Supersymmetric BKP systems and their symmetries

In this paper, we construct the additional symmetries of the supersymmetric BKP(SBKP) hierarchy. These additional flows constitute a B type $SW_{1+\infty}$ Lie algebra because of the B type reduction of the supersymmetric BKP hierarchy. Further we generalize the SBKP hierarchy to a supersymmetric two-component BKP (S2BKP) hierarchy equipped with a B type $SW_{1+\infty}\bigoplus SW_{1+\infty}$ Lie algebra. As a Bosonic reduction of the S2BKP hierarchy, we define a new constrained system called the supersymmetric Drinfeld-Sokolov hierarchy of type D which admits a $N=2$ supersymmetric Block type symmetry.


Introduction
In the study of integrable hierarchies, it is interesting to find their symmetries and identify the algebraic structure of the symmetries. Among these symmetries, the additional symmetry is an important type which contains dynamic variables explicitly and these additional flows do not commutes with each other. Additional symmetries of the Kadomtsev-Petviashvili(KP) hierarchy were introduced by Orlov and Shulman [1] which contain one important symmetry, i.e. the so-called Virasoro symmetry. These symmetries form a centerless W 1+∞ algebra closely related to matrix models by means of the Virasoro constraint and string equations [2,3,4,5]. Two sub-hierarchies as the BKP hierarchy and CKP hierarchy [6,7,8,9,10,11] have been shown to possess additional symmetries, with consideration of the reductions on the Lax operators.
Various generalizations and supersymmetric extensions [12] of the KP hierarchy have deep implications in mathematical physics, particularly in the theory of Lie algebras. In [13,14], the theory of the super Lie algebras was surveyed by considering super Boson-Fermion Correspondences. One important supersymmetric extension is the supersymmetric Manin-Radul Kadomtsev-Petviashvili (MR-SKP) hierarchy [15] which contains a lot of integrable super solitary equations equipped with the super-pseudodifferential operators. Apart from the Manin-Radul one, Mulase supersymmetrize the KP hierarchy by constructing a hierarchy called the Jacobian SKP hierarchy which does not possess a standard Lax formulation [16]. This hierarchy has strict Jacobian flows, i.e. it preserves the super Riemman surface about which one can also see [17]. The additional symmetries for super hierarchies were firstly found in the paper [18] by constructing the standard Orlov-Schulman additional nonisospectral flows. Later the additional symmetry of the MR-SKP hierarchy was studied by Stanciu [19]. The ghost symmetries, hamiltonian structures and extensions of the MR-SKP hierarchy were studied as well as reductions of the MR-SKP hierarchy [20,21]. Later the supersymmetric BKP (SBKP) hierarchy was constructed in [22]. After that this series of super hierarchies were seldom studied in mathematical physics partly because of their extreme complexities.
For the symmetry of the two-component BKP hierarchy, there is a series of works such as [7,23,24,25]. In the paper [26], we construct the generalized additional symmetries of the two-component BKP hierarchy and identify its algebraic structure. Besides, the D type Drinfeld-Sokolov hierarchy was found to be a good differential model to derive a complete Block type infinite dimensional Lie algebra. About the Block algebra related to integrable systems, we did a series of works in [27]- [29]. In this paper, we will construct the additional symmetries of the supersymmetric BKP hierarchy. These additional flows constitute a B type SW 1+∞ Lie algebra. Further we generalize the SBKP hierarchy to a supersymmetric two-component BKP hierarchy (S2BKP) hierarchy and derive its algebraic structure. As a reduction of the S2BKP hierarchy, in this paper a new supersymmetric Drinfeld-Sokolov hierarchy of type D will be constructed and proved to have a super Block type additional symmetry.
In (1+1) dimensional sypersymmetric integrable systems, starting from 1980s, there is also a series of work about superversions of Korteweg-de Vries(KdV), Toda-KdV equations and so on [30]- [33]. Most of them are related to the conformal field theory and string theory [31,32]. The generalized KdV equations and Toda lattice equations are particularly interesting integrable nonlinear systems in connection with conformal field theories. Their Virasoro symmetry can be extended to a W n algebra which is known to arise from the Hamiltonian structure of the generalized KdV equation [34] by incorporating conserved currents of higher spins. The supersymmetric version of the Drinfeld-Sokolov reduction of the Toda-KdV theories gives the generators of the related super W -algebra with the commutation relations provided by the associated Hamiltonian structure [35]. In this paper we will extend the Lie algebraic method of Drinfeld and Sokolov [34] to the sypersymmetric case and develop a Lie superalgebraic method for a supersymmetric D type Drinfeld-Sokolov hierarchy. We further derive that the supersymmetric Drinfeld-Sokolov hierarchy of type D possesses a N = 2 supersymmetric Block type Lie algebraic structure.
This paper is arranged as follows. In the next section we recall some necessary facts of the SBKP hierarchy. In Sections 3, we will give the additional symmetries for the SBKP hierarchy. The ghost symmetry of the SBKP hierarchy will be devoted to Section 4 and 5 using the techniques in [20]. Further in Section 6 and 7, we generalize the SBKP hierarchy to a S2BKP hierarchy and derive its B type SW 1+∞ SW 1+∞ algebra. As a Bosonic reduction of the supersymmetric two-component BKP hierarchy, we define a new constrained system called the supersymmetric Drinfeld-Sokolov hierarchy of type D which possesses a N = 2 supersymmetric Block type Lie algebra in the following two sections. Finally, we will give a short conclusion and a further discussion.

