Symmetry, Integrability and Geometry: Methods and Applications CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae ⋆

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school [Date E., Jimbo M., Kashiwara M., Miwa T., J. Phys. Soc. Japan 50 (1981), 3806-3812] (see also [Kac V.G., van de Leur J.W., Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 369-406]). We present appropriate bosonization formulae. We show that in the context of the CKP theory certain orthogonal polynomials appear. These polynomials are polynomial both in even and odd (in Grassmannian sense) variables.


Introduction
In this paper we develop the ideas of Date, Jimbo, Kashiwara and Miwa [2,3]. In [3] it was pointed out that the tau function of the CKP hierarchy may be presented as the vacuum expectation value of bosonic fields φ n which act in a bosonic Fock space, denoted by F in the present paper. As it was shown in [3] the higher CKP flows are induced by the action of bosonic current algebra operators J n , n > 0. We shall show that in contrast to the familiar fermionic approach the action of the currents J −n on the vacuum state does not generate the whole Fock space F where the original bosonic operators φ n act. To generate the whole Fock space we need to add an additional fermionic field which is a sort of super-counterpart to the bosonic current. This problem is studied in Section 2. In Subsection 2.1 we introduce Fermi operators whose action on the vacuum vector complete the action of the current algebra J n to obtain the whole Fock space F . A bosonization formula which expresses the original bosonic field φ(z) in terms of the current algebra J n and of the fermion field θ(z) is suggested (2.11). (As a byproduct of this relation we obtain an equality (2.29) which relates Pfaffian and Hafnian expressions which earlier appeared in a quite different context [7].) Here we show that this fermionic field is a superpartner of currents and it naturally creates a dependence of CKP tau function on auxiliary odd Grassmannian parameters. Though we present a bilinear equations written in terms of super vertex operator we do not construct Lax equations with respect to odd parameters. In Section 4 we introduce new orthogonal polynomials in many variables, C λ (t), which appear as a result of This paper is a contribution to the Special Issue "Geometrical Methods in Mathematical Physics". The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html the 'bosonization' of the basis Fock vectors of F , see (4.4). These polynomials depend both on CKP higher times t n (n odd) and the above-mentioned Grassmannian odd parameters. In certain sense these polynomials play a role similar to the role of the Schur functions in the theory of KP [16] and TL [17], and the role of the projective Schur functions in the theory of BKP [12,19], namely, CKP tau function may be presented as a series in these polynomials over partitions (see Subsection 4.6). However in contrast to the KP and BKP cases these polynomials are not CKP tau functions themselves. At the end some combinatorial properties of C λ are discussed.

CKP bosonic tau function
In this section we follow a suggestion in [3] and describe a CKP hierarchy of PDEs starting from a collection of free bosons. This imitates their approach in the BKP case, where one starts with neutral fermions. However, this hierarchy, although related to Lie algebra c ∞ , differs from the usual CKP hierarchy, for which one takes a reduction of the KP hierarchy by assuming that the Lax operator satisfies L * = −L; such Lax operators come from certain KP tau functions, which are fixed by some involution and where one puts the even times to zero, see e.g., [3] or [1] for more details. The hierarchy described in this paper is different and is not related to the usual CKP which comes from a reduction of KP. In the latter case one has a realization of c ∞ , for which the level is positive. Our construction realizes c ∞ with a negative level.

Bose-Fermi correspondence in the CKP case
We follow a suggestion of Date, Jimbo, Kashiwara and Miwa in their paper [3] and introduce free bosons, but for convenience of notation we shift the index by 1 2 . So φ i with i ∈ 1 2 + Z satisfy commutation relations: The Fock space F , respectively F * , is defined by so that F has as basis the vectors with j 1 > j 2 > · · · > j n−1 > j n > 0 and m i positive integers. Defining we have a direct sum decomposition of F : It is straightforward to check that the dimension of F k is given by the partition of k into positive elements of 1 2 + Z. Define the formal character as Then where the normal ordering is defined by In other words J n = 0 for n even and one has the following familiar commutation relations The elements :φ i φ j : form a representation of the Lie algebra c ∞ , see e.g. [9]. However, we want to stress that its level (the value of its central element) is negative. Note also that in the commutation relations (2.7) we have the factor − n 2 instead of the usual n 2 . It is clear that By a similar argument as before, again since these are bosons, we can apply an element J −n infinitely many times to |0 . Since the degree of J n is −n we obtain that the action of this Heisenberg algebra on the vacuum vector produces in the dim q F the partition function of partitions in only odd numbers: Now we calculate, using (2.4), This part should be explained by something else and we expect it to be fermions, at least anticommuting variables. The factor 1 + q k is related to a fermion of degree k. This is how we get these elements and calculate their commutation relations. We first calculate Now, using (2.7) we see that Hence setting

