Symmetry, Integrability and Geometry: Methods and Applications Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows ⋆

We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy the $2n$-component KP hierarchy. The hierarchy equations can be formulated in terms of pseudo-differential equations for $n \times n$ matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato-Wilson relations. A reduction process leads to the AKNS, two-component Camassa-Holm and Cecotti-Vafa models and the formalism provides simple formulas for their solutions

In memory of Vadim Kuznetsov. One of us (JvdL) f irst met Vadim at a seminar on Quantum Groups held at the Korteweg-de Vries Institute in Amsterdam in 1993. Vadim was then still a post-Doc. Later meetings at several other conferences, e.g. in Cambridge, Oberwolfach and Montreal provided better opportunities to learn Vadim's kind personality and his wonderful sense of humor. Both authors of this paper have the best recollection of Vadim from meeting at the conference "Classical and Quantum Integrable Systems" organized in 1998 in Oberwolfach by Werner Nahm and Pierre van Moerbeke. A memory of one pleasant evening spent with him in the setting of a beautiful conference center library clearly stands out. Vadim joined us and Boris Konopelchenko after a (successful) play of pool game against Boris Dubrovin. The library has a wonderful wine cellar and we were all having a good time. Vadim was in great mood and filled the conversation with jokes and funny anecdotes. When thinking of him, we will always remember in particular that most enjoyable evening.
a Grassmannian approach to constructing tau functions in terms of expectation values of certain Fermi operators constructed using boson-fermion correspondence. This formalism provides a systematic way of constructing multi-component tau functions for multi-dimensional Toda models, which, in this paper, embed both positive and negative flows.
The set of Hirota equations for the tau functions is obtained by taking expectation value of both sides of bilinear commutation relation defined on a Clifford algebra with Fermi operators ψ ± (i) l , satisfying the relations for l, k ∈ Z + 1/2, i, j = 1, . . . , n, λ, µ = +, −. We rewrite Hirota equations in terms of formal pseudo-differential operators acting on matrix wave functions derived from the tau functions. This method gives rise to a general set of Sato-Wilson equations.
Next, we impose a set conditions on the tau function which define a reduction process. Under this reduction process the pseudo-differential equations for the wave functions describe flows of dressing matrices of the multi-dimensional Toda model. In the case of 2×2 matrix wave functions these equations embed the AKNS model and the two-component version of the Camassa-Holm (CH) model for, respectively, positive and negative flows of the multi-dimensional 2 × 2 Toda model. Section 2 is meant as an informal review of semi-infinite wedge space and Clifford algebra. In this setup, in Section 3, we formulate the multi-component tau functions as expectation values of operators satisfying the bilinear identity. This formalism contains the Toda lattice hierarchy as a special case. Section 4 shows how to rewrite the formalism in terms of pseudo-differential operators acting an wave function. We arrive in this way in general equations of Sato-Wilson type. The objective of the next Section 5 is to introduce a general reduction process leading to a multi-dimensional Toda model with positive and negative flows acting on n × n matrix wave functions explicitly found in terms of components of the tau functions from Section 3. The Grassmannian method provides in this section explicit construction of matrices solving the Riemann-Hilbert factorization problem for GL n .
As shown in Section 6 the model obtained in Section 5 embeds both AKNS and the 2-component Camassa-Holm equations. Further reduction uses an automorphism of order 4 and as described in Section 7 reduces flow equations to Cecotti-Vafa equations. In Section 8, we use the Virasoro algebra constraint to further reduce the model by imposing homogeneity relations on matrices satisfying Cecotti-Vafa equations.
The particular advantage of our construction is that it leads to solutions of the AKNS and the 2-component Camassa-Holm equations and Cecotti-Vafa equations in terms of relatively simple correlation functions involving Fermi operators defined according to the Fermi-Bose correspondence. These solutions are constructed in Section 9 and 10, respectively.
2 Semi-inf inite wedge space and Clif ford algebra Following [9], we introduce the semi-infinite wedge space F = Λ 1 2 ∞ C ∞ as the vector space with a basis consisting of all semi-infinite monomials of the form Define the wedging and contracting operators ψ + j and ψ − j (j ∈ Z + 1/2) on F by These operators satisfy the following relations (i, j ∈ Z + 1/2, λ, µ = +, −): hence they generate a Clifford algebra, which we denote by Cℓ.
