The "Ghost" Symmetry of the BKP hierarchy

Symmetry [1] plays an important role in the study of the integrable system. Many crucial properties of the integrable system, such as the Noether conserved laws, Hamiltonian structure, Darboux transformation and reduction, are closely connected with symmetries. There are several kinds of symmetry of the integrable system. For instance,in the well-known KP theory, there is an important symmetry called "ghost" symmetry [2]. By identifying the "ghost" symmetry with the k-th time flow, the constrained KP hierarchy (cKP) [3][4][5][6][7][8][9][10][11] can be easily defined. In this paper, we shall focus on the study of the "ghost" symmetry of the BKP hierarchy. The "ghost" symmetry was first introduced by W.Oevel [2] in studying the solutions of the cKP hierarchy. Then it was extensively studied in [5,[12][13][14][15][16]. In the KP hierarchy, the "ghost" symmetry is closely related with a squared eigenfunction potential (SEP), which is associated to a pair of arbitrary eigenfunction Φ(t) and adjoint eigenfunction Ψ(t) by means of following definition [2]: Here1 M n = L n + and L is a Lax operator of the KP hierarchy. The predecessor of SEP was in fact the Cauchy-Baker-Akhiezer kernel introduced in [17], which is an important object for a study of vector fields action on Riemann surfaces and Virasoro action on tau functions. In [15], Aratyn et al gave a systematic study for the SEP and the "ghost" symmetry in KP case. By using SEP as a basic building block in the definition of the KP hierarchy, they established a new way to reformulate the theory of the KP hierarchy called SEP method. The crucial fact of the SEP method is that there exists a spectral representation for any eigenfunction of the KP hierarchy with SEP as an spectral density. They also showed that the "ghost" symmetry [5,13], which is generated by SEP, has close relation with the additional symmetries of the KP hierarchy [18][19][20][21][22][23][24].In fact, SEP can be regarded as a generating function for the additional symmetries of the KP hierarchy when both eigenfunction Φ(t) and adjoint eigenfunction Ψ(t) defining the SEP are Baker-Akhiezer (BA) functions. In present work, we would like to consider the "ghost" symmetry for the BKP hierarchy. Here BKP hierarchy [25] is an important reduction of the ordinary KP hierarchy under the constraints on the Lax operator L * = -∂L∂ -1 . In contrast with the KP hierarchy, the SEP of the BKP hierarchy can not generate directly a symmetry flow due to the BKP constraints L * = -∂L∂ -1 . Thus, we have to find a new potential, which is used to generate the "ghost" symmetry of the BKP hierarchy and is expected to be expressed by SEP. So this new potential is called the B-type of the squared eigenfunction potential(BSEP). Fortunately, as we shall show, the BSEP was first introduced by Loris [16] in the study of symmetry reduction of the BKP hierarchy. Similar to the case of the KP hierarchy [15], before giving the "ghost" symmetry of the BKP hierarchy, we need to study SEP first. Starting from the BKP bilinear identity, we shall show that there is also a spectral representations for the eigenfunctions of the BKP hierarchy, i.e., any eigenfunction of BKP hierarchy can be represented as a spectral integral over BA wave function with a spectral density expressed in terms of SEP. Then according to the differential Fay identity of the BKP hierarchy, we get the expression of the basic SEP(the one whose defining eigenfunctions are BA functions). Thus we can give the general expressions of SEP for the BKP hierarchy with the spectral representation. We then point out the importance of the spectral representations by showing that it can in fact provide another definition of the BKP hierarchy. In other words, we get an equivalent formulation of BKP hierarchy. We also call it SEP method for the BKP