The supersymmetric BKP hierarchy
Let us firstly recall some basic facts [22] on the supersymmetric BKP system which is well defined by two Lax operators.
A is assumed as an algebra of smooth functions of a spatial coordinate x, a grassmann variable θ and their super-derivation denoted as D = ∂ θ + θ∂. This algebra A has the following multiplying rule Here the value |Φ| means the super degree of the operator Φ which shows the operator Φ is Fermionic or Bosonic. The supersymmetric derivative D satisfies the supersymmetric analog of the Leibniz rule where a is a homogeneous element of A. We introduce the even and odd time variables (t 2 , t 3 , t 6 , t 7 , ·) and the following definition of even and odd flows We recall that the supercommutator is defined as The bracket has a property as [X, This family of infinite odd and even flows satisfy a nonabelian Lie superalgebra whose commutation relations are For any operator A = i∈Z f i D i ∈ A and homogeneous operators P, Q, its nonnegative projection, negative projection, adjoint operator are respectively defined as Also for the operator D k , the adjoint operator is defined as Basing on definitions in [22], the Lax operator of the supersymmetric BKP hierarchy has a form as The supersymmetric BKP hierarchy is defined by the following Lax equations One can rewrite the operator L in a dressing form as We call the eq.(2.13) the B type condition of the supersymmetric BKP hierarchy. Given L, the dressing operator Φ is determined uniquely up to a multiplication to the right by operators with constant coefficients. The dressing operator Φ takes values in a B type Volterra group. The supersymmetric BKP hierarchy (2.10) can also be redefined as with k ≥ 1.
With the above preparation, it is time to construct additional symmetries for the supersymmetric BKP hierarchy in the next section.

Additional symmetries of the supersymmetric BKP hierarchy
In this section, we are to construct additional symmetries for the supersymmetric BKP hierarchy by using the Orlov-Schulman operators whose coefficients depend explicitly on the time variables of the hierarchy. The Orlov-Schulman operators M i and auxiliary operator Q are constructed in the following dressing structure Then one can get the following lemma.
Proof. For the proof, one can do the following direct calculation For Q, the following identities hold Then it is easy to get the following lemma by dressing structures.
Proof. The dressing structure will lead to and further to Then we get and using eq.(2.14) we can derive The other identities can be proved using the similar dressing techniques.
From now on, we will introduce the following operator B mklp defined as where k, m ≥ 0; l, p = 0, 1. This operator is the generator of the additional symmetry of the SBKP hierarchy which shows the difference between generators of the SKP and SBKP hierarchies. For the SKP case in [15], in the construction of B mklp , it contains only one term. Then the following proposition can be got.
Proof. The lemma can be proved by dressing the following identities by Φ To prove that B mnlp satisfies the B type condition, we need the following lemma.
Lemma 3.4. The operators M i satisfy the following conjugate identities, the following calculations , will lead to this lemma. The anti adjoint property of Q can be proved in a similar way.
It is easy to check the following proposition holds basing on the Lemma 3.4 above.
Proposition 3.5. The operator B mklp satisfies a B type condition, namely Proof Using the Proposition 3.4, the following calculation will lead to this proposition Basing on above proposition, it is reasonable to define additional flows of the supersymmetric BKP hierarchy as Proposition 3.6. The flows (3.31) commute with the flows of the supersymmetric BKP hierarchy. Namely, one has which holds in the sense of acting on Φ.
Proof The proposition can be checked case by case with the help of eq.(3.26). For example, This proposition tells us that the additional flows of the supersymmetric BKP hierarchy are in fact its symmetries whose algebraic structure can be shown in the following proposition. Proof. The isomorphism is given by which further lead to One can find the above construction keeps ξ commuting with z.