Introduce the following elements of F (m ∈ Z): It is clear that F is an irreducible Cℓ-module such that Think of the adjoint module F * in the following way, it is the vector space with a basis consisting of all semi-infinite monomials of the form The operators ψ + j and ψ − j (j ∈ Z + 1/2) also act on F * by contracting and wedging, but in a different way, viz., We introduce the element m| by m| = · · · ∧ v m+ 5 , such that 0|ψ ± j = 0 for j < 0. We define the vacuum expectation value by 0|0 = 1, and denote A = 0|A|0 .
Note that (ψ ± k ) * = ψ ∓ −k and that with i 1 < i 2 < · · · < i k < 0 and j 1 < i 2 < · · · < j ℓ < 0 form dual basis of F and F * , i.e., We relabel the basis vectors v i and with them the corresponding fermionic operators (the wedging and contracting operators). This relabeling can be done in many different ways, see e.g.
[10], the simplest one is the following (j = 1, 2, . . . , n): v (j) k = v nk− 1 2 (n−2j+1) , and correspondingly: Notice that with this relabeling we have: Define partial charges and partial energy by Total charge and energy is defined as the sum of partial charges, respectively the sum of partial energy. Introduce the fermionic fields (0 = z ∈ C): Next, we introduce bosonic fields (1 ≤ i, j ≤ n): where : : stands for the normal ordered product defined in the usual way (λ, µ = + or −): One checks (using e.g. the Wick formula) that the operators α (ij) k satisfy the commutation relations of the affine algebra gl n (C) ∧ with the central charge 1, i.e.: and that α (ij) k |m = 0 if k > 0 or k = 0 and i < j.
The operators α k satisfy the canonical commutation relation of the associative oscillator algebra, which we denote by a: Note that α (j) 0 is the operator that counts the j-th charge. The j-th energy is counted by the operator The complete energy is counted by the sum over all j of such operators. In (8.1) we will define another operator L 0 , which will also count the complete energy. In order to express the fermionic fields ψ ±(i) (z) in terms of the bosonic fields α (ii) (z), we need some additional operators Q i , i = 1, 2, . . . , n, on F . These operators are uniquely defined by the following conditions: They satisfy the following commutation relations: We shall use below the following notation |k 1 , k 2 , . . . , k n = Q k 1 1 Q k 2 2 · · · Q kn n |0 , k 1 , k 2 , . . . , k n | = 0|Q −kn n · · · Q −k 2 2 Q −k 1 1 , such that k 1 , k 2 , . . . , k n |k 1 , k 2 , . . . , k n = 0|0 = 1.
Also observe that (Γ (j) We have
If the operator A satisfies (3.1), then so does A. Following Okounkov [14] we calculate in two ways and in a similar way Using this we rewrite (3.3): For n = 1 this is the Toda lattice hierarchy of Ueno and Takasaki [15]. We now rewrite the left-hand side of (3.4) to obtain a more familiar form. For this we replace z by z −1 and write u, resp. v for t ′ , resp. s ′ , we thus obtain the following bilinear identity. Proposition 1. The tau functions satisfy the following identity: From the commutation relations (2.1) one easily deduces the following Another way of obtaining solutions is as follows. We sketch the case n = 2 (see e.g. [7]) which is related to a two matrix model. Let dµ(x, y) be a measure (in general complex), supported either on a finite set of products of curves in the complex x and y planes or, alternatively, on a domain in the complex z plane, with identifications x = z, y =z. Then, for each 1 ≤ j, k ≤ n and λ, ν = +, − the operator A = e B with B = ψ λ(j) (x)ψ ν(k) (y) dµ(x, y) satisfy (3.1). If one chooses j = 1, k = 2 and λ = +, ν = − and defines for the above A then these tau-functions satisfy (3.5).

Wave functions and pseudo-dif ferential equations
We will now rewrite the equations (3.5) in another form. Note that for these are the equations of the 2n-component KP hierarchy, see [9]. One obtains equation (66) of [9] if one chooses are the 2n-component KP tau-functions, viz for (α, −β) = ρ and (γ, −δ) = σ they satisfy the 2n-component KP equations: In [9] one showed that one can rewrite these equations to get 2n × 2n matrix wave functions.
