Pfaffian and Determinantal Tau Functions

Adler, Shiota and van Moerbeke observed that a tau function of the Pfaff lattice is a square root of a tau function of the Toda lattice hierarchy of Ueno and Takasaki. In this paper, we give a representation theoretical explanation for this phenomenon. We consider 2-BKP and two-component 2-KP tau functions. We shall show that a square of a BKP tau function is equal to a certain two-component KP tau function and a square of a 2-BKP tau function is equal to a certain two-component 2-KP tau function.


Introduction
Sato and Sato [27] and Date, Jimbo, Kashiwara and Miwa [4] introduced and described in the beginning of the 1980s the KP hierarchy in various setting. They introduced a tau function, which is a fundamental object in this theory. It is an element in a G L ∞ group orbit and as such a solution of the KP hierarchy. Around the same time Date, Jimbo, Kashiwara and Miwa introduced in [5] also a new hierarchy of soliton equations, which they called the KP hierarchy of type B or BKP hierarchy. The corresponding tau function is an element of the B ∞ group orbit and hence a solution to this new BKP hierarchy. In a straightforward calculation, they show that this BKP tau function is the square root of a certain KP tau function. Their proof of this phenomenon is fundamental for the contents of this paper. Here, we show that this method is also applicable for the observation of Adler, Shiota and van Moerbeke [1,2], that a tau function of the Pfaff lattice is a square root of an Ueno, Takasaki [32] Toda lattice tau function. In this introduction, we will recall the work of Date, Jimbo, Kashiwara and Miwa [4] and explain the relation BKP versus KP. The involutions that are used in their work, which provide the relation between the two tau functions of KP and BKP, also give the relation between the various tau functions of the Pfaff and Toda lattice. This means that the construction of the KP group element out of the BKP group element is the same. One finds in both cases that g KP = h BKPĥBKP , whereĥ BKP is constructed out of h BKP using one of the involutions.
The Pfaff lattice of Adler and van Moerbeke [1,2] was discovered by Jimbo and Miwa ( [10], section 7) it was rediscovered by Hirota and Ohta [13] as the coupled KP hierarchy and studied in a paper by Kac and one of the authors [15] as the charged DKP hierarchy. The Pfaff lattice studied in [16] is slightly bigger than the one studied by Adler, Shiota and van Moerbeke in [2] it is the charged BKP hierarchy of [15], called the large BKP in [25] to make difference to the small BKP of [5]. Here, we study this charged or large BKP.
In addition, we also consider the large 2-BKP which is two-sided evolution (via positive and negative parts of current modes, like the Toda lattice hierarchy of Ueno and Takasaki [32] in contrast to the one-sided KP). There we have two sets of higher times, t andt. This is the B-analog of Takasaki's 2-DKP of [29] (which he called the Pfaff lattice by analogy with Pfaff lattice of Adler and van Moerbeke). Also, we use two discrete variables for BKP (and also for 2-BKP): τ m,n (t,t).
In this introduction and the first 5 sections we will only describe the one-sided case, to avoid technicalities. In Section 6 we will introduce the two-sided case, which will be important for matrix models.
For matrix models, we need the semi-infinite Toda and Pfaff lattices. The fermionic expressions for the related tau functions is considered in the Appendix A.1 where a set of examples is written down. In this Section, we generalize the observation of Adler, van Moerbeke and Shiota [1,2] about the relation between Toda and Pfaff lattices.
But, first, we will start with the recollection of the work of Date, Jimbo, Kashiwara, and Miwa, on KP and BKP hierarchies.
Fermions and KP hierarchy Consider, following [4] and [10] charged fermions where [x, y] + = x y + yx is anticommutator. The elements f i and f † i form the basis of a vector space, which we denote by V . Introduce the spin modules (right and left Fock spaces) with vacuum vectors, |0 , 0|, such that Let g ∈ G L ∞ which may be written as where : , and a i j are some complex numbers. The elements : f i f † j : together with 1 form a basis of the Lie algebra gl ∞ , see [4], [10] for more details. Then, is the KP hierarchy in the fermionic form, see [4]. One turns this equation into a hierarchy of differential equation using the boson-fermion correspondence, see, e.g., [14] for more details.
However, the tau function can be be calculated in a different way, define the oscillator algebra We have where then, Now, Then τ KP (t), defined by the following expectation value is a solution of the KP hierarchy.
Neutral fermions and the BKP hierarchy. Following [10], we introduce an involution ω on the Clifford algebra, defined by The fixed points of ω in the vector space V with basis f i , f † i are the elements The elementŝ (13) are elements in the −1 eigenspace of ω. b j andb j form a new basis of V , we have The fixed points in gl ∞ are 1 and the elements These elements form a Lie algebra of type b ∞ . If one considers the action of these elements on the vacua of the spin modules, one obtains a level two representation of b ∞ (Note that here we allow also certain infinite sums of these elements).
Note also that only the α k for k odd are fixed by ω.
It is straightforward to check that The following observation is crucial. The elements : b j b k : together with 1, or the elements :b jbk : together with 1, separately form the Lie algebra b ∞ . These two Lie algebras that have C1 as intersection, commute. If one considers the action of these elements separately on the vacua, one obtains a level one representation. The corresponding module in terms of the b j orb j is called the spin module of type B. One has and As in the KP case we now consider an element in the group h ∈ B ∞ , such an element may, e.g., be written as Then, is the BKP hierarchy in the fermionic form, see, e.g., [10]. One turns this into a hierarchy of differential equations using twisted vertex operators, see [15] for more details. The corresponding BKP tau function can be obtained as follows. Define the twisted oscillator algebra Then, γ n = 0 if n is even. We have and then, The BKP tau function τ BKP (t) is defined by the following expectation value and is a solution of the BKP hierarchy. A crucial observation is the fact that in this construction we could have replaced the b n byb n and in this way would have obtained the same result.

