Virasoro and W-constraints for the $q$-KP hierarchy

The origin of q-calculus (quantum calculus) [1,2] traces back to the early 20th century. Many mathematicians have important works in the area of q-calculus and qhypergeometric series. The q-deformation of classical nonlinear integrable system (also called q-deformed integrable system) started in 1990's by means of q-derivative ∂ q instead of usual derivative ∂ with respect to x in the classical system. As we know, the q-deformed integrable system reduces to a classical integrable system as q goes to 1. Several q-deformed integrable systems have been presented, for example, qdeformation of the KdV hierarchy [3,4,5], q-Toda equation [6], q-Calogero-Moser equation [7]. Obviously, the q-deformed Kadomtsev-Petviashvili (q-KP) hierarchy is also a subject of intensive study in the literature from [8] to [13]. The additional symmetries, string equations and Virasoro constraints [14,15,16,17,18,19] of the classical KP hierarchy are important since they are involved in the matrix models of the string theory [20]. For example, there are several new works [21,22,23,24,25] on this topic. It is quite interesting to study the analogous properties of qdeformed KP hierarchy by this expanding method. In [11], the additional symmetries of the q-KP hierarchy were provided. Recently, additional symmetries and the string equations associated with the q-KP hierarchy have already been reported in [11,13]. The negative Virasoro constraint generators {L -n , n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained in [13] by the similar method of [18]. Our main purpose of this article is to give the complete Virasoro constraint generators {L n , n ≥ -1} and W-constraints {w m , m ≥ -2} for the p-reduced q-KP hierarchy by the different process with negative part of Virasoro constraints given in [13]. The method of this paper is based on Adler-Shiota-van Moerbeke (ASvM) formula. This paper is organized as follows. We give a brief description of q-calculus and q-KP hierarchy in Section 2 for reader's convenience. The main results are stated and proved in Section 3, which are the Virasoro constraints and W-constraints on the τ function for the p-reduced q-KP hierarchy. Section 4 is devoted to conclusions and discussions. q-CALCULUS AND q-KP HIERARCHY At the beginning of the this section, Let us recall some useful facts of q-calculus [2] in the following to make this paper be self-contained. The Euler-Jackson q-difference ∂ q is defined by and the q-shift operator is θ ( f (x)) = f (qx). It is worth pointing out that θ does not commute with ∂ q , indeed, the relation as q approaches 1 is the ordinary differentiation ∂ x ( f (x)). We denote the formal inverse of ∂ q as ∂ -1 q . The following q-deformed Leibnitz rule holds where the q-number (n) q = q n -1 q-1 and the q-binomial is introduced as Let (n) q ! = (n) q (n -1) q (n -2) q • • • (1) q , the q-exponent e q (x) is defined by Similar to the general way of describing the classical KP hierarchy [14,19], we will give a brief introduction of q-KP hierarchy and its additional symmetries based on [10,11]. The Lax operator L of q-KP hierarchy is given by where The corresponding Lax equation of the q-KP hierarchy is defined as here the differential part 3) can be generated by dressing operator S = 1 + ∑ ∞ k=1 s k ∂ -k q in the following way Dressing operator S satisfies Sato equation The q-wave function w q (x,t; z) and the q-adjoint function w * q (x,t; z) of q-KP hierarchy are given by which satisfies following linear q-differential equations here the notation P| x/t = ∑ i P i (x/t)t i ∂ i q is used for a q-pseudo-differential operator of the form P = ∑ i p i (x)∂ i q , and the conjugate operation " * " for P is defined by q , (PQ) * = Q * P * for any two q-PDOs. Furthermore, w q (x,t; z) and w * q (x,t; z) of q-KP hierarchy can be expressed by sole function τ q (x;t) [10] as w * q = τ q (x;t + [z -1 ]) τ q (x;t) e 1/q (-xz)e -ξ (t,z) = e 1/q (-xz)e -ξ (t,z) e τ q e q (xz)e ξ (t,z) ≡ ŵq e q (xz)e ξ (t,z) . The following Lemma shows there exist an essential correspondence between q-KP hierarchy and KP hierarchy. Lemma 1. [10] Let , be a solution of the classical KP hierarchy and τ be its tau function. Then τ q (x,t) = τ(t + [x] q ) is a tau function of the q-KP hierarchy associated with Lax operator L in eq.