Constraint on the Multi-Component CKP Hierarchy and Recursion Operators

1 Introduction

The KP hierarchy plays a very important role in mathematics and physics, particularly in string theory. For example, KP hierarchy has important application in matrix models [1, 2]. As we all know, the Witten’s conjecture [3] proved by Kontsevich [4] said that the matrix integral(well-known Kontsevich integral) satisfies the string equation of the two-constrained KP hierarchy by considering one point. In addition, the partition function of the grand canonical ensemble turns out to be a tau-function of KP hierarchy by considering the matrix model describing D-particles in four dimensions [5]. KP hierarchy has two sub-hierarchies, BKP hierarchy [6] and CKP hierarchy [7]. There are many important results about BKP hierarchy and CKP hierarchy. Such as additional symmetries of a two-component BKP hierarchy, Block algebra in two-component BKP hierarchy [8]. Gauge transformations for the constrained BKP hierarchy and constrained CKP hierarchy were constructed in [9].

Recently, multi-component KP [10, 11] and multi-component Toda systems [12, 13] attract more and more attention because its widely use in the fields of multiple orthogonal polynomials and non-intersecting Brownian motions. The multi-component KP hierarchy was discussed with application on representation theory and random matrix model in [10, 11]. The multi-component 2D Toda hierarchy was considered from the point of view of the Gauss-Borel factorization problem, the theory of multiple matrix orthogonal polynomials, non-intersecting Brownian motions, and matrix Riemann-Hilber problems [14]. As a sub-hierarchy, a kind of definition of multi-component BKP hierarchy was given in [15]. The (n,1)-reduced DKP hierarchy was constructed in a different way [16]. Date, Jimbo, Kashiwara, Miwa generalized BKP hierarchy to multi-component cases in the form of bilinear equations [17].

Recently, the multi-component KP [18] and Toda hierarchies have already been studied, especially for the multi-component CKP hierarchy, we should begin work from it.

As we all known that possessing infinite number of symmetries is a common property of the classical integrable systems. A lot of results on concrete forms of symmetries were founded [19–21]. Using recursion operator to generate symmetries of integrable systems is an effective tools [22, 23]. Besides, in order to establish the Hamiltonian structure of integrable systems, we can also use recursion operator [19, 24, 25]. The recursion operator has many important application in integrable systems, so it is necessary to construct the recursion operator for integrable systems.

In the articles [25, 26–28], several different methods are used to construct recursion operators. In the articles [27, 29], recursion operators for the commutative 1-constrained cBKP hierarchy and the commutative 1-constrained cCKP hierarchy have been given. The purpose of this article is to give recursion operators of the two-component cCKP hierarchy and to show the relation between the two-component cCKP hierarchy and the cCKP hierarchy.

We arrange this article as follows. We find the Lax operator and give a new definition of the multi-component constrained CKP hierarchy in Section 2. In Section 3, use the similarity way, we give the Lax equations of the two-component constrained CKP hierarchy. The recursion relation for the two-component constrained CKP hierarchy is given in Sections 4 and 5. At last, in Section 6, we discuss the contact between two-component constrained CKP hierarchy and cCKP hierarchy.

2 Lax Equations of the Multi-Component Constrained CKP Hierarchy

By considering CKP condition on the multi-component constrained CKP, i.e. L^*=–L, the multi-component constrained CKP (McCKP) hierarchy can be defined by the following Lax operator

In the following context, we take n=1 for simplicity, we consider the Lax operator of the multi-component constrained CKP (McCKP) hierarchy as

where q and r are N×N matrix functions.

The eigenfunction q and the adjoint eigenfunction r of the McCKP hierarchy are defined, respectively, by

It is obvious that the Lax operator L satisfies the C type condition L^*=–L. We can write the operator L in a dressing form as

where

and H satisfy

The equation of the McCKP hierarchy is

where

and

In order to get qtnj and rtnj, we need the following definition.

Definition 2.1If the matrix operator B is a differential operator and has form B: = ∑n = 0∞∂nan, then we define B∗g(x) = ∑m = 0∞(−1)m(∂mg(x))am.

3 Lax Equations of the Two-Component Constrained CKP Hierarchy

In this part, we give the Lax equations of the two-component constrained CKP hierarchy. The Lax operator has form

where q and r are 2×2 matrix functions.

We consider

where α and β are 2×2 matrix functions and C₁ satisfy C1∗ = C1.

Then the corresponding Lax equations is

where

As the same time, we consider

where α and β are 2×2 matrix functions and C₂ satisfy C2∗ = C2.

Then the corresponding Lax equations is

where

So we have C₁+C₂=E, the eigenfunction α and the adjoint eigenfunction β are defined, respectively, by

We call that (12) and (15) are the Lax equations of the two-component constrained CKP hierarchy.

