$\tau$-Functions, Birkhoff Factorizations and Difference Equations

$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $\tau$-functions, given as matrix elements of the action of the loop group of ${\rm GL}_{2}$ on two-component fermionic Fock space, give solutions of a $Q$-system. An obvious generalization using the loop group of ${\rm GL}_3$ acting on three-component fermionic Fock space leads to a new system of 4 difference equations.


Introduction
Many integrable differential equations can be transformed to simpler, bilinear form by introducing new dependent variables called τ -functions. In practice, these τ -functions are given as matrix elements of infinite-dimensional groups or Lie algebras, etc.
For instance, the famous Korteweg-de Vries (KdV) equation is transformed by the substitution u = 2 ln(τ ) xx into Hirota bilinear form where D u is the Hirota operator so that D u σ · τ = σ u τ − στ u . See [16] for details and many more examples. In the case of the KdV equation, the τ -function is a matrix element for the action of the loop group of GL 2 on one-component fermionic Fock space, see for instance [10,20,26].
To produce the integrable equations from τ -functions, one introduces an intermediate object, the Baker function. It satisfies linear equations, and the compatibility of these equations gives the integrable hierarchy.
In this paper we are interested in integrable difference (as opposed to differential) equations. Still, we follow very much the setup sketched above for the KdV hierarchy.
In the first part of this paper, we introduce a collection of τ -functions as matrix elements of the action of loop group elements for GL 2 , depending on discrete variables 1 c i , which play a similar role as the higher KdV times t 2k+1 , k > 1. These τ -functions are of the form τ where k, α are discrete variables. In fact, these τ -functions turn out to be (see Theorem 2.1) Hankel determinants, well known since the 19th century in the theory of orthogonal polynomials, see, e.g., [17].
We then define Baker functions. In this case, they are 2 × 2 matrices depending on a spectral parameter z, on the discrete variables k, α (and on the c i ): where S ± (z) = (1−S/z) ±1 are the shift fields, constructed from the elementary shift S : C[c k ] → C[c k ], defined as the multiplicative map such that S(1) = 0, S(c k ) = c k+1 . The shift fields S ± (z) play a similar role here that the vertex operator Γ(z) does in the theory of the KdV hierarchy. Next, we introduce linear equations for the 2 × 2 Baker functions: Here, the connection matrices V are 2×2 matrices depending on the spectral parameter. These connection matrices can be explicitly expressed in terms of the τ -functions.
We then show that compatibility of these equations leads to the discrete zero-curvature equations Since we can give explicit expressions for the connection matrices V in terms of the τ -functions, we obtain the following basic system: After applying a change of variables, one can see that this is precisely the A ∞/2 Q-system, see, e.g., [12]. We refer to this system as the 2Q-system, as it is obtained from the representation theory of the central extension of the loop group of GL 2 .
In the second part of the paper we generalize our derivation of the 2Q-system by using the loop group of GL 3 , obtaining τ -functions τ (α,β) k, (c i , d i , e i ), where k, , α, β ∈ Z and the c i , d i , e i are coordinates on the lower triangular subgroup of the loop group of GL 3 . We can explicitly calculate these τ -functions, see Theorem 3.2, but their formula is much more complicated than the simple Hankel determinants in the 2 × 2 case. Next we introduce Baker functions, now 3 × 3 matrices depending on a spectral parameter, and the linear equations for the Baker functions.
Again, we can explicitly calculate the connection matrices in terms of τ -functions, see Lemma 3.7. Compatibility of the equations satisfied by the connection matrices in this case gives us a system of four equations (Theorem 3.8), which we will refer to as the 3Q-system. For all α, β ∈ Z and k, ≥ 0 (1. 3) The first two of these new equations are generalizations of T -system equations. More precisely, for fixed β, after a change of variables, the first equation is a T -system equation. Similarly, in the second equation, after applying a change of variables, we obtain a T -system equation for fixed α. It is known that Q-and T -systems are related to many areas in mathematics and physics. See for instance [11,12] for relations to cluster algebras, and [24] for applications in integrable systems.
In particular it is known, see [15], that some particular solutions of T -systems are q-characters of Kirillov-Reshetikhin modules [22,23]. It is therefore natural to ask if a similar representation theoretic meaning of particular solutions of the new 3Q-system, (1.3), exists.
In another direction, the τ -functions for the Q-system are, as we mentioned above, determinants of Hankel matrices, which appear in the theory of orthogonal polynomials [17], and in the Toda lattice [21]. Again one can wonder what the meaning of the 3Q-system is from the point of view of orthogonal polynomials and Toda lattices. We will see in Section 3 that the τ -functions of the 3Q-system depend on the choice of a lower triangular matrix Here C(z), D(z), E(z) are series in z. When we look at the special case where E(z) = 0, we obtain τ -functions that are determinants of block Hankel matrices, related to bi-orthogonal polynomials, and 4-band Toda lattices (see, e.g., [5]). See [1] for a preliminary report on this.
We hope that it is clear from this paper that the theory of Q-systems and T -systems, with their many applications, is just the tip of an iceberg. For any n > 2, there are nQ-systems and nT -systems, which are generalizations of the 2Q-and 2T -systems. In this paper, we discuss the construction of the nQ-systems for n = 2, 3. See [2] for more general hierarchies.
2 The 2 × 2 case 2.1 2 × 2 τ -functions and Q-system We have an action of the central extension, GL 2 , of the loop group GL 2 = GL 2 (C((z))) on twocomponent fermionic Fock space, F (2) , the semi-infinite wedge spaced based in C 2 ⊗ C z, z −1 , see, e.g., [31] for n-component fermions, and [19] for the construction of central extensions of Lie algebras and corresponding groups. Some of this material is reviewed in Appendix A. Let π : GL 2 → GL 2 be the projection onto the non-centrally extended loop group. We will consider the action of a group element, g C ∈ GL 2 , where on the vacuum vector v 0 of F (2) : Here where c i ∈ C. In order for π(g C ) to belong to GL 2 (C((z))) we would need to impose the condition that c i = 0 for i 0. However, sometimes it is useful to think of the c i s as formal variables. In that case, π(g C ) belongs to the invertible elements in gl 2 (C((z)))[[c i ]], and will be a 2 × 2 matrix with coefficients given by series that are infinite in both directions in z.
In order to define our τ -functions, we need to define fermionic translation operators which we denote Q i , i = 0, 1. In (A.9), we will carefully define these operators and their action on F (2) , in terms of wedging and contracting operators, for α, β ∈ F (2) . Here, e k a = e a z k , where e a denotes the standard basis vectors of C 2 , e 0 = 1 0 , The projections of the Q i s onto the loop group GL 2 are given by We also need the translation element We define shifts on the series C(z) by We similarly define the shifted group element g (α) We then have 4) and the same relation with π applied, The fundamental objects in the theory of the Toda lattice (see, e.g., [21]) and Q-systems are the τ -functions defined by Here , is the bilinear form on semi-infinite wedges. (For more details, see Appendix A.2, where we will define a basis for F (2) . , is the bilinear product with respect to which these basis vectors are orthonormal.) The τ -functions in the 2 × 2 case are determinants of Hankel matrices.
Here Res w = Res w 1 Res w 2 · · · Res w k , and the residue Res z (a(z)) is the coefficient For the proof of this, see Appendix B.1. The simple form of the τ -functions allows us to apply the Desnanot-Jacobi identity (cf. [9]) to obtain [12] the following difference equations, referred to as the 2Q-system, satisfied by our τ -functions: For all k ≥ 0 and for all α ∈ Z, The disadvantage of obtaining the difference equations in this way is that it is not at all apparent how to generalize this to the 3 × 3 situation, in which the formulas for the τ -functions are much more complicated. We thus present another way of obtaining our 2 × 2 difference equations.