hierarchy. Next, after BSEP is systematically studied, we define the "ghost" symmetry flows ∂ α for the BKP hierarchy by means of its action on the Lax operator L and the dressing operator W . Furthermore, actions of ∂ α on the eigenfunction Φ and τ function are given by BSEP. At last,we consider applications for above theory. We shall first derive the bilinear identities for the cBKP hierarchy [16,27,28] with the SEP method. And then by letting eigenfunctions in the BSEP be BA functions,we get the relation between the "ghost" symmetry and the additional symmetry: in this case, the BSEP becomes a generating function for the additional symmetries of the BKP hierarchy. With the help of this fact, we shall give a simple and straightforward proof for the Adler-Shiota-van Moerbeke formula of the BKP hierarchy [29][30][31][32]. This paper is organized in the following way.In section 2,some basic facts about the BKP hierarchy are reviewed. Then,SEP for the BKP hierarchy is studied in detail in section 3. After some interesting properties of the BSEP studied in section 4, the "ghost" symmetry for BKP is showed in section 5. At last, we devote section 6 to two applications on the spectral representation and the "ghost" symmetry. Here, we shall review some basic facts about the BKP hierarchy [25]. The BKP hierarchy can be defined in Lax form as where the Lax operator is given by with coefficient functions u i depending on the time variables t = (t 1 = x, t 3 , t 5 , • • • ) and satisfies the BKP constraint It can be shown [25] that the constraint ( 4) is equivalent to the condition (B 2n+1 ) [0] = 0 . The Lax equation ( 2) is equivalent to the compatibility condition of the linear system 2 where ψ BA (t, λ) is called BA wave function. The whole hierarchy can be expressed in terms of a dressing operator W , so that and the Lax equation is equivalent to the Sato's equation with constraint Let the solutions of the linear system (5) be the form where ξ(t, λ) = ∞ i=0 t 2i+1 λ 2i+1 and w(t, λ) = 1 + w 1 /λ + w 2 /λ 2 + • • • . Then ψ BA (t, z) is a wave function of the BKP hierarchy if and only if it satisfies the bilinear identity [25] dλλ where dλ ≡ ∞ dλ 2πi = Res λ=∞ and t = (t 1 = x, t 3 , t 5 , • • • ). 2 For a differential operator A and a function f , A(f ) denotes the action of A on f. In the BKP hierarchy,if Φ(or Ψ) satisfies we shall call Φ (or Ψ) eigenfunction (or adjoint eigenfunction) of the BKP hierarchy. Obviously, ψ BA (t, λ) is also an eigenfunction. The relation between the eigenfunctions and adjoint eigenfunctions can be seen from the fact B * 2n+1 ∂ = -∂B 2n+1 . This fact implies that any eigenfunction Φ gives rise to an adjoint eigenfunction Ψ = Φ x . In particular,we have ψ * BA (t, λ) = -λ -1 ψ BA (t, -λ) x , where ψ * BA (t, λ) ≡ W * -1 (e -ξ(t,λ) ). Moreover, from the bilinear identity (9), solutions of the BKP hierarchy can be characterized by a single function τ (t) called τ -function such that [25] where This implies that all dynamical variables {u i } in the Lax operator L can be expressed by τ -function, which is an essential character of the KP and BKP hierarchy. Moreover, another important property of τ function of the BKP is the following Fay like identity. Proposition 1. [31] (Fay identity) The tau function of the BKP hierarchy satisfies: where (s 1 , s 2 , s 3 ) stands for cyclic permutations of s 1 , s 2 and s 3 . Proposition 2. [31] (Differential Fay identity)For the BKP hierarchy, Note that these identities are indeed different from the counterpart of the KP hierarchy because of the BKP constraint (4). In the next context, we shall show it is for the same reason that the SEP of the BKP hierarchy can not generate directly the symmetry flow. As mentioned in Introduction, we hope to get a new potential-BSEP from the SEP of the BKP hierarchy. So we shall study some interesting properties of the SEP of