] T t : a, b, c, d r, s, t, u, x, y a, b, qc, qd r, s, qt, qu, x, ȳ, q = abtu cdrs, x = ab (ȳ + qt)(ȳ + qu) x (ȳ + r)(ȳ + s), ȳ = rs y (x + c)(x + d) (x + a)(x + b). a discrete dynamical system on (x, y) plane. (nonintegrable in general) 3

4 When q = 1, q-p VI is integrable. x ab (y + t)(y + u) x (y + r)(y + s), y rs y (x + c)(x + d) (x + a)(x + b), with a constraint abtu cdrs = 1. conserved elliptic curve: (x+a)(x+b)y 2 +{(r+s)x 2 +ab(t+u)}y +rs(x+c)(x+d) = Hxy. We will generalize this to hyper-elliptic curves. 4

5 Plan of the talk 1. Derive a simple form of Lax pair and evolution equations of q-garnier system 2. Generalize QRT system to hyper-elliptic curve 3. Two dual formulations of q-garnier system from q-kp 5

6 1. Lax pair and evolution equations of q-garnier system 6

7 Lax pair for q-garnier (contiguous type) L 2 : F (x)ȳ(x) + G(x)y(x) A(x)y(qx) = 0, L 3 : qx F (x)y(qx) + G(x)ȳ(qx) qtb(x)ȳ(x) = 0. A(x) = N+1 i=1 F (x) = N i=0 (x a i ), B(x) = f i x i, N+1 i=1 G(x) = ct + N i=1 (x b i ), g i x i + x N+1. Shift of parameters: (ā i, b i, c, t) = (a i, b i, c, qt). We have 2N(+1) dynamical variables f i, g i. normalization of f 0,..., f N is a gauge factor) ȳ(x) ȳ(qx) y(x) y(qx) (Overall 7

8 L 2 (or L 3 ) describing a deformation of L 1 : A(x)F ( x q )y(qx) R(x)y(x) + tb(x q )F (x)y(x q ) = 0, where R(x) is a polynomial of degree 2N + 1. Compatibility of L 2, L 3 (or L 1 ) q-garnier system: xf (x) F (x) = ta(x)b(x) for G(x) = 0, G(x)G(x) = ta(x)b(x) for F (x) = 0. 8

9 Rewrite the L 1 equation as [ A(x)F ( x q )T R(x) + tb(x q )F (x)t 1] y(x) = 0, where T x = qxt. In q 1, the L 1 equation is divisible by F (x) and becomes A(x)T U(x) + tb(x) T = 0. This is an algebraic equation in commuting variables (x, T ) of bi-degree (N + 1, 2). a hyper-elliptic curve of genus g = N. (Seiberg-Witten curve) We will see that autonomous q-garnier system = generalized QRT system on hyper-elliptic curve. 9

10 2. QRT system and its hyper-elliptic generalization 10

11 QRT system [Quispel-Roberts-Tompson (1989)] [Tsuda (2004)] It is an integrable mapping on plane preserving a family of curves of bi-degree (2, 2): C u : φ 1 (x, y) + u φ 2 (x, y) = 0. We have two flips r x, r y 7 6 r x : (x, y) ( x, y), r y : (x, y) (x, ȳ). 5 4 Iterations of r x, r y = QRT system. The curve C u is conserved. We generalize this to hyper-elliptic curves

12 Consider a family of curves of bi-degree (N + 1, 2) C u : A(x)y U(x) + tb(x) y = 0, passing through the following 2N + 6 points: y = y = 0 (with one constraint) x = 0 x = Coefficients of U(x) = N+1 i=0 u i x i are free (except for u 0, u N+1 ) N-dimensional family of hyper-elliptic curves of g = N. 12

13 To fix the parameters u i, we need N points Q i = (x i, y i ). D = {Q 1,, Q N } = dynamical variables. However, the evolution equation (addition formula) on hyperelliptic curve is not bi-rational in (x i, y i )-coordinates. Following Mumford, we represent the divisor D by a pair of polynomials F (x), G(x) (degree N, N + 1) such that F (x i ) = 0, y i = G(x i) A(x i ). one can define two bi-rational flips r x : (F, G) ( F, G), r y : (F, G) (F, Ḡ). 13

14 y-flip: C u is deg = 2 in y hyper-elliptic involution y i ȳ i, y i ȳ i = tb(x i) A(x i ). Since y i = G(x i) A(x i ) for F (x i) = 0, we have G(x)Ḡ(x) = ta(x)b(x) for F (x) = 0. This corresponds to the equation for G(x) of q-garnier system. 14

15 x-flip: Note that G(x) A(x)y U(x) + tb(x) y /. y = G(x) A(x) = G(x) 2 U(x)G(x) + ta(x)b(x) =: P (x). From Q i C u (and some additional conditions), one can define F (x) by P (x) = xf (x) F (x), i.e. xf (x) F (x) = ta(x)b(x) for G(x) = 0. This corresponds to the equation for F (x) of q-garnier system. Hence, autonomous q-garnier = hyper-elliptic QRT. 15

16 Numerical example. N = 2 case log-log plot of (x 1, y 1 ), (x 2, y 2 ) 16

17 Conserved curve (amoeba) 17

18 Tropical limit (N = 3) y = y = x = 0 x = 18

19 Variations. So far, the 2N + 6 points are on four lines: can be generalized to any bi-degree (2,2) curve. In the most generic case, we obtain hyper-elliptic QRT system of elliptic difference type. (5d 6d lift in gauge theory) Its non-autonomous deformation gives an elliptic Garnier system. Equivalent (?) to the one constructed by Rains- Ormerod [Rains-Ormerod(2016)]. 19