Birkhoff factorization
Define an element of the central extension of the loop group of GL 2 : and assume that it has a Birkhoff factorization [27,28] (see also Appendix C.1): k an invertible z independent matrix. This assumption is justified precisely when the matrix element τ (α) [29]. In their paper, Segal-Wilson treat essentially the case of n = 1 of the theory of n-component fermionic Fock space used in our current paper, although they emphasize the connection to the geometry of infinite Grassmannians, whereas we put the theory of fermion operators in the forefront. Segal-Wilson explain that the vanishing of the τ -function detects that the corresponding element W = gH + of the infinite Grassmannian is not in the big cell. Being in the big cell for W = gH + is equivalent to g having a Birkhoff factorization. We leave it to the reader to check that this picture still holds for arbitrary n.
Now we want to display the negative component of π g [k](α) . To calculate this we make some extra structure explicit.
Let N be the subgroup of elements of GL 2 of the form (2.1). We can think of the coefficients c k as coordinates on N , so First define shifts acting on B. These are multiplicative maps given on generators by We will often write S ± for S ±1 . We also define shift fields. These are multiplicative maps given by We sketch the proof in Appendix C.2. Now that we have expressed the negative component of the Birkhoff factorization in terms of matrix elements of the centrally extended loop group, we no longer need the central extension and we will simplify notation: for the remainder of Section 2, we will write g T for π(T ) and Q a for π(Q a ), a = 0, 1. In particular, in the rest of this section we write is not zero. Here we are dropping the π as discussed above. It is easy to see that for k < 0 such a factorization is not possible. Indeed, assume for simplicity that α = 0, and consider Here the subscripts −, 0+ on a two sided infinite series in z denote the terms containing negative, respectively non-negative powers of z.
The existence a Birkhoff factorization for T k g C for k < 0 reduces then to the existence of a Birkhoff factorization of the left factor since the right hand side factor of T k g C already belongs to the non-negative loop group.
If we could write this factor as Γ − Γ 0+ , then we would have Γ(Γ 0+ ) −1 = 1 0 0 1 + O z −1 . In particular, looking at the second column of this matrix equality, we see that this would mean (since the entries of Γ 0+ and its inverse would contain only non-negative powers of z) that there are power series f (z), g(z) ∈ C [[z]] so that It is clear that for k < 0 such series f (z), g(z) do not exist (we would need f (z) to be zero, but there is no g(z) in C[[z]] such that g(z)z −k = 1 + O z −1 ). The argument for α = 0 is similar.