( 3), where Define Γ q and Orlov-Shulman's M operator [11] for q-KP hierarchy as M = SΓ q S -1 and The the additional flows for each pair {m, n} are difined as follows or equivalently The additional flows [∂ * mn , ∂ k ] = 0 but do not commute with each other, so they are additional symmetries [12]. (M m L n ) - serves as the generator of the additional symmetries along the trajectory parametrized by t * m,n . Theorem 1. [13] If an operator L does not depend on the parameters t n and the additional variables t * 1,-n+1 , then L n is a purely differential operator, and the string equations of the q-KP hierarchy are given by In this section, we mainly study the Virasoro constraints and W-constraints on τ-function of the p-reduced q-KP hierarchy. To this end, two useful vertex operators X q (µ, λ ) and Y q (µ, λ ) would be introduced. The vertex operator X q (µ, λ ) is defined in [11] as We can also denote the vertex operator X q (µ, λ ) by where the symbol :: means that we keep t i be always left side of ∂ j , and α(λ The following lemma is given without proof. Lemma 2. Taylor expansion of the X q (µ, λ ) on µ at the point of λ is The first items of There is Adler-Shiota-van Moerbeke (ASvM) formula [11] for q-KP hierarchy as where the operator Y q (µ, λ ) is the generators of additional symmetry of q-KP hierarchy as ASvM formula is equivalent to the following equation The following theorem holds by virtue of the ASvM formula. Theorem 2. ( W Proof. Consider the condition ∂ * m,n+m ŵq = 0, from eq.( 8), and denote τq = G(z)τ q , The operator G(z) has the property, which is where c is constant. Combining eq.( 16) with eq.( 18) finishes the proof. Now we consider the p-reduced q-KP hierarchy, by setting (L p ) -= 0, i.e. L p = (L p ) + . From Lax equation of q-KP hierarchy, the p-reduced condition means that L is independent on t j p as ∂ j p L = 0, j = 1, 2, 3, • • • and τ q is independent on t j p as ∂ j p τ q = 0, j = 1, 2, 3, • • •. Based on theorem 2, the Virasoro constraints and W-constraints for the p-reduced q-KP hierarchy will be obtained. Let n = kp in theorem 2 and denote First of all, for m = 0, eq.( 17) in theorem 2 becomes Let c = 0, we have that α kp τ q = ∂ τ q ∂t kp = 0, it is just the condition L p = (L p ) + for p-reduced q-KP hierarchy. For m = 1, it is Theorem 3. The Virasoro constraints imposed on the tau function τ q of the p-reduced q-KP hierarchy are ), and L n satisfy Virasoro algebra commutation relations Proof. Following the results in eq.( 20) and eq.( 21), we have Theorem 4. Let the W-constraints on the tau function τ q of the p-reduced q-KP hierarchy are and they satisfy following algebra commutation relations For m ≥ 3, using the similar technique in theorem 3 and 4, we can deduce the higher order algebraic constrains on the tau function τ q of the p-reduced q-KP hierarchy. Remark 1. As we know, the q-deformed KP hierarchy reduces to the classical KP hierarchy when q → 1 and u 0 = 0. The parameters ( t1 , t2 , One can further identify t 1 + x with x in the classical KP hierarchy, i.e. t 1 + x → x. The deformation as q goes to 1 of Virasoro constraints and W-constraints for the p-reduced q-KP hierarchy are identical with the results of the classical KP hierarchy given by L.A.Dickey [16] and S.Panda, S.Roy [18]. To summarize, we have derived the Virasoro constraints and W-constraints of the preduced q-KP hierarchy in theorem 3 and 4 respectively. The results of this paper show obviously that the Virasoro constraint generators {L n , n ≥ -1} and W-constraints {w m , m ≥ -2} for the p-reduced q-KP hierarchy are different with the form of the KP hierarchy. Furthermore, we also would like to point out the following interesting relation between the q-KP hierarchy and the KP hierarchy and it seems to demonstrate that q-deformation is a non-uniform transformation for coordinates t i → ti , which is consistent with results on τ function [10] and the q-soliton [12] of the q-KP hierarchy. Here Ln [16,18] are Virasoro generators of the KP hierarchy.