Ghost symmetry of supersymmetric BKP hierarchy
In this section, we will give another special symmetry which does not contain time variables explicitly. Before that, we firstly need to define the super-Baker-Akhiezer function (super-BA) and adjoint super-BA function as The following property can be found Then we can prove that The B type condition implies that the adjoint super-BA function Ψ BA can be in fact the supersymmetric derivative of its corresponding super-BA function Φ BA , i.e.
Then we can also define super-eigenfunctions of the supersymmetric BKP hierarchy as The super-eigenfunctions have the following spectral representation in term of integrals of Baker-Akhiezer functions as The B type condition also implies that the adjoint eigenfunction ψ can be chosen as the supersymmetric derivative of its corresponding eigenfunction φ, i.e. ψ = φ [1] . (4.10) The supersymmetric tau function of the supersymmetric BKP hierarchy can be defined by the residue (the coefficient before D −1 ) of supersymmetric Lax operators as D 4k−2 D ln τ = resL 4k−2 , D 4k−1 D ln τ = resL 4k−1 . (4.11) Define two eigenfunctions φ 1 , φ 2 and the following operator which is used to generate the ghost flows. According to (4.13) one can find the operator B g satisfies the B type condition, i.e. (4.14) Then the ghost flow of the supersymmetric BKP hierarchy can be defined as following where functions φ 1 , φ 2 are the eigenfunction and adjoint eigenfunction of the supersymmetric BKP hierarchy. The following proposition will tell you the above flow is a symmetry of the supersymmetric BKP hierarchy. Proof. Note that the derivatives φ [1] 1 , φ [1] 2 are in fact adjoint supersymmetric eigenfunctions. The commutativity between ghost flows and supersymmetric BKP flows is in fact equivalent to the following Zero-Curvature equation which includes the following detailed proof 1 )] = 0.
In the above proof the P 0 (A) means the coefficient over the term D 0 of the operator A.

The supersymmetric two-component BKP hierarchy
Let us firstly define the supersymmetric two-component BKP hierarchy by two Lax operators. Now we introduce the even and odd time variables (t 2 , t 3 , t 6 , ·;t 2 ,t 3 ,t 6 , ·) and the following definition of even and odd flows This two families of odd and even flows satisfy a nonabelian Lie superalgebra whose commutation relations are The two Lax operators of the supersymmetric two-component BKP hierarchy will be defined in forms as such that We call eqs.(5.6) the B type condition of the supersymmetric two-component BKP hierarchy. The supersymmetric two-component BKP hierarchy is defined by the following Lax equations: which is equivalent to the following equations with k ∈ Z + . One can write the operators L andL in a dressing form as satisfy Given L andL, the dressing operators Φ andΦ are determined uniquely up to a multiplication to the right by operators with constant coefficients. The dressing operators Φ andΦ take values in two separated B type Volterra groups. The supersymmetric two-component BKP hierarchy can also be redefined as with k ∈ Z + .

Additional symmetries of the supersymmetric two-component BKP hierarchy
In this section, we are to construct additional symmetries for the supersymmetric two-component BKP hierarchy by using the Orlov-Schulman operators whose coefficients depend explicitly on the time variables of the hierarchy.
With the same dressing operators given in the eq.(5.14), Orlov-Schulman operators M i ,M i , Q,Q are constructed in the following dressing structure where Q = ∂ θ − θ∂. Then one can derive the following lemma similarly as the case of the singlecomponent supersymmetric BKP hierarchy. where i = 0, 1.
Proof. For the proof, we do the following calculation Further one can derive For Q, we can get Then it is easy to get the following lemma using the above lemma and dressing structures.
From now on, we will introduce the following two operators B mklp andB mklp , given any pair of integers (m, k, l, p) with m, k ≥ 0, l, p = 0, 1, as As a corollary, the following proposition can be got. To prove that B mklp andB mklp satisfy the B type condition, we need the following lemma.
will lead to this lemma. It is easy to check the following proposition holds basing on the Lemma 6.4 above.
Proposition 6.5. Operators B mklp andB mklp satisfy the B type condition, namely Proof Using the Proposition 6.4, the following calculation will lead to the first identity of this proposition The second identity can be proved in a similar way. Now we can define the following additional equations as Similarly, we will get the following proposition.
The other cases can be proved in similar ways. This is the end of this proposition.
Similarly as the SBKP hierarchy, the algebraic structure of the additional symmetry of the S2BKP hierarchy will be talked about in the next proposition.
Proposition 6.7. The algebra of additional symmetries of the two-component SBKP hierarchy is isomorphic to the Lie algebra of super quasi-differential operators, which is isomorphic (as a Lie algebra) to SW 1+∞ SW 1+∞ .
Proof. The isomorphism is given by which further lead to One can find the above construction keeps ξ commuting with z andξ commuting withẑ.
If we do a (4n,2)-reduction from the supersymmetric two-component BKP hierarchy, a reduced hierarchy called the supersymmetric D type Drinfeld-Sokolov hierarchies with a supersymmetric Block type additional symmetry which will be discussed in the next section.