Commutation relations of the θ(z)'s
We now want to calculate the commutation relations of these θ(z)'s given in (2.11). For this we first rewrite the commutation relations (2.1) as follows: Note also that The first equation of (2.14) is obtained in the following way. First using (2.7) one has Combining this with (2.15) one obtains , see (2.9), we obtain the second relation in (2.14) as follows: The third formula is proved in a similar way. We will also use the following identities which can be found in V. Kac's book [8]: Now replacing z and y by −y and −z respectively, gives where H(y) is as in (2.5) (not the one in (2.28)), and D y = y∂ y + ∂ y y is the Euler operator. Now write Note that there is no conflict with the J's defined in (2.5), since here the J's have indices in 1 2 + Z. It is clear that the above commutation relation (2.16) in modes gives (compare with (2.7)). From (2.13) we also have [J n , J m ] = 0, n ∈ 1 + 2Z, m ∈ 1 2 + Z.
Thus we can combine the (anti)commutation relations of all J's as follows:

Even and odd times
Since the elements J −k with k ∈ 1 2 + Z anticommute among themselves, they can only appear once in Such a J −k explains the factor 1 + q k in the q-dimension formula (2.8). One can identify the J n 's, for n < 0 with even and odd times, i.e., with commuting variables t j , 0 < j ∈ 1 + 2Z, and Grassmann variables t j 2 , 0 < j ∈ 1 + 2Z, and identify Fock space F with the space (or some completion of it, since we take exponentials), where one has in particular, t 2 j = 0 for j ∈ 1 2 + Z . We will write t = t 1 , t 1 give the field exactly in commuting and anticommuting variables t k . Now using the free boson-(boson+fermion) correspondence, i.e., using the vertex operator expressions for the fields (where we used (2.12), (2.11), (2.17) and (2.19)), in the following subsections we shall express the bilinear identity as a hierarchy of differential equations. A similar expression for (2.20) was also found in [9].

The CKP bilinear equation
Following [3] we define the operator that S commutes with the action of :φ i φ j : on the tensor product F ⊗ F of the Fock space F and The CKP Hirota equation is [3]: where g is for instance given by (see [3]): We rewrite (2.21) as We could now use the isomorphism σ to define this hierarchy in terms of the times t. However we will not do that yet, but concentrate first in the next subsection on the form of σ(g|0 ).

The CKP tau function
Now let ODP ev be the set of all partitions in an even number of odd parts, where a part may appear at most once. We call them "Odd Partitions of even length with Distinct parts" (ODP ev ); later on we also need "Odd Partitions of odd length with Distinct parts" (ODP odd ), their union ODP = ODP ev ∪ ODP odd , and the partition 0. Hence for 0 = α ∈ ODP ev one has where all α i ∈ 1 + 2Z ≥0 , and we assume Introduce for such partition 0 = α ∈ ODP ev Then we can rewrite g|0 as Note that α ∈ ODP ev otherwise 0|g|0 = 0. We also rewrite (2.23) as It is clear that we can also write Then clearly . Now substitute the vertex operator expression for the fields σφ(z j )σ −1 . We thus obtain (assu- (2.25) Then setting Π(z) := −1≤i<j≤k where Pf stands for the Pfaffian. The last equality follows from Wick's theorem and from We will now show that f (t) is equal to Note that from this for k odd, due to Wick's rule for bosons we also have where Hf stands for the Hafnian. The Hafnian of a symmetric matrix A of even order is defined as follows where the sum runs over all permutations σ of {1, . . . , 2k} satisfying As one can see the Hafnian contains 1 · 3 · 5 · · · · · (2k − 1) =: (2k − 1)!! terms.

Remark 1.
Comparing this with f 0 in (2.26), we have a new proof for the identity (2.29) of [7].
Since g|0 , where g is given by (2.22), is a possibly infinite linear combination of which can be obtained by taking residues of the expression in (2.24), one deduces that (2.32)

A CKP wave function
Now we want to study φ(z)g|0 . Consider the expression where (2.20) was used. Clearly, one also has Now substitute this in (2.23), omitting the tensor symbol and writing s j for t j in the right-hand side of the tensor product, we obtain, that for every α, β ∈ ODP odd the coefficient of ξ α η β is equal to Now we want to express g α (t, z) in terms of the τ β (t)'s. It will be convenient to introduce some more notation here. Let where all α i ∈ 1 + 2Z ≥0 , and we assume then we define an "addition" as follows Note that the notation α + β was used differently in [11] where it was defined as ( In a similar way the subtraction α\α i for α i ∈ α is defined by Then for hence k odd, we find In particular for α = (1) = 1 we find We now want to calculate g α (t, z) as some expectation value. Using (2.33), we have where we have used (2.32).
Now concentrating on (2.35) and divide this by (−)

Bilinear identity in super notations
In the previous section we obtained a wave function. In this section our approach will be slightly different. We want to superize, i.e., obtain a supersymmetric wave function that also include the Grassmannian times t i with i ∈ 1 2 + Z and the corresponding bilinear equation for this super wave function (3.11). Let us point out that in this way we shall re-write results of the previous section using super notations. We regard this an important step, which might be very fundamental for the further development of the theory. However, unfortunately we were not able to obtain Lax equations with respect to odd Grassmannian times in this setting.

