Virasoro constraints and W-constraints for the q-kp hierarchy

Transcription

1 Virasoro constraints and W-constraints for the q-kp hierarchy Kelei Tian XŒX Jingsong He å t University of Science and Technology of China Ningbo University July 21, 2009

2 Abstract Based on the Adler-Shiota-van Moerbeke (ASvM) formula, the Virasoro constraints for the p-reduced q-deformed Kadomtsev-Petviashvili (q-kp) hierarchy are established, and then the Virasoro constraint generators are obtained. The another main purpose of this article is to give the W-constraints for the p-reduced q-kp hierarchy constrained by the string equation. q-derivative q-kp hierarchy Virasoro constraints for the p-reduced q-kp hierarchy W-constraints for the p-reduced q-kp hierarchy

3 q-derivative The q-derivative q is defined by q (f (x)) = f (qx) f (x) x(q 1) and the q-shift operator is θ(f (x)) = f (qx), for 0 < q < 1. Let q 1 denote the formal inverse of q. In general the following q-deformed Leibnitz rule holds: q n f = ( ) n θ n k ( q k f ) q n k, n Z (2) k k 0 q where the q-number and the q-binomial are defined by (n) q = qn 1 q 1 ( ) n = (n) q(n 1) q (n k + 1) q, k q (1) q (2) q (k) q ( ) n = 1, 0 q (1)

4 For a q-pseudo-differential operator(q-pdo) of the form P = n i= p i q, i we separate P into the differential part P + = i 0 p i q i and the integral part P = i 1 p i q. i The q-exponent e q (x) is defined as follows e q (x) = n=0 x n (n) q!, (n) q! = (n) q (n 1) q (n 2) q (1) q. Its equivalent expression is of the form (1 q) k e q (x) = exp( k(1 q k ) x k ). (3) k=1

5 q-kp Hierarchy Let L be one q-pdo given by L = q + u 0 + u 1 1 q + u 2 2 q +, (4) which is called Lax operator of q-kp hierarchy. There exist infinite number of q-partial differential equations relating to dynamical variables {u i (x, t 1, t 2, t 3,, ), i = 0, 1, 2, 3, } and can be deduced from generalized Lax equation, L t n = [B n, L], n = 1, 2, 3,, (5) which are called q-kp hierarchy. Here B n = (L n ) + = n b i q i means the positive part of q-pdo, and we will use L n = L n L n + denote the negative part. i=0

6 L in (5) can be generated by dressing operator S = 1 + k=1 s k q k in the following way Dressing operator S satisfies Sato equation L = S q S 1, (6) S t n = (L n ) S, n = 1, 2, 3,. (7) The wave function for q-kp hierarchy is defined by w q (x, t; z) = Se q (xz) exp( t i z i ) i=1 Lw q = zw q

7 The adjoint wave function is w q (x, t; z) = (S ) 1 x/q e 1/q ( xz) exp( and the notation P x/t = i P i(x/t)t i i q is used for P = i p i(x) i q. t i z i ), i=1

8 Theorem (Iliev P) There exists the tau function of q KP hierarchy i.e. τ q s.t. w q = τ q(x; t [z 1 ]) τ q (x; t) e q (xz)exp( t i z i ), (8) i=1 here wq = τ q(x; t + [z 1 ]) e τ q (x; t) 1/q ( xz)exp( [z] = (z, z2 2, z3 3, ). t i z i ), (9) i=1

9 Theorem (Iliev P) If τ is the tau function of the KP hierarchy then τ q (x; t) = τ(t + [x] q ) (10) is the tau function of q KP hierarchy, where [x] q = (x, (1 q)2 2(1 q 2 ) x 2 (1 q)3, 3(1 q 3 ) x 3, ).

10 Additional symmetries of the q-kp hierarchy Define Γ q and M ( Tu M.H.) Γ q = (1 q)i [it i + (1 q i ) x i ] q i 1 (11) i=1 Dressing [ k k q, Γ q ] = 0 gives: M SΓ q S 1 (12) k M = [B k, M] (13) k (M m L n ) = [B k, M m L n ] (14)

11 Define the additional flows or equivalently S t m,n = (M m L n ) S (15) L t m,n The additional flows mn = = [(M m L n ), L] (16) t m,n commute with the hierarchy, i.e. [ mn, k ] = 0 but do not commute with each other, they are the additional symmetries of the q-kp hierarchy.

12 String equation of the q-kp hierarchy Theorem When L p = (L p ) + and 1, p+1 S = 0, the string equation of the p-reduced q-kp hierarchy is [L p, 1 p (ML p+1 ) + ] = 1, p = 2, 3,. (17)

13 Process OLD Additional flows String equation ( n+1,1 ) Virasoro constraint (L n, n = 1, 2, 3, ) Additional flows ŵ τ This paper Virasoro constraints and W-constraints by Adler-Shiota-van Moerbeke (ASvM) formula

14 Vertex operator The vertex operator X q (µ, λ) is introduced by X q (µ, λ) = e q (xµ)eq 1 (xλ)exp( t i (µ i λ i µ i λ i ))exp( i ). i i=1 Vertex operator X q (µ, λ) also can be denoted as i=1 (18) X q (µ, λ) =: exp(α(λ) α(µ)) : (19) where α(λ) = α n λ n n, and α n = n for n > o, α 0 = 0 α n = n t n + (1 q) n x n for n < o,. 1 q n The Symbol :: means that we keep t i be always the right side of j.