20 An elliptic Garnier system. [x] = (x, p/x; p) L 2 : F (x)[ k x 2]ȳ(x q ) G(x)A(k x )y(x q ) + G(k x )A(x)y(x), L 3 : F (x)[ k x 2]y(x) G(x)B(k x )y(x) + G(k x )B(x)y(x q ), A(x) = G(x) = x N+3 i=1 N+1 i=1 [ x a i ], B(x) = [ x µ i ], N+1 i=1 N+3 i=1 [ x b i ], F (x) = wx N µ i = l, k = k/q, l = ql. i=1 [ x ][ k λ i λ i x ], It has special solutions given in terms of elliptic HG : τ = det[ 2N+10 V 2N+9 ]. (N = 1 ell-e (1) 8. [Noumi-Tsujimoto-Y.(2013)]) 20

21 One can also consider various degenerate configurations of 2N + 6 points. Example. Differential Garnier system: (5d 4d reduction in gauge theory) 21

22 3. Two dual formulations from q-kp 22

23 (m, n)-reduced Lax operator for q-kp. Ψ(qz) = A(z)Ψ(z), A(z) = d(t)x m (z) X 1 (z), X i (z) = x i,1 1 x i, x i,n 1 1 r i z d(t) = diag(t 1,, t n ). x i,n W (A (1) m 1 ) W (A(1) n 1 ) symmetry [Kajiwara-Noumi-Y (2002)] ( Tropical R-map, Yang-Baxter map.) Interpretation [Inoue-Lam-Pylyavskyy (2016)] as cluster integrable system., 23

24 A duality : m n. This duality is known as Spectral duality [Milonov-Morozov-Runov-Zenkevich-Zotov (2012)], Fiber-base duality [Mitev-Pomoni-Taki-Yagi (2014)],. Ψ(qz) = A(z)Ψ(z) = d(t)x m (z) X 2 (z)x 1 (z)ψ(z). We put Ψ 1 = Ψ, Ψ i+1 = X i Ψ i (1 i m), and ψ i,j = (Ψ i ) j, then ψ i+1,j = x i,j ψ i,j + ψ i,j+1, ψ m+1,j = t 1 j T z ψ 1,j, ψ i,n+1 = r i zψ i,1. These relations are symmetric under the exchange: m n, ψ i,j ψ j,i, x i,j x j,i, r k t 1 k, z T z.

25 Two Lax form of q-garnier system Ψ(qz) = A(z)Ψ(z), Ψ(z) = B(z)Ψ(z). (i) (m, n) = (2, 2N + 2) case: [Suzuki (2015)] A(z) =.. + z. (ii) (m, n) = (2N + 2, 2) case:(= [Sakai (2005)], up to gauge) A(z) = + z + + z N + z N+1. 24

26 Case (i): We consider the deformation A(z) = X 2 (z)x 1 (z), B(z) = X 1 (z), d(t) = 1. The time evolution equation ( compatibility condition ĀA(z)B(z) = B(qz)A(z)) is explicitly written as follows: (the Yang-Baxter-map) P 1 = n k=1 x j = x j P j+1 P j, ȳ j = y j P j P j+1, k 1 x j j=1 n y j j=k+1, P j+1 = µπ(p j ), where π(x j ) = µ δ j,nx j+1 (similar for y j ), µ = r 2 qr 1, r 1 = r 2, r 2 = qr 1. 25

27 Case (ii) A(z) = r r 1 2 X 2N+2 X 1, B(z) = 0 1 z y 1 x 1 Then, for F (z) = A(z) 12, G(z) = A(z) 11, we have. zf (z) F (z) = det A(z) for G(z) = 0, G(z)Ḡ(z) = det A(z) for F (z) = 0. 26

28 Summary We studied the q-garnier system (and elliptic Garnier system) from a geometric points of view. Simple form of the Lax pair and evolution equations are given. Autonomous limit is interpreted as generalized QRT system. As reduction of q-kp, two dual Lax formulations are obtained. Thank you! 27

29 Special solutions. [Nagao-Y (2016)] Consider the configurations where the points are in special position (ratio q Z ) (1) 1 1 (2) Then the q-garnier system has terminating HG solutions such that the τ-functions are given by determinants of (1) q-appell-lauricella functions φ D of N-variables, (2) generalized q-hypergeometric functions N+1 φ N. 28

30 The idea of the derivation : Padé method. (1) Padé approximation (differential grid): ψ(x) = N+1 i=1 (a i x; q) = P m(x) (b i x; q) Q n (x) + O(xm+n+1 ), x 0. (2) Padé interpolation (on q-grid): ψ(x) = c log q x N i=1 (a i x; q) = P m(x) (b i x : q) Q n (x), (x = 1, q,..., qm+n ). Key fact. The functions y(x) = P m (x), ψ(x)q n (x) give the solutions of the Lax linear equations L 2, L 3 (and L 1 ). 29

31 Schur function representation of τ-function q-garnier: τ m,n = det [ p m i+j ] n i,j=1, differential Garnier: t k = N+1 i=1 exp( k=1 b k i ak i k(1 q k ). a i = s i q α i, b i = s i, q 1, hence t k = t k x k ) = N+1 i=1 k=0 α i s k i k. p k x k, 30