the BKP hierarchy in this section. For any pair of (adjoint) eigenfunctions Φ(t), Ψ(t),there exists a function S(Φ(t), Ψ(t)) called SEP, determined by the following equations, In particular,for n = 0,we have, One can see that this definition is the same as the one [2] in the KP hierarchy except even number flows are frozen. There are two properties of SEP for the BKP hierarchy. Lemma 3. If Φ(t) and Ψ(t) are BKP eigenfunction and adjoint eigenfunction respectively, then one has the following relation: Proof:We only prove the first identity since the proof of the second one is similar. Because ) and we find Lemma 4. If Φ 1 and Φ 2 are two eigenfunctions of the BKP hierarchy, then Proof: is an eigenfunction of the BKP hierarchy ,then where the time t ′ is taken at some arbitrary fixed value. In other words, Φ(t) owns a spectral representation in the form of with spectral densities given by SEP,that is, Proof: Denote the RHS of ( 22) by I(t, t ′ ).Then by the BKP bilinear identity (9), one finds that . By considering ( 16), we have Remark 1:Here we only give the spectral representation for eigenfunctions.As for the adjoint eigenfunctions,the spectral representation can be derived similarly by considering ( 9) and ( 17),that is, 3 Some useful formulas below are needed in the proof. where A is a pseudo-differential operator,and a is a function. However,because of the relation between the eigenfunctions and adjoint eigenfunctions,we must show that our spectral representations for BKP hierarchy are compatible. In fact,any adjoint eigenfunction Ψ for BKP can be written as the derivative of an eigenfunction Φ, that is, Ψ = Φ x . So with the help of ( 22), ( 9), (18) and So our representation is consistent with Ψ = Φ x , which shows it is necessary to only study the spectral representation of the eigenfunctions for the BKP hierarchy. x ,so we can rewrite (22) as Our results ( 24) and ( 25) can be regarded as a natural reduction from corresponding ones [15] of the KP hierarchy by considering BKP constraints L * = -∂L∂ -1 and Ψ = Φ x . Remark 3:In particular, is given from ( 22) by setting Φ(t) = ψ BA (t, µ). Now we shall use the above obtained spectral representation to get general expressions of SEP.Before this we will use the differential Fay identity (13) to get S(ψ BA (t, µ), ψ BA (t, -λ) x ), which is a basic and useful SEP of the BKP hierarchy. According to the proposition 2, sets 1 = λ -1 and s 2 = -µ -1 , we can find Taking into account of the following identity, Furthermore, in order to get S(ψ BA (t, µ), ψ BA (t, -λ) x ), by using ( 8) and ( 11), we first calculate Note is used in the first equality above. Taking (27) into the last term of above formula, then Note the first term cancels the fourth term of the first equality above. So we have, Next, we shall give the expression of another basic SEP-S Φ(t), ψ BA (t, -λ) x . According to the spectral representation of Φ(t) in (23), then Thus we only need to compute the underlied part above. To this end, with the help of e ξ(-2[µ -1 ],-λ) = µ+λ µ-λ in (28),we first calculate Here, the delta-function is defined as and the following property of delta-function is used: given a function as is seen from z i n (z/λ) n = λ i n (z/λ) n+i . Thus taking (30) back into the underlined part above, then since the definition of SEP up to the term independent of t. Similarly, we can get the expressions of S(ψ BA (t, λ), Φ x (t)) and S(Φ 1 (t), Φ 2x (t)) by considering Φ x (t) = dλλ -1 ϕ(λ)ψ BA (t, λ) x , see appendix A and B. Thus we have the following corollary. Corollary 6. If Φ(t), Φ 1 (t), Φ 2 (t) are eigenfunctions of the BKP hierarchy, then Remark 4: Note that (35) is also derived by Loirs [16] by a different method. Remark 5:According to proposition 5 and corollary 6, we have In fact,the inverse of proposition 5 is also correct and it provides another formulation of the BKP hierarchy, that