Here we want to obtain two n × n matrix wave functions. We assume from now on that (4.1) holds for j = 0. Denote (s = 0, 1): 1 + y 1 then some of the above functions also depend on x 0 and x 1 we will add these variables to these functions and write e.g.
. Introduce for k = 0, 1 the following differential symbols ∂ k = ∂/∂x k ∂ ′ k = ∂/∂y k . Introduce the wave functions, here x is short hand notation for x = (x 0 , x 1 ) Then (3.5) leads to One can also deduce the following 6 equations, for a proof see the appendix A:
In particular Substituting this in (5.8) we obtain Note first that from (5.9) one deduces that Clearly also Using (4.5) we rewrite (A.8): N.B. This summation starts with 0. In particular In a similar way we deduce, using (4.6), from (A.10): N.B. This summation starts with 1.
In particular We will now combine (4.7), (4.8), (5.3)-(5.6) and (5.10)-(5.13). For this purpose we replace ∂ 0 by the loop variable z and ∂ 1 by the loop variable z −1 . We write and and thus obtain for fixed α and β (P (j) (x, t, u, z) = P (j) (α, β, x, t, u, z)): Which are the generalized AKNS equations (2.5)-(2.10) of [1]. Write and from now on in this section We thus obtain the following equations: From this one easily deduces the following equations for M 0 : Note that if we defineP i = M −1 0 M i , then from (5.16) we get: Note that for x 0 = x 1 = 0 we have (5.18)
In view of the fact that the determinant of the matrix (M 0 EM −1 0 )/4 is equal to the constant, −1/16, we choose to parametrize this matrix in terms of two parameters, A and f , which enter expressions for B and C as follows: Recalling equation (6.4) we easily find Using the first two identities of equation (6.6) and the fact that we derive expressions for re f ± qe −f as follows: and Taking a derivative of relation re f − qe −f = f y with respect to variable s we find, in view of (6.7), that: where we defined: Comparing two expressions (6.9) and (6.10) for the quantity re f + qe −f we find the following relation 2gA = f y 4 − A y . (6.12) By adding and subtracting (6.10) and (6.8) we get Plugging these expressions into the second relation in eq. (6.7) yields: Thus, Plugging definition (6.11) of g into relation (6.12) leads to: Inserting on the left hand side of the above identity the value of A from equation (6.13) and multiplying by −4 yields where we again used value of A from equation (6.13). Note that equation (6.14) is written solely in terms of f . For a quantity u defined as: with κ being an integration constant, it holds from relation (6.14) that Let us now denote the product f 2 s f y by m. Then from relations (6.15) and (6.16) we derive Performing an inverse reciprocal transformation (y, s) → (x, t) defined by relations: for an arbitrary function F , we find that equations (6.18), (6.19) and (6.20) become in terms of the (x, t) variables. Equations (6.21)-(6.23) were introduced by Liu and Zhang in [13] and are called the two-component Camassa-Holm equations (see also [4]).
The relation (6.14) is equivalent to the following condition which first appeared in [4]. Comparing equations (6.13) and (6.15) we find that These relations give u, f s in terms of the AKNS quantities r, q, τ 0 0 .

Homogeneity
Sometimes one wants to obtain solutions of the Cecotti-Vafa equations that satisfy certain homogeneity condition (see e.g. [5]). For this we introduce the L 0 element of a Virasoro algebra. The most natural definition in our construction of the Clifford algebra is the one given in terms of the oscillator algebra.
It is straightforward to check that Moreover, one also has Assume from now on that our operator A that commutes with Ω is homogeneous of degree p with respect to L 0 , i.e., We then calculate It is straightforward to see that this is equal to On the other hand using and we also have: in particular Assume now that we also have imposed the first and second reduction, then one has and thus also Putting all t for t j = t (j) 1 and u j = u (j) 1 . Note that in Section 10 we will construct explicit solutions of (7.4). These solutions are however not homogeneous, so they do not satisfy (8.2). We will construct such solutions in a forthcoming publication.