A relation between KP and BKP tau functions
We follow Jimbo and Miwa [10] or rather You [34] and define an isomorphismˆon the B-type Clifford algebrâ Now, and γ n +ˆ(γ n ) = γ n +γ n = α n , n odd.
Let g = hĥ, whereˆh =ĥ and h ∈ B ∞ , then h andĥ commute, Now consider the expression where t k andt k are two sets of parameters, the times of the BKP hierarchy. The above expression (31) can be reduced, i.e., one can move the exponentials to the right such that they act as 1 on the right vacuum |0 . Since γ k commutes withĥ andγ k commutes with h and the action of γ k on h is the same asγ k onĥ (because 1 andb I is the same element but then with the b j s replaced byb j s. Using the action on the left and right vacuum of the elements, we find that Now note that also hence Now using (30), we find for g = hĥ. Since h is of the form (20) we find, using (29), that The element (−) m : f n f † −m : −(−) n : f m f † −n : is fixed by ω, hence g ∈ G L ∞ is an element in the level two representation of b ∞ , which one gets by considering the action of the ω invariant elements in gl ∞ . In other words, this τ KP (t 1 , 0, t 3 , 0, t 5 , 0, . . .) is an element in the B ∞ group orbit of the vacuum, where we consider B ∞ as a subgroup of G L ∞ and not as a group acting on the spin module of type B. The latter one is related to the level 1 representation of b ∞ .
A relation between KP and BKP wave functions Recall from (7) the formula for f (z), the KP wave function is defined as Using the boson-fermion correspondence (see, e.g., [4,14] or (67) in the next section), which is based on (8), then gives Recall b(z) from (25) and define analogouslyb(z) = j∈Zb j z j , then using (12) and (13) Now let, as before, g = hĥ, then Using (26) and the known bosonization formula (6.5) of [10], and a similar formula with "hats" ((7.4) in [10]), we obtain If we now divide this by we obtain an expression for W KP (t odd , z). Define Again using (26), one deduces Now comparing (44) and (46), we obtain:

Another Realization of the KP Hierarchy
To describe the relation between the Toda lattice hierarchy (or two-component KP) and the Pfaff lattice (or large BKP), we need another realization of both spin modules. We start with the KP hierarchy and relabel the fermions f i and f † i as follows The minus sign in (52) will be convenient later on when we apply the involution ω to this new realization. Then, for a, b = 1, 2 all other anti-commutation relations are zero. The action on the vacuum is given by Now, the KP equation (4) may be rewritten as where g is the same as in (4) It may also be written in the form where g (0,0) , g (1,−1) , g (1) and g (−1) do not contain neither ψ, nor ψ † . Then, g (0) is even and g (±1) are odd in the Z 2 grading of the Clifford algebra. Define a two-component oscillator algebra (cf (5)) We have and introduce right and left Fock vectors which are highest weight vectors for the oscillator algebra (59): for any a, n, m and k = 0, 1.
We have the orthogonality condition n, m, k|1|n , m , k = δ n,n δ m,m δ k,k where n, n , m, m ∈ Z, k, k = 0, 1. Now, we can write Boson-fermion correspondence. Now, we set the following correspondence. The left Let σ be the corresponding isomorphism. Introduce the fermionic fields then, where θ 2 = 0, and Here, for a, b = 1, 2 and a = b.

B-Case in this New Realization
Recall the involution ω (see (11)) on the Clifford algebra. Using the relabeling (48)-(52), this induces It is straightforward to check that Now define a new basis in our Clifford algebra consisting of elements that are fixed by ω and of elements x with ω( Then, all other elements anticommute. Also, Spin module for b ∞ . Here, we consider the elements ψ j , ψ † j , ϕ, which are the elements invariant under ω. Recall all other elements anticommute and The b ∞ spin module splits into two parts, V0 ⊕ V1, where V0 has the highest weight vector |0 = |0 , and V1 has the highest weight vector |1 = √ 2ϕ|0 . Both V0 and V1 are irreducible highest weight modules for b ∞ , which is formed by the quadratic elements of the Clifford algebra together with 1.
From now, we shall focus on V0.
As before one can define an oscillator algebra B In V0, one has the following highest weight vectors if one restricts to the B where similar to (61) Let h be a group element of B ∞ . Then we can consider h|0 , clearly such an element decomposes h|0 = n∈Z h n |n As for G L ∞ there is a boson-fermion correspondence given by for the fermionic fields (cf. (66)) and (67) induces Then, σ B (h|0 ) = τ B . And using (96) one finds Note that pθ = −θ p. An element h ∈ B ∞ is an even element in the Clifford algebra. We have One obtains Note that h = a + √ 2bϕ where a is an even element expressed in ψ i and ψ † i , and b is an odd element expressed in ψ i and ψ † i . Then, Now clearly: Also, Equation (105) turns into the large BKP hierarchy or Pfaff lattice:

A Relation Between the Tau Functions
In the same way as the small BKP is related to the KP hierarchy, the large BKP (or Pfaff lattice) is related to the two-component KP (or Toda Lattice hierarchy).
In fact, one can use the same involutionˆof (28) on the Clifford algebra, it induceŝ Then,ˆ( We want to consider e l s l (β l +ˆ(β l )) h ·ˆ(h) |0 = e l s l β l h · e l s lβlĥ |0 Now, Thus, and hence e s l (β l +β l ) = e l s l α (1) l e − l s l α (2) l First, note that Thus, The following Lemma will be useful: If n < 0, then we have Next, we consider the case If n > 0, then: Now, use that Thus, If n < 0, then Which finishes the proof of the lemma Thus, and Thus,

A Relation Between the Wave Functions
We introduce the two-component KP wave function W ±(a) n (t, z) by We introduce next the BKP wave function V ± n (z). To do that, we first observe that (109) can be rewritten in the matrix form and We will now show that two-component KP wave functions W ±(0) n (t (1) , t (2) , z) evaluated at t (1) j = s j = −t (2) for group elements, g = hĥ may be expressed in terms of the BKP wave functions V ± n (z).

