Pfaffian and determinantal tau functions I

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 [26] and Date, Jimbo, Kashiwara and Miwa [4] introduced and described in the beginning of the 1980's 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 GL ∞ 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 [31] 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 by 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 [24] 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 [31] in contrast to the one-sided KP). There we have two sets of higher times, t andt. This is the B-analogue of Takasaki's 2-DKP of [28] (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 2-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 Section 7 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] where [x, y] + = xy + 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 ∈ GL ∞ which may be written as where : , and a ij 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 by 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 [α n , α m ] − = nδ n+m,0 where [x, y] − = xy − yx is commutator. Define 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 elementsb 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 also form the Lie algebra b ∞ . 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 As in the KP case we now consider an element in the group h ∈ B ∞ , such an element may e.g. be written as is the BKP hierarchy in the fermionic form, see e.g. [10]. One turns this into a hierarchy of differential equations by 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 γ n |0 = 0|γ −n = 0, n ≥ 0 .
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 [33] and define an automorphismˆon the B type Clifford algebrâ and γ n +ˆ(γ n ) = γ n +γ n = α n , n odd. Then and since the elements h and γ n commute withĥ andγ n 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 ∈ GL ∞ is an element in the level 2 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 GL ∞ 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 W KP (t, z) = wKP(t,z) τKP(t) , with

Another realization of the KP hierarchy
In order to describe the relation between the Toda lattice hierarchy (or 2-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 (48) will be convenient later on when we apply the involution ω to this new realization.
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) rewritten as g = e a,b=1,2 n,m∈Z a ab nm ψ (a) n ψ †(b) m + a=1,2 n∈Z a a0 nm ψψ †(α) n + a=1,2 n∈Z a 0a nm ψ (a) n ψ † +a 00 ψψ † .
Remark 1. One can identify eq. (52) with the Hirota equation for the three-component KP hierarchy [4], [10], [14] restricted on the class of g which does not depend on ψ 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 2-component oscillator algebra (cf (5)) We for any a, n, m and k = 0, 1.

B-case in this new realization
Recall the involution ω (see (11)) on the Clifford algebra. Using the relabeling (44)-(48), 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 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 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 irriducible highest weight modules for b ∞ , which is formed by the quadratic elemnts 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 (57) Let h be a group element of B ∞ . Then we can consider h|0 , clearly such an element decomposes As for GL ∞ there is a boson-fermion correspondence given by for the fermionic fields (cf. (62)) and (63) induces Then σ B (h|0 ) = τ B . And using (92) one finds Note that pθ = −θp.
An element h ∈ B ∞ is an even element in the Clifford algebra.
We have Since 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: Equation (101) 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 2-component KP (or Toda Lattice hierarchy). In fact one can use the same involutionˆof (28) on the Clifford algebra, it inducesˆ( We want to consider The following lemma will be useful: If n < 0 then we have Next we consider the case If n > 0 then: Which finishes the proof of the lemma And τ Remark 3. It follows from Proposition 1 that the square of the BKP function is related to the two-component KP Green function Kn(x, y, t (1) , t (2) ) := n + 1, n − 1|e m>1 t (1) as follows (Vn(t, z)) 2 = Kn(z, z, t, −t) Let us note that two-component KP is useful to study matrix models. The Green function K(x, y) is widely used for computing various correlation functions. However, these models do not possess the property of factorization g = hĥ.

A relation between the wave functions
We introduce the 2 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 (105) can be rewritten in the matrix form res We will now show that two-component KP wave functions W (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 2 component KP wave function, via To prove this we calculate = (−z) n e j>0 t (1) j z j n, −n, 0|e j>0 t (1) Now set g = hĥ and t (2) . Then Thus we obtain which is the first formula of (−1) The other formulas can be obtained in a similar way. From these formulas and Proposition 1 one easily deduces Proposition 2.

