Classical and quantum aspects of ultradiscrete solitons. Atsuo Kuniba (Univ. Tokyo) 2 April 2009, Glasgow

Transcription

1 Classical and quantum aspects of ultradiscrete solitons Atsuo Kuniba (Univ. Tokyo) 2 April 29, Glasgow

2 Tau function of KP hierarchy ( N τ i (x) = i e H(x) exp j=1 ) c j ψ(p j )ψ (q j ) i (e H(x) = time evolution op. involving β 1, β 2,...) τ i (x) = det(1 + F ) = j N F jj + 1 j 1 <j 2 N F j 1 j 1 F j1 j 2 F j2 j 1 F j2 j 2 +, F jl = c jq j p j q l ( pj q j ) i 1 β m q j β m p j m

3 Ultradiscrete (tropical) limit lim ɛ log τ i(x) ɛ + with an elaborate ɛ-tuning of the parameters c j, p j, q j, β m leads to a tropical tau function associated with combinatorial data called Rigged Configuration in Bethe ansatz. Solitons in tau function Strings in Bethe ansatz c j ψ(p j )ψ (q j )

4 λ Example from sl n=4 µ (1) µ (2) µ (3) Rigged configuration (µ, r) = (λ, (µ (1), r (1) ),..., (µ (n 1), r (n 1) )) µ (a) : configuration (Young diagram) r (a) : rigging (integers attached to µ (a) ) (+ selection rule) Charge of rigged configuration c(µ, r) := 1 2 n 1 a,b=1 C ab min(µ (a), µ (b) ) min(λ, µ (1) ) + min(λ, µ) = ij min(λ i, µ j ), (C ab ) = Cartan matrix of sl n ) r = i r i n 1 a=1 r (a) Regard rigged conf. = {strings}, string = row attached with rigging.

5 Tropical tau function min (ν,s) τ i (λ) = min (ν,s) {c(ν, s) + ν(i) } (1 i n) extends over the power set of (µ, r). e.g., µ (1) µ (2) 1 µ (3) 3 2 Proposition ([K-Sakamoto-Yamada 27] Tropical Hirota eq. ) τ k,i 1 + τ k 1,i = max( τ k,i + τ k 1,i 1, τ k 1,i 1 + τ k,i λ k ), where τ k,i = τ i (λ 1,..., λ k ), τ k,i = τ k,i r (a) r (a) +δ a1 µ (1)

6 Rigged configuration originates in string hypothesis in Bethe ansatz Example from sl 2 (Heisenberg chain) H = L (σ x k σx k+1 + σy k σy k+1 + σz k σz k+1 ) k=1 Bethe equation for length L = 6 chain with 3 down spins. ( ) u1 + i 6 = (u 1 u 2 + 2i)(u 1 u 3 + 2i) u 1 i (u 1 u 2 2i)(u 1 u 3 2i), ( ) u2 + i 6 = (u 2 u 1 + 2i)(u 2 u 3 + 2i) u 2 i (u 2 u 1 2i)(u 2 u 3 2i), ( ) u3 + i 6 = (u 3 u 1 + 2i)(u 3 u 2 + 2i) u 3 i (u 3 u 1 2i)(u 3 u 2 2i).

7 i 2.2i.472 i i.944 i i.472 i i 1 2

8 Kerov-Kirillov-Reshetikhin (1986) gave a canonical bijection Bethe root {rigged configurations} KKR Bethe vector {highest paths} (1 =, 2 = ) which may viewed as a combinatorial analogue of Bethe ansatz.

9 Higher rank example (sl 4 ) {rigged configurations} 1:1 {highest paths} λ µ (1) 1 µ (2) 3 2 µ (3) Bethe roots Bethe vectors highest path = b 1 b 2... b L b i = row shape (λ i ) semistandard tableau. (+ highest condition)

10 Example of KKR algorithm from sl Top left rigged configuration KKR

11 Theorem.([KSY]) Image of the KKR map is given by (λ, (µ (1), r (1) ),..., (µ (n 1), r (n 1) )) KKR b 1... b L x k,1 x k,n {}}{{}}{ b k = ( ,..., n... n) }{{} λ k (semistandard tableau), x k,i = τ k,i τ k 1,i τ k,i 1 + τ k 1,i 1 We will see that this is an analogue of for KdV eq. u = 2 2 log τ x 2

12 Crystals for U q (ŝl n) B l = { i 1,..., i l semistandard} equipped with crystal structures. u l := B l is the (classically) highest element. An element of B λ1 B λ2 is called a path.

13 Combinatorial R R : B l B m Bm B l, x y ỹ x x i x i = y i ỹ i = Q i (x y) Q i 1 (x y) (i mod n), x i = of letter i in tableau x Q i (x y) = min { k 1 1 k n j=1 x i+j + (y i : similar), n y i+j } i th local energy. j=k+1 Example : will be denoted by or simply

14 Energy E i of path b 1 b k (i mod n) E i (b 1 b k ) := Sum of Q i (x y) attached to all vertices in u b 1 b 2 b k Theorem.([KSY] Tropical fermionic formula ) Suppose b 1 b L KKR (µ, r) {τ k,i }. Then, E i (b 1 b k ) = τ k,i (1 k L)

15 U q (ŝl n) vertex model at q = T l : B 1 B 1 B 1 B 1 B 1 B 1 b 1 b 2 b 3 b 1 b 2 b 3 B l u l b 1 b 2 b 3 b 4 b 1 b 2 b 3 b 4 T 1, T 2,... : commuting family of time evolutions (deterministic transfer matrices)

16 Example of time evolution T 2 : The dynamics on vertical edges reproduces Box-ball system with carrier (Takahashi, Satsuma, Matsukidaira) = empty box, 2, 3, 4 = colored balls.