PROPOSITION 2. A BKP wave function is related to a two-component KP wave function, via
To prove this, we calculate w (1) n (t (1) , t (2) , z) = n + 1, −n, 0|e j>0 t (1) j α j +t (2) jα j ψ (1) Now, set g = hĥ and t (1) j = s j = −t (2) . Then Thus, we obtain which is the first formula of The other formulas can be obtained in a similar way. From these formulas and Proposition 1, one easily deduces Proposition 2.

2-BKP and two-component 2-KP (two-component Toda lattice) tau functions. Consider
which may be considered as the two-component 2-KP tau function, or, the same, two-component Toda lattice tau function. 1 The Hirota equations for the tau function (140) may also be found in the Appendix A. 3 In what follows, we shall put t  where g = hĥ.
For proof of Proposition 3, we note that from Lemma 1 it follows that which allows to repeat all the steps of the derivation of the Proposition 1.

An Example: Toda Lattice (TL) and B-Type Pfaff Lattice (BPL)
In many applications (like random matrices or random partitions), the semi-infinite Toda lattice and semi-infinite Pfaff lattice are of use. Here, we relate the semiinfinite Pfaff lattice of B-type to the semi-infinite Toda lattice.

Semi-infinite Toda lattice.
First, let us recall that the Hirota equation for the Toda lattice tau function [10,32] and for the two-component KP tau function [10] coincide up to a sign factor, see the relation (9.7) in [10] and the Theorem 1.12 in [32].
Here, we shall consider the semi-infinite Toda lattice which may be presented as (1) , t (2) The tau function (153) is the tau function τ (0) . This choice provides the semi-infinity of the TL equation which is where we put τ TL N (t (1) , t (2)  and Wick's rule (see [10]), the tau function (153) may be presented in the determinantal form: where m i j (t (1) , t (2) ) = k,l≥0 M i+k, j+l s k (t (1) )s l (−t (2) ) As we see τ TL 1 = m 00 . The tau function (153) may also be written in Takasaki form [28,30,31] as τ TL N (t (1) , t (2) where The series (157), where M is specified, appears in various problems of random matrices, random partitions and in counting problems. Another way to present tau functions of the semi-infinite TL (which is of use in many models of random matrices, see [12] and Examples in Appendix A.1 below) is τ TL n,m (t (1) , t (2) ,t (1) ,t (2) where dμ(z 1 , z 2 ) is a bi-measure which should be specified according to the problem we are interested in. There are additional parameters here,t (1) ,t (2) and m, which are hidden parameters of the Toda lattice solutions. This is a particular case of the tau function (140). From (159), we obtain τ TL n,m (t (1) , t (2) ,t (1) ,t (2) ) = δ n,m for n ≤ m.

(162)
and It is well-known that the determinant of a skew symmetric N × N matrix vanishes if N is odd. The square root of a skew symmetric matrix of even size is the Pfaffian (see Appendix A.5) of this matrix. Let us call the sum of a skew symmetric matrix and a symmetric matrix of rank 1 the quasi-skew symmetric matrix. The square root of a quasi-skew symmetric matrix may be identified with a Pfaffian of some different matrix: Thus, for both odd and even N , we have We have the following COROLLARY 1.
The first applications of BKP tau functions (167) were given in [16] to describe orthogonal and symplectic ensembles of random matrices. In Appendix A.1, one may find further examples.

Outlook
In our next paper [18], the analog of the Proposition 1 for the multicomponent BKP tau functions [15] will be written down. Also, we shall consider the relations between various matrix integrals and between series over partitions which result from Propositions 1 and 5.
The relation between realizations (161) and (167) is given by In B-case relations (155),(156) read as As we see the matrix m is quasi-skew symmetric. Then, for both odd and even N , we have from Proposition 5 According to Lemma 2, τ B N is a certain Pfaffian (this result may be obtained also from the Wick theorem applied to the fermionic expectation value (167).