The two-sided BKP (2-BKP) and two-component Toda lattice
In this section we consider a two sided version of some of the previous constructions. Remark 4. We have For the proof of the Remark 4 we notice that i ) g † and the fact that the pairing of two vectors and of two corresponding dual vectors coincides. Later we shall omit the argument g on the left hand side of (139).
In what follows we shall put l = 0, t In the same way as Proposition 1 was obtained we get 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.
The other way to prove it is just to modify g in the Proposition 1 by a certain right factor whose action on |0, 0, 0 recreates the vector (−1) Semi-infinite Toda lattice. First, let us recall that the Hirota equation for the Toda lattice tau function [10], [31] 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 [31]. Here we shall consider the semi-infinite Toda lattice which may be presented as τ TL n (t,t) = (−) (see also Appendix A.1).
Using the relations and Wick's rule (see Appendix A.4), the tau function (153) may be presented in the determinantal form: As we see τ TL 1 = m 00 . The tau function (153) may also be written in Takasaki form [27], [29], [30] as where Series (157), where M is specified, appear in various problems of random matrices, random partitions and in counting problems. Another way to present tau functions of the semi-infinite TL tau functions (which is of use in many models of random matrices, see [12] and Examples below) is (159) where dµ(z 1 , z 2 ) is a bi-measure which should be specified according to a problem we are interested in. There are additional parameters here,t (1) ,t (2) and m, which are hidden parameters of Toda lattice solutions. This is a particular case of the tau function (139).
From (159) we obtain τ TL n,m (t (1) , t (2) ,t (1) ,t (2) ) = δ n,m for n ≤ m, and [12]) both the fermionic representation for the partition function of the ensemble of normal matrices (about this ensemble see [3] and [21]), and the partition function of the complex Ginibre ensemble (about Ginibre ensemble of complex matrices see Ch. 15.1 in [20]). In this examplez is the complex conjugate of z.
Here δ (2) is the two-dimensional delta function.
Example 1.2. The choice dµ(z 1 , z 2 ) = e z1z2 dz 1 dz 2 , z 1,2 ∈ R yields (see [12]) the fermionic representation for the partition function of the two-matrix model introduced in [9] (the relation of the two-matrix model and TL see in [6] and different fermionic representation see in [7]). In case we take z 1,2 ∈ S 1 we obtain the model of two unitary matrices [34].
In case the bi-measure dµ and the matrix M are related by the moment's transform then the tau functions (153) and (153) may be equated as follows Now the tau function (153) written here depends on the additional parameterst (1) ,t (2) , m. Expression for τ TL 1 given by (160) allows to obtain all τ TL N , N > 1, in the reccurent way from TL Hirota equation (154).
where τ B N (t) = N |e i∈Z βiti e 1 2 i,j Aij ψiψj + √ 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 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 for odd and for even N , we have We have the following where τ B n,m (t,t) = n|e i∈Z βiti e and where dω(z 1 , z 2 ) = −dω(z 1 , z 2 ) and dν(z) are measures.
Example 2.8. For the Ginibre ensemble of real matrices (see Ch. 15.3 in [20]) the choice of the measure is more complicated, and may be found in [23].
The relation between realizations (164) and (170) is given by As we see the matrix m is quasi-skew symmetric one. 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 (170).
We obtain a generalization of the result presented in [1], [2] where the relation (177) was achieved for the case of even N and for skew symmetric matrices m. This case may referred as the D-type Pfaff lattice (DPL), see [28].

Outlook
In our next paper [18] the analogue 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 5 and 1.
solves Hirota equation which up to the sign factor (−) n+n ′ in the first integral is (179) if we change z → z −1 in the second integral in (182). This brings us to the relation (152).
A.2 Hirota equation for the two-sided two-component KP.
The A. 3 Hirota equations for the two-sided BKP.
Wick's relations. Let each of w i be a linear combination of Fermi operators: Then the Wick formula is l|ŵ 1 · · ·ŵ n |l = Pf [A] i,j=1,...,n if n is even 0 otherwise (196) where A is n by n antisymmetric matrix with entries A ij = l|ŵ iŵj |l , i < j.