Super vertex operator
Recall the super commutation relations (2.18) and the super times (2.18) and the definition of Γ(t) in (2.27). It is convenient to introduce an auxiliary parameter ζ, which is a Grassmannian variable, an odd counterpart to z: ζ 2 = 0, zζ = ζz, deg z = 2 deg ζ = −1.
Below we consider the action of odd negative powers of D on the exponentials e zx+ζξ , provided we define ∂ n e zx := z n e zx , n ∈ Z. To do this we write D 1−2n = D∂ −n . In such a way we write D 2n · e ϕ(t,z,ζ) = z n e ϕ(t,z,ζ) , D 2n+1 · e ϕ(t,z,ζ) = z n −ζ + ξD 2 e ϕ(t,z,ζ) , n ∈ Z; (3.13) in particular ζe ϕ(t,z,ζ) = − D − ξD 2 · e ϕ(t,z,ζ) . (3.14) Let us introduce where negative powers of D are to be understood in the sense of (3.13). From (3.14), (3.15) and (3.12) it follows that In the Lemma below star means the conjugation in the algebra of super PDOs with properties (ab) * = ±b * a * , where − is taken iff both a and b are odd. We have (∂  16) where + and − are taken if g is respectively even and odd. Proof . Let where P (0) n are odd and P (1) n , Q n are even. Then Using (3.16), we find that the right-hand side of (3.17) is equal to Next consider the left-hand side of (3.17). We have The evaluation of the Ber of the product of these two results in (3.19). A similar calculation yields (3.18).
Taking into account where ∂ := ∂ ∂t 1 = D 2 , and ϕ(t, z, ζ) is as in (3.3), we obtain where the subscript − means the taking of projection on series with negative powers. Thus are differential operators. These equations are basically equivalent to the set of equations (2.40).

Related symmetric functions
In this section we want to introduce polynomial functions, C λ , related to the basis vectors |λ of the bosonic Fock space F as the image of the mapping σ described in the Subsection 2.3. These are polynomials in Grassmannian even and odd variables t. In super Miwa variables z i , ζ i , i = 1, . . . , k these functions are symmetric with respect to the action of the permutation group S k on the set of pairs (z i , ζ i ), and polynomial with respect to variables ζ i and x i := z −1 i . These polynomials may be considered as analogues of the celebrated Schur (and projective Schur) functions which are related to Fock space of charged (resp. neutral) fermions. The theoretic field construction of new functions allows to derive certain properties which are similar to the properties of the Schur and the projective Schur functions.

Polynomials C λ
Let us introduce suitable notations for the basis of bosonic Fock vectors in F (see (2.3)) and in F * labeled by partitions whose parts are odd numbers where λ = (λ 1 , λ 2 , . . . , λ k ) is a set of odd numbers and λ 1 ≥ λ 2 ≥ · · · ≥ λ k > 0, (λ) := k = 1, 2, . . . . The set of partitions with odd parts will be denoted by OP. Note that the above vectors λ| differ from the vector α| as defined in (2.32), that is the reason why we write λ here to avoid this confusion.
In this section we shall use, besides the parts λ i of partitions λ, also the variables n i = 0, 1, 2, . . . related to odd numbers λ i as follows λ i =: 2n i + 1. (4. 2) The sum |λ| := λ 1 + · · · + λ k is called the weight of the partition λ. Here the factor d λ for λ ∈ OP is defined by where m i = m i (λ) is the number of parts of λ equal to i (or, the same, the multiplicity of i).
(Then we can denote the partition by its frequency notation λ = (1 m 1 3 m 3 5 m 5 · · · )). For instance we see that, for partitions λ and µ, we have the ortho-normality condition λ | µ = δ µ,λ , λ, µ ∈ OP. ; . . . ) where deg t j := j. For example We evaluate C λ in case all (Grassmannian) odd variables vanish: t n+ 1 2 = 0, n = 0, 1, 2, . . . . (Recall that in this case we denote t as t = (t 1 , t 3 , . . . ).) Remark 2. Weighted polynomial functions are often presented as symmetric functions of some variables x 1 , . . . , x N , where the number of variables may be irrelevant. In our case we put where all even-labeled t n vanish. For n odd we write x n i , n = 1, 3, . . . , . . , N . Below by polynomial functions in Miwa variables we mean polynomial functions in the variables x i = 1 z i . Then C λ (t) vanishes if (λ) is odd 1 . If (λ) is even, then by Wick's theorem where OP e is the set of all partitions with even number of odd parts, and as we shall see where s λ is the Schur function, and (n|m) is a one-hook partition in the Frobenius notations, see [11]. Indeed, first, from where h n are complete symmetric functions [11]. Now from (2.1), (2.2) we obtain while Schur function evaluated on a one-hook partition is (see Chapter I, § 3, Example 9 in [11]) s (n 1 |n 2 ) (t) = h n 1 +1 (t)e n 2 (t) − h n 1 +2 (t)e n 2 −1 (t) + · · · + (−) n 2 h n 1 +n 2 +1 (t), where e n are elementary symmetric functions. Then taking into account that for t = (t 1 , t 3 , . . . ) of form (4.5) we get the equality h m (t) ≡ e m (t), we obtain (4.6).
Thus we get It follows from (4.7) and from s λ (−t) = (−1) |λ| s λ tr (t) that Next, if all Grassmannian odd variables except t 1 2 vanish, we obtain Recall that t = (t 1 , t 3 , t 5 , . . . ) and that λ i are related to n i via (4.2). In general case, where the odd Grassmannian variables do not vanish, we can see that C λ is of even Grassmannian parity in case (λ) is even, and it is of odd Grassmannian parity in case (λ) is odd. We can write Hf s (n i |n j ) (t) + C e if (λ) even, where C e and C o are polynomials in odd Grassmannian variables of the order (λ), C e starts with quadratic terms, while C o starts with cubic ones. This follows from the consideration of (2.24) in Subsection 2.1.