is, Proposition 7. Given a function ψ(t, λ) which has the form ψ(t, λ) = e ξ(t,λ) Σ ∞ j=0 ω j (t)λ -j with ω 0 = 1 and ξ(t, λ) as in (8),where multi-time t = (t 1 = x, t 3 , • • • ) and λ is the spectral parameter,let us assume that ψ(t, λ) has the following spectral representation: for two arbitrary multi-times t and t ′ ,where the function S(t; λ, µ) is defined such that ∂ ∂t 1 S(t; λ, µ) = ψ(t, µ)ψ(t, -λ) x .Then,(38)is equivalent to the Hirota bilinear identity (9),so in this way ψ(t, λ) becomes BA functions of the associated BKP hierarchy. The proof for one side of the equivalence that Hirota bilinear identity (9) imply the spectral representation (38), is contained in the proof of (26). So we only need to show that (38) implies (9).To the end, by differentiating both side of (38)w.r.t.t ′ 1 , then, By letting t ′ = t, and considering ψ(t, λ)ψ(t, -λ) = 1 + O(λ -1 ), we have C = 1. Thus ψ(t, λ) satisfies dλλ -1 ψ(t, λ)ψ(t ′ , -λ) = 1,i.e., the Hiorta biliner equations of the BKP hierarchy. By now,we have established the SEP method for the BKP hierarchy,which provides another formulation of the BKP hierarchy. Based on the useful properties of the SEP given the last section, we are now in a position to discuss a new potential Ω-BSEP [16], which will be used to generate the "ghost" flow of the BKP hierarchy in the next section. We first provide three expressions of Ω for different eigenfunctions, and then give their identities. BSEP is also defined as a function of a pair of BKP eigenfunctions Φ 1 and Φ 2 : The definition of BSEP can be up to a constant of integration. It is obvious that Ω(Φ 1 , Φ 2 ) = -Ω(Φ 2 , Φ 1 ) and that Ω(Φ, 1) = Φ (since 1 is an eigenfunction). So according to ( 29)and ( 30),we have Remark 6: As the definition of BSEP can be up to the term independent of t, we can omit the terms independent of t in (40). That is, We would like to mention there is another expression for Ω(ψ BA (t, µ), ψ BA (t, -λ)),i.e., Here the vertex operator [25] is defined as follows, where the columns : • • • : indicate Wick Normal ordering w.r.t the creation/annihilation "modes"t l and ∂ ∂t l ,respectively. Thus according to the definition of the Vertex operator (43 ) and the wave function ( 8) and ( 11),we can easily get So ( 42) is true. As for Ω(Φ(t), ψ BA (t, -λ)), according to the definition of Ω, two identities (34) and ( 35), then Ω(Φ(t), ψ BA (t, -λ)) can expressed by the form of Note that ( 46) implies (41) in Remark 6 as we expected. Remark 7: In fact,with the help of spectral representation (23) for the BKP hierarchy and the expression for Ω(ψ BA (t, µ), ψ BA (t, -λ)) (41) , Ω(Φ(t), ψ BA (t, -λ)) is derived alternatively as We further show a more general Ω of eigenfunctions Φ 1 and Φ 2 , Next,we would like to show three identities on Ω of the BKP hierarchy. Lemma 8. For the BKP hierarchy, where Proof: First of all, we move all terms in right side hand of (48) to the left, take ψ BA (t, λ) in ( 8) and ( 11) into it, then (28), removing e ξ(t,λ)+ξ(t,µ) For convenience, denote the left hand side of above equality by C. Secondly, we shall prove indeed C = 0 from the Fay identity( 12) of the BKP hierarchy, thus (48) is proved. To this end, by letting s 0 = 0 in Fay identity(12), then Then,after shifting i.e., C = 0, as we claimed before. This is the end of the proof. After the preparation above,now we can give two important identities of the BSEP below. Proposition 9. Under shift of the times t of the BKP hierarchy, BSEP obeys Proof: By a straightforward calculation, then is derived from (49). Remark 8: In fact,these identities above have been given in Loris' paper [16], but here we give another proof and our proof is much easier. After the preparation above, now we can define the "ghost" symmetry flows generated by the BSEP through its action on the Lax operator. We shall further show its actions on the dressing