4 The Recursion Operator for the Two-Component Constrained CKP Hierarchy I

In this section, we give the recursion operator for the two-component constrained CKP hierarchy. In order to finish our work conveniently, we need the following lemma:

Lemma 4.1The following four identities hold true

Using these identities, we can get the following important theorems:

Theorem 4.2The recursion operators of tm1 flows for the (12) are like this:

where A_ij and B_ij (i=1, 2; j=1, 2, 3, 4) are complicated which can be seen in Appendix I.

Here, we just give the proof of (21),

Proof. We can calculate the L² as following:

with

L(q) = qx + q(∫rq) + r⊤(∫q⊤q),L(r⊤) = rx⊤ + q(∫rr⊤) + r⊤(∫q⊤r⊤),L∗(q⊤) = − qx⊤ − (∫q⊤q)r − (∫q⊤r⊤))q⊤,L∗(r) = − rx − (∫rq)r − (∫rr⊤)q⊤.

Therefore,

B21 = L2C1, C1 = E11 + α∂−1β − β⊤∂−1α⊤.

And, we have

B21 = E11∂2 + (αβ − β⊤α⊤)∂ + 2qrE11 + 2r⊤q⊤E11 + 2αxβ + αβx− 2βx⊤α⊤ − β⊤αx⊤ + q∂−1(C1∗L∗(r)) + L(q)∂−1(C1∗r)+ r⊤∂−1(C1∗L∗(q⊤)) + L(r⊤)∂−1(C1∗q⊤) + A∂−1β + B∂−1α⊤,

with A = αxx + 2qrα + 2r⊤q⊤α + q(∫L∗(r)α)) + r⊤(∫L∗(q⊤)α) + L(q)(∫rα) + L(r⊤)(∫q⊤α).

And B = − βxx⊤ − 2qrβ⊤ − 2r⊤q⊤β⊤ − q(∫L∗(r)β⊤) − r⊤(∫L∗(q⊤)β⊤) − L(q)(∫rβ⊤) − L(r⊤)(∫q⊤β⊤).

Denote An1 as (Bn1)−,n=1, 2, …. By considering the CKP condition and (12), we have the following results:

then

For (27), let us firstly calculate ((B21)+(Am1))+. We set Am1 = ∂−1a1 + ∂−2a2 + … .

So ((B21)+(Am1))+ = E11∂a1 + E11a2 + (αβ − β⊤α⊤)a1.

The identity Res∂[Bm1, B11] = 0 yields

The left term of (31) equals Res∂Ltm1 = (qr + r⊤q⊤)tm1, and the right term of (31) yields Res∂[∂, ∂−1a1 + ∂−2a2 + …] = a1x. So, a₁ can be expressed as

In order to compute a₂, we should use identity Res∂[Bm1, (B21)2] = 0. By considering the similar identity

The left term of (33) equals Res∂Ltm12 = (qxr − qrx + rx⊤q⊤ − r⊤qx⊤)tm1, and the right term of (33) yields Res∂[∂2 + 2qr⊤ + 2rq⊤, ∂−1a1 + ∂−2a2 + …] = −a1xx + 2a2x + 2qra1 − 2a1qr + 2r⊤q⊤a1 − 2a1r⊤q⊤. We can easily get

a2 = 12∫(qxr − qrx + rx⊤q⊤ − r⊤qx⊤)tm1 + 12(qr + r⊤q⊤)tm1− ∫[qr∫(qr + r⊤q⊤)tm1]+∫[∫(qr + r⊤q⊤)tm1qr]− ∫[r⊤q⊤∫(qr + r⊤q⊤)tm1 ] + ∫[∫(qr + r⊤q⊤)tm1r⊤q⊤]

Hence, we can directly calculate ((B21)+(Am1))+ as following:

((B21)+(Am1))+ = E11∂(∫(qr + r⊤q⊤)tm1)  + 12E11∫(qxr − qrx + rx⊤q⊤ − r⊤qx⊤)tm1 +12E11(qr + r⊤q⊤)tm1 − E11∫[qr∫(qr + r⊤q⊤)tm1] +E11∫[∫(qr + r⊤q⊤)tm1qr] − E11∫[r⊤q⊤∫(qr + r⊤q⊤)tm1] +E11∫[∫(qr + r⊤q⊤)tm1r⊤q⊤] + (αβ − β⊤α⊤)(∫(qr + r⊤q⊤)tm1).