15 Taylor expansion of the X q (µ, λ) on µ at the point of λ is here X q (µ, λ) = n= (µ λ) m m=0 m! n= λ m n W (m) n, λ m n W (m) n = m µ X q (µ, λ) µ=λ.

16 The first items of W (m) n are W (o) n = δ n,0, W (1) n = α n, W (2) n = ( n 1)α n + i+j=n : α i α j : W (3) n = (n + 1)(n + 1)α n + i+j+k=n : α i α j α k : 3 (n + 2) : α i α j : 2 i+j=n

17 ASvM Formula There is ASvM formula for q-kp hierarchy: X q (µ, λ)w q (x, t; z) = (λ µ)y q (µ, λ)w q (x, t; z). (20) where the operator Y q (µ, λ) is the generators of additional symmetry of q-kp hierarchy as Y q (µ, λ) = (µ λ) m m=0 m! n= λ m n 1 (M m L m+n ). (21) ASvM formula is equivalent to the following equation n+m,mτ q = W (m+1) n (τ q ) m + 1 (22)

18 Consider the condition n+m,mŵ q = 0, we have ŵ q (G(z) 1) n+m,mτ q τ q = 0. (23) Using the ASvM formula we get ( W n (m+1) m + 1 c)τ q = 0, m = 0, 1, 2, 3 (24) This equation is the key part of this method.

19 Virasoro constraints for p-reduced q-kp hierarchy For m = 0, ( W n (1) c)τ q = 0. (25) 1 It is just the condition L p = (L p ) + for p-reduced q-kp hierarchy. For m = 1, it is i.e. ( 1 2 ( W n (2) 2 i+j=n c)τ q = 0, : α i α j : c)τ q = 0 (26)

20 Let n = kp, and denote t i = t i + (1 q)i i(1 q i ) x i, i = 1, 2, 3,. Virasoro constraints of the p-reduced q-kp hierarchy are here L n τ q = 0, n = 1, 0, 1, 2, 3,, L 1 1 p L 0 1 p n t n + 1 t n p 2p n = p + 1 n 0(modp) i+j=p ij t i t j, n t n + ( p t n p ), n = 1 n 0(modp)

21 L k 1 p n t n + 1 t n+kp 2p n = 1 n 0(modp) i + j = kp i, j 0(modp) L n satisfy Virasoro algebra commutation relations ij t i t j. [L n, L m ] = (n m)l (n+m), m, n = 1, 0, 1, 2, 3,, (27)

22 W-constraints for the p-reduced q-kp hierarchy For m = 2, it is i.e. ( 1 3 ( W n (3) 3 i+j+h=kp c)τ q = 0, : α i α j α h : c)τ q = 0 (28)

23 W-constraints for the p-reduced q-kp hierarchy Let w m i + j + h = mp i, j, h 0(modp) : α i α j α h :, m 2, which are the W-constraints for the p-reduced q-kp hierarchy, i.e. w m τ q = 0, m 2, and they satisfy following algebra commutation relations [L n, w m ] = (2n m)w n+m, n 1, m 2

24 Summary and Discussion m = 1 Virasoro constraints m = 2 W-constraints m = 3, 4, 5, higher algebra constraints in a similar manner The q-kp hierarchy tends to the classical KP hierarchy when q 1 and u 0 = 0, and then t 1 + x x. The result of this paper is consistent with the classical ones given by L. A. Dickey and S. Panda, S.Roy. We have known the Virasoro constraints and W-constraints for the KP, BKP(Tu M.H.), q-kp hierarchy, What about CKP hierarchy?

25 string theory: Matrix Models!Seiberg-Witten theory partition function tau function (Douglas M.) solving Virasoro constraints in matrix models(h-th/ ) Kontsevich-Hurwitz partition function(arxiv: )

26 References V. Kac, P. Cheung, Quantum Calculus(New York: Springer-Verlag,2002 ). P. Iliev, Tau function solutions to a q-deformation of the KP hierarchy, Lett. Math. Phys.44(1998), J.S.He,Y.H. Li, Y.Cheng, q-deformed KP hierarchy and its constrained sub-hierarchy. SIGMA 2 (2006), 060. H.F Shen, M.H. Tu, On the string equation of the BKP hierarchy, arxiv: L. A. Dickey, Soliton Equations and Hamiltonian Systems (2nd Edition)(World Scintific, Singapore, 2003).

27 M.H. Tu, q-deformed KP hierarchy: its additional symmetries and infinitesimal Bäcklund transformations, Lett. Math. Phys. 49(1999), M. Douglas, Strings in less than one dimension and the generalized KdV hierarchies Phys. Lett. B 238, (1990). A.Alexandrov, A.Mironov, A.Morozov, Solving Virasoro Constraints in Matrix Models, arxiv:hep-th/ A.Mironov, A.Morozov, Virasoro constraints for Kontsevich-Hurwitz partition function, arxiv: S. Panda, S. Roy, The Lax operator approach for the Virasoro and the W-constraints in the generalized KdV hierarchy, Internat. J. Modern Phys. A8(1993),

28 Thank you!