operator, eigenfunction Φ(t) and τ function. Given a set of eigenfunctions Φ 1a , Φ 2a , a ∈ {α},the "ghost" symmetry of the BKP hierarchy is defined in the following way Next,we need to show that,the definition above is consistent with the BKP constraint (4) and ∂ α commutes with ∂ t 2n+1 . In other words, ∂ α is indeed a kind of symmetry flow of the BKP hierarchy. For simplicity in the next context, we introduce an operator Proof: According to the definition of A, and using a identity Furthermore, using the definition of ∂ α , a simple computation leads to because of the above identity on A. This means ∂ α L * is consistent with BKP constraint (4). Proposition 11. ∂ α commutes with ∂ t 2n+1 . Proof : We first claim the following equations hold for A and ∂ α , which will be proved latter. With the help of above equation, a simple calculation infers L] using Jacobi identity = 0, which shows ∂ α commutes with ∂ t 2n+1 . Therefore, the remaining part of the proof is to show our claimed statement (52). First of all, the definition of the "ghost" flows ∂ α L = [A, L] implies obviously ] .Thus,we have Secondly, the derivative of A with respect to t 2n+1 is given by Taking (10) into it, then Note Φ 1a,x and Φ 2a,x are two adjoint eigenfunctions. Furthermore 4 5 , Thus we have At last, according to (53)-( 54) , ( 52) is obtained. Next,let's see the action of the above "ghost" flows on the eigenfunctions Φ: Proposition 12. The "ghost" symmetry is the compatible condition of the linear problems Proof: The main idea of the proof is to use (52), which is equivalent to ghost symmetry flow (51), to get the commutativity of the flows ∂ α ∂ t 2n+1 Φ = ∂ t 2n+1 ∂ α Φ. So, according to (18),we can rewrite (56) into and then By a tedious but straightforward calculation, we have(see appendix C) Thus by substituting (59) into (58),we get We now consider the commutativity of two "ghost" symmetries generated by different pairs of eigenfunctions {Φ 1a , Φ 2a } a∈{α} and {Φ 1b , Φ 2b } b∈{β} , and their corresponding flows are ∂ α L = [A, L] and Proposition 13. If two "ghost" symmetry flows ∂ α and ∂ β are generated by A and A ′ above, then Proof: By using the relation Collecting terms in AA ′ according to ∂ -1 Φ 1b,x ,∂ -1 Φ 2b,x , Φ 2a ∂ -1 and Φ 1a ∂ -1 in order, then using (57), we have At last,let us see the action of "ghost" flow on the τ function. a∈{α} Ω(Φ 2a (t), Φ 1a (t))τ (t) using the expression of ψ BA (t, λ) in ( 8) and (11). Remark 9: So starting from the "ghost" symmetry, we find that Cτ + 1 2 a∈{α} Ω(Φ 2a (t), Φ 1a (t))τ (t) is a new τ function of the BKP hierarchy. This transformation is also given by Loris [16] started from bilinear identity. Remark 10:The symmetry reduction of BKP hierarchy,which is now called constrained BKP(cBKP) hierarchy [16] , is just to identify ∂ α with -∂ t 2n+1 ,i.e. or a∈{α} Ω(Φ 2a (t), Φ 1a (t))τ (t) = 1 2 a∈{α} Ω(Φ 1a (t), Φ 2a (t))τ (t). To conclude this section, we would like to stress that the "ghost " symmetry (51) of the BKP hierarchy is indeed different from the counterpart in the KP hierarchy. This difference is due to the BKP constraint (4). Moreover, the BSEP provides a convenient tool to show it. In this section, we shall show two applications for previous results. a∈{α} [Φ a∈{α} [-Φ a∈{α} ∆ λ Ω(Φ 2a (t), Φ 1a (t))ψ BA (t, λ). using (49) So we have ∂ α τ (t) = 1 2 = ∂ t 2n+1 ψ BA (t, -λ) + ∂ α ψ BA (t, -λ) = ∂ t 2n+1 ψ BA (t, λ) + 1 2 a∈{α} [Φ 2a (t)Ω(Φ 1a (t), Ψ BA (t, -λ)) -Φ 1a (t)Ω(Φ 2a (t), Ψ BA (t, -λ))] using (56) = ∂ t 2n+1 ψ BA (t, -λ) + 1 2 a∈{α} [-Φ -ψ BA (t, λ)ψ BA (t ′ , -λ)Φ 1a (t ′ )Φ 2a (t ′ + 2[λ -1 ])] = a∈{α} [Φ 2a (t ′ ) dλλ -1 ψ BA (t, λ)ψ BA (t ′ , -λ)[ 1 2 Φ 1a (t ′ + 2[λ -1 ]) + 1 2 Φ 1a (t ′ )] -(1 ↔ 2) In this paper,we use the notations:( i ai∂ i )+ = i≥0 ai∂ i , ( i ai∂ i )-= i<0 ai∂ i , ( i ai∂ i ) [k] = a k , Res( i ai∂ i ) = a-1 and ( i ai∂ i ) * = i (-∂) i ai.