Supersymmetric D type Drinfeld-Sokolov hierarchy
Assume a new Lax operator L which has the following relation with two Lax operators of the supersymmetric two-component BKP hierarchy introduced in the last section L = L 4n =L 2 , n ≥ 2. (7.1) Then the Lax operators of the supersymmetric two-component BKP hierarchy will be reduced to the following Lax operator of the supersymmetric D type Drinfeld-Sokolov hierarchy whose Bosonic case can be seen in [24,25,26] One can easily find the Lax operator L of the supersymmetric D type Drinfeld-Sokolov hierarchy will satisfy the following B type condition This Lax operator L of the supersymmetric D type Drinfeld-Sokolov hierarchy has the following dressing structure [25] are pseudo supersymmetric differential operators that also satisfy the following B type condition The dressing structures inspire us to define two fractional operators as Two fractional operators L (7.8) The supersymmetric D type Drinfeld-Sokolov hierarchy being considered in this paper is defined by the following Lax equations: The dressing operators Φ andΦ are same as the ones of the supersymmetric two-component BKP hierarchy. Given L, the dressing operators Φ andΦ are uniquely determined up to a multiplication to the right by operators of the form (7.5) and (7.6) with constant coefficients. The supersymmetric D type Drinfeld-Sokolov hierarchies can also be redefined as the following Sato equations with k = 4i − 2, 4i − 1, i ∈ Z + . After the above preparation, we will show that this supersymmetric D type Drinfeld-Sokolov hierarchy has a nice Block symmetry as its appearance in the Bigraded Toda hierarchy [27].

Supersymmetric Block symmetries of supersymmetric D type Drinfeld-Sokolov hierarchies
In this section, we will put the constrained condition eq.(7.1) into the construction of the flows of the additional symmetry which form a N = 2 supersymmetric extension of the well-known Block algebra [36].
With the dressing operators given in eq.(7.8), we introduce two new Orlov-Schulman operators as following It is easy to see the following lemma holds. and which can be simplified to To make the operators used in the additional symmetry satisfying the B type condition, we need to prove the following B type property of M i −M i which is included in the following lemma.
which can further lead to In the above calculation, the commutativity between L and M 0 −M 0 is already used. Till now, the proof is finished.
One can also get For the supersymmetric D-type Drinfeld-Sokolov hierarchy, we define the additional operator B lplp mk as where l, p,l,p = 0, 1; = a,b=l,p,l,p ab, = a=l,p,l,p a. One can get the following proposition. Proof Using the Proposition 6.4, the following calculation will lead to That means it is reasonable to define the additional flow of the supersymmetric D type Drinfeld-Sokolov hierarchy as Whether these additional flows are symmetries of the supersymmetric D type Drinfeld-Sokolov hierarchy will be answered in the next proposition. where l, p,l,p = 0, 1; m, k ∈ Z + , n = 4i − 2, 4i − 1, i ∈ Z + , which hold in the sense of acting on Φ,Φ or L.
Proof According to the definition, and using the actions of the additional flows and the flows of the D type Drinfeld-Sokolov hierarchy on Φ, we have The other cases of this proposition can be proved in similar ways.
The above proposition indicates that eq.(8.16) is a symmetry of the supersymmetric D type Drinfeld-Sokolov hierarchy. Further we can prove that the following identities hold true Using same techniques used in [27], the following theorem can be derived. which holds in the sense of acting on Φ,Φ or L.
Proof The similar proof for eq.(8.19) can be found in our paper [26]. This is one kind of supersymmetric extensions of the Block algebra because it is involved with the supersymmetric variables and supersymmetric derivative D. However, its algebra structure is still not clear now, which deserves further study in the future.

Conclusions and Discussions
Our earlier papers show that the Block type algebras appear not only in Toda type difference systems but also in differential systems such as two-BKP hierarchy, D type Drinfeld-Sokolov hierarchy [26]. The above results show that in their corresponding supersymmetric systems, there also exists the hidden N = 2 Block type supersymmetric algebraic structures. These results further show that the Block type infinite dimensional Lie algebra has a certain of universality in integrable hierarchies.
Although the supersymmetric two-component BKP hierarchy and supersymmetric Drinfeld-Sokolov hierarchy of type D might be not available in the fermionic string theory now comparing with the application of the classical KP hierarchy to the bosonic string, they have a great advantage of showing their superconformal structures. In this paper, we also show the structure of a super Block algebra of the supersymmetric two-BKP hierarchy and its reduced hierarchy. The superconformal algebra may appear in the related Hamiltonian structures by considering the reductions of super-BKP hierarchies like reductions of the super-KP system to super-KdV system. This may be an interesting subject for our future study which may relate the supersymmetric BKP systems in this paper to problems in physics. There are also some other interesting subjects such as the relation of the hierarchies introduced in this paper with the quantum spin chains as in [37]. These directions might be included in our future study.