Orthogonality
One can verify the equality, using (2.27) On the other hand due to (4.9) and to (2.18) we obtain 0| Γ(t)Γ(t) |0 = e From the last equality we obtain . . . and therefore Let f (t) and g(t) be series in the variables {t i }. We introduce the following scalar product (4.14) In particular due to (2.18) where [a] denotes the integer part of a (i.e., a = [a] + where 0 ≤ ≤ 1 for a < 0). It follows from (4.13) and (4.3) that the polynomials form an orthonormal basis in the scalar product (4.14):

Combinatorial meaning of
Each partition with odd parts may be presented as λ = (1 m 1 3 m 3 5 m 5 · · · ) where m i is the multiplicity of the number i (that means that the partition λ contains the part equal to i m i times). The length (λ) of the partition λ is equal to i=1,3,5,...
Let us visualize this, in a similar way as in the papers [5,18], as the one-dimensional semiinfinite lattice of cites (baskets) in our case numbered by odd positive integers. A basket number i (i = 1, 3, 5, . . . ) contains m i identical balls (and therefore the multiplicity m i may be also called the occupation number). Nonequivalent distributions of balls is in one-to-one correspondence with partitions from the set OP. (The length of a partition is equal to the number of balls, the ratio of the weight of the partition and the length of the partition may be considered as the location of the mass center of the balls).
Let us consider the following discrete time random process describing the creation of λ ∈ OP or, the same, of ball configurations. It starts with a given partition, say, µ. (The case where µ = 0 describes the configuration where all baskets are empty at time t = 0). At each discrete time instant t = 1, 2, 3, . . . one of the following two possible events occurs with equal probability (A) either two balls are created in the leftmost basket (basket number 1), or (B) a ball chosen at random in any of baskets, say, in basket number i, is moved to the nearest basket to the right (to the basket number i + 2). It is clear that at each time step the weight of the related partition increases: |λ| → |λ| + 2, thus |λ| = 2t. A problem is to find a number of ways to create a given distribution λ of the balls in baskets along the process described above in t = 1 2 |λ| steps. We denote this number N µ→λ .
If t = t + t then Γ(t) = Γ(t )Γ(t ) by inserting the unity operator λ∈P |λ λ| between Γ(t ) and Γ(t ) we obtain This property is quite similar to the property of the Schur functions (see (5.9) in I of [11]).
One may relate C λ/µ to the numbers N µ→λ described in the previous subsection.

CKP tau function and polynomials C λ
First of all we note that C λ (t) is not a solution of the Hirota bilinear equations, and, therefore is not a CKP tau function. However, due to (4.11) CKP tau functions (3.6) are series in C λ (t) as follows where g λ = λ| g |0 . where the sum ranges over all λ ∈ OP whose parts have even multiplicities, m i =: 2k i , i.e., of the form λ = 1 2k 1 3 2k 3 5 2k 5 · · · . The numbers U λ are defined as (4.17) The right-hand side of (4.16) may be compared to sums over partitions in [6] and in [14] dealing with tau functions of neutral and charged BKP hierarchies, respectively. where λ = (λ 1 , . . . , λ k ) is a partition, and x λ = x λ 1 · · · x λ k , and where P ev is a set of all partitions with even number of parts.