Considering the term (A21(Bm1)+)+, we write it as A21(Bm1)+ − (A21(Bm1)+)−. Using the identity (19), we can compute the second term

(A21(Bm1)+)− = [(q∂−1(C1∗L∗(r)) + L(q)∂−1(C1∗r)+r⊤∂−1(C1∗L∗(q⊤)) + L(r∗)∂−1(C1∗q⊤) + A∂−1β + B∂−1α⊤)(Bm1)+]−= q∂−1(Bm1)+∗C1∗L∗(r) + L(q)∂−1(Bm1)+∗C1∗r + r⊤∂−1(Bm1)+∗C1∗L∗(q⊤)+L(r⊤)∂−1(Bm1)+∗C1∗q⊤+ A∂−1(Bm1)+∗β + B∂−1(Bm1)+∗α⊤.

Next, we calculate the following term,

(Bm1)+∗C1∗L∗(r) = (Bm1)+∗(B11)∗(r) = (B11)∗(Bm1)+∗(r) + [(Bm1)+∗, (B11)∗](r) = −(B11)∗rtm1 − (B11)tm1∗r = −E11rxtm1 − (αβ−β⊤α⊤)rtm1 − (∫rtm1L(α))β+(∫rtm1L(β⊤))α⊤  −(∫rtm1q)C1∗r − (∫rtm1r⊤)C1∗q⊤ − αtm1βr − αβtm1r  +βtm1⊤α⊤r + β⊤αtm1⊤r − (∫rL(α)tm1)β − (∫rL(α))βtm1  +(∫rL(β⊤)tm1)α⊤ + (∫rL(β⊤)αtm1⊤ − (∫rqtm1)C1∗r  −(∫rq)(C1∗r)tm1 − (∫rrtm1⊤)C1∗q⊤ − (∫rr⊤)(C1∗q⊤)tm1.

Similarly, we can get

(Bm1)+∗C1∗L∗(q⊤) = (Bm1)+∗(B11)∗(q⊤) = (B11)∗(Bm1)+∗(q⊤) + [(Bm1)+∗, (B11)∗](q⊤) = −(B11)∗qtm1⊤ − (Bm1)tm1∗q⊤ = −E11qxtm1⊤ − (αβ − β⊤α⊤)qtm1⊤ − (∫qtm1⊤L(α))β + (∫qtm1⊤L(β⊤))α⊤  −(∫qtm1⊤)qC1∗r − (∫qtm1⊤r⊤)C1∗q⊤ − αtm1βq⊤ − αβtm1q⊤  +βtm1⊤α⊤q⊤ + β⊤αtm1⊤q⊤ − (∫q⊤L(α)tm1)β − (∫q⊤L(α))βtm1  +(∫q⊤L(β⊤)tm1)α⊤ + (∫q⊤L(β⊤)αtm1⊤ − (∫q⊤qtm1)C1∗r  −(∫q⊤q)(C1∗r)tm1 − (∫q⊤rtm1⊤)C1∗q⊤ − (∫q⊤r⊤)(C1∗q⊤)tm1.

And also we can get

(Bm1)+∗C1∗(r) = C1∗(Bm1)+∗(r) + [(Bm1)+∗, C1∗](r) = −C1∗rtm1 − (C1∗)tm1r = −E11rtm1 − (∫rtm1α)β + (∫rtm1β⊤)α⊤  −(∫rαtm1)β − (∫rα)βtm1 + (∫rβtm1⊤)α⊤ + (∫rβ⊤)αtm1⊤.

(Bm1)+∗C1∗(q⊤) = C1∗(Bm1)+∗(q⊤) + [(Bm1)+∗, C1∗](q⊤) = −C1∗qtm1⊤ − (C1∗)tm1q⊤ = −E11qtm1⊤ − (∫qtm1⊤α)β + (∫qtm1⊤β⊤)α⊤  −(∫q⊤αtm1)β − (∫q⊤α)βtm1 + (∫q⊤βtm1⊤)α⊤ + (∫q⊤β⊤)αtm1⊤.

After bringing these results into (21), we get the recursion flow of q. Using the same method, we can get the other recursion operators.

5 The Recursion Operator for the Two-Component Constrained CKP Hierarchy II

In the same way, we consider the following equations:

and we will get the following theorem:

Theorem 5.1The recursion operators of tm2 flows for the (15) are like this:

where A_ij and B_ij (i=1, 2; j=1, 2, 3, 4) are complicated which can be seen in Appendix II.

6 Conclusions and Discussions

In this article, the Lax equations of McCKP hierarchy and two-component cCKP were defined and we construct recursion operators for two-component cCKP hierarchy. What’s more, we find that McCKP hierarchy, MCKP hierarchy, and cCKP hierarchy have the following reduction chain:

In our next work, we will try to use the same method to get recursion operators for other multi-component hierarchies. By considering the potential application of CKP hierarchy in matrix models in string theory, this constrained multicomponent CKP hierarchy deserves further studying. We believe that the recursion operator constructed in article should be useful in constructing gravitational descendants for the n-point functions of the fields in Topological Landau–Ginzburg Theory by considering the dispersionless topological recursion relation such as [30].