17 Theorem.([KSY]) (1) Tropical tau function = Energy of crystal (math) previous theorem = Baxter s corner transfer matrix for box-ball system (phys) Let b 1 b 2 b L KKR (µ, r) {τ k,i }. Then, τ k,i = of balls in b 1 b 2 b k time evolution T (2) Tropical Hirota equation = eq. of motion of box-ball system.

18 main combinatorial object Bethe ansatz rigged configuration Corner transfer matrix energy (charge) in crystal role in box-ball system action-angle variable tau function dynamics linear bilinear

19 Dynamics of box-ball system in terms of rigged configuration t = : t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: λ (1 48 ) µ (1) µ (2) 4t t t µ (3) configuration conserved quantity (action variable) rigging linear flow (angle variable) KKR bijection direct/inverse scattering map (separation of variables)

20 Summary so far classical KP tau Ultradiscretization (top down) Tropical tau Comb. Bethe ansatz (bottom up) quantum Charge (energy of crystal) KKR theory = inverse scattering scheme of box-ball system on lattice [K-Okado-S-Takagi-Y 26] Initial value problem solved and General N-soliton solution constructed for ŝl n symmetric tensor reps. [KSY 27] (ŝl 2 2-dim. rep. case also by [Mada-Idzumi-Tokihiro 28])

21 Periodic generalization (sl 2 case) T l : B 1 B 1 B 1 B 1 B 1 B 1 b 1 b 2 b L b 1 b 2 b L b 1 b 2 b 3 b 4 b L B l u u B l b 1 b 2 b 3 b 4 b L Choice s.t. u = u : periodic box-ball system (Yura-Tokihiro 22) Example of T 3 : ( B 1 = { 1, 2 } )

22 Action-angle variables any path highest path rigged conf. cyclic shift KKR b 1...b L b d+1...b L b 1...b d (µ, I) (not unique) µ = (µ 1,..., µ g ) I = (I µ1,..., I µg ) µ 1 µ g 1 I µg 1 I µ1 p i := L 2 j µ min(i, j) µ g I µ g Lemma. (For simplicity assume µ 1 > > µ g ) µ is unique and invariant under {T l } (action variable) (I + d h 1 )/AZ g is unique (angle variable), where h l = (min(l, i)) i µ Z g, A = ( δ ij p i + 2 min(i, j) ) i,j µ

23 P(µ) := {paths whose action variable = µ} J (µ) := Z g /AZ g iso-level set set of angle variables Φ : P(µ) J (µ) by Φ(b 1... b L ) := (I + d h 1 )/AZ g Theorem. ([KT-Takenouchi 26] Tropical Abel-Jacobi map) Φ is a bijection and P(µ) T l Φ J (µ) T l is commutative. P(µ) Φ J (µ) where T l (J) = J + h l on J (µ). Nonlinear dynamics becomes straight motion in J (µ) = Z g /AZ g, which is an tropical analogue of Jacobi variety.

24 Solution of initial value problem (inverse method) direct scattering Φ linear flow T 1 3 T mod AZ (answer) inverse scattering Φ

25 Tropical Riemann theta (z R g ): Θ(z) := min n Z g{t nan/2 + t nz} Theorem. ([K-Sakamoto 26] Tropical Jacobi inversion ) J (µ) P(µ) (µ, I) b 1 b 2... b L ( {1, 2} L ) is given by b k = 1 + Θ ( ) ( ) J kh 1 Θ J (k 1)h1 Θ ( ) ( ) J kh 1 + h + Θ J (k 1)h1 + h, with J i = I i 1 2 p µ i. Also obtained in [Mada-Idzumi-Tokihiro 28]. Higher spin generalization, Θ-formula for Carrier [KS28].

26 Tropical period matrix A originates in Bethe ansatz at q =. (KKR is q = 1.) U q (ŝl 2) Bethe equation at q = under string hypothesis: Ax constant vector mod AZ g linear! x (R/Z) g : Bethe root Bethe root x 1:1 J J (µ) = Z g /AZ g via Ax = J. Generic dynamical period is the minimum N s.t. ( N (velocity vector) AZ g N = LCM 1, j det A det A[j] ). A is known for affine g [K-Nakanishi 22]. Conjectures for generic dyamical period, etc.

27 A (1) 2 path = B 21 1 l LCM of = period under T l 1 1, 21, 21, 21, , 29, , , , 22, , , , 5, , , A (1) 3 path = B 3 B 2 B 1 B 3 B 2 B 1 B 2 (r, l) LCM of = period under T (r) l 38 (1,1) 1, 39, 95 6, 95 6, 38 31, 38 27, (1,2) 1, 39, 95 12, 95 12, 19 31, 19 27, (2,1) 1, 13, 95 4, 95 4, , 19 9, (2,2) 1, 5, 38 3, 38 3, 76 41, 76 21, (2,3) 1, 6, 95 11, 95 11, 95 34, 95 48, (3,1) 1, 13, 95 2, 95 2, , 38 9,

28 I haven t a slightest idea of what people did with it. H. Bethe