Matrix Baker functions and connection matrices
Next, we define the Baker functions. These are elements of the loop group defined by Since the Baker functions are all invertible, they are related by connection matrices belonging to the loop group. Define Γ [k](α) . We are interested in connection matrices that implement nearest neighbor steps on the lattice of Baker functions. So define elementary connection matrices , so that we have k . (2.9) Pictorially: Walking around the triangles in this diagram, we see that we get factorizations of all elementary connection matrices. In particular, k+1 . (2.10) Such factorizations are well known in the theory of integrable systems, see for instance Adler [3], Sklyanin [30]. They go back to the work of Darboux in the 19th century, see for example [25]. We study the elementary connection matrices more explicitly.
Proof . The first expression for the elementary connection matrices (in terms of negative components g To derive the second expression for the elementary connection matrices in terms of positive components, g we use (see also (2.4), or rather (2.5)) Rearranging factors then proves the second form for the connection matrices.
Remark 2.5. Note that the second equality in the above lemma tells us that the elementary connection matrices contain only z k for k ≥ 0. This allows us to easily calculate these connection matrices in terms of τ -functions, as we can ignore any (often complicated) terms that would contribute only negative powers of z.
First note that Theorem 2.2 allows us to expand g [k](α) − and its inverse up to order z −1 as and we expand the shift fields in partial shifts Proof . As an example, we calculate V , using (2.11): dropping all terms containing z −1 or lower, see Remark 2.5 for why this is justified. Here x, y are some expressions in the τ -functions which we will determine by noting that det V We see that k−1 . This proves the lemma for V giving equations for the h Theorem 2.7. The equations (2.12) are equivalent to the 2Q-system Bringing all terms under the same denominator and then rearranging terms, we see that this is equivalent to We thus need only prove that the equality holds for k = 0. But this is just which is true since τ So we have rederived the 2Q-system, see the equations (2.1), using the Birkhoff factorization.

τ -functions
We now discuss the generalization to the 3 × 3 case, proceeding very similarly to the 2 × 2 case. We have an action of the central extension GL 3 of the loop group GL 3 = GL 3 C z −1 on three-component fermionic Fock space F (3) . See, e.g., [31] for n-component fermions, and [19] for the construction of central extensions of Lie algebras and corresponding groups. Some of this material is reviewed in Appendix A. Let π : GL 3 → GL 3 be the projection onto the non-centrally extended loop group and consider the action of the group element, g C,D,E ∈ GL 3 , where on the vacuum vector of F (3) . Here where the x i are formal variables and the vacuum vector, v 0 is, analogous to the 2 × 2 case, As in the 2 × 2 case, we have fermionic translation operators Q i , 0 ≤ i ≤ 2. The action of these Q i s on F (3) is defined carefully in the appendix (see (A.9)). Their projections onto the loop group GL 3 are given by the following (commuting) matrices We also have the translation elements We define shifts on the series X(z) by It is convenient to allow our series to be infinite in both directions. This causes no issues of convergence, if we think of the coefficients of these series to be formal variables. For example, if Analogous to the shifted group elements of the 2 × 2 case, see (2.1), are the shifted group elements We then have (using , and the same relations with π applied, Similarly to the 2 × 2 case, the fundamental objects in the 3 × 3 theory are the τ -functions defined by Here v 0 is the vacuum vector in the three-component fermionic Fock space F (3) , and , is the bilinear form, see Appendix A.2. (As in the 2 × 2 case, in the appendix we define a basis for F (3) , and , is the bilinear form with respect to which these basis vectors are orthonormal.) Note that if we introduce another translation group element T 3 = Q 2 Q −1 0 then we can write nonuniquely where k = n c + n d , = n d + n e , and we take n c , n d , n e ≥ 0. .
Here and from now on, we use the convention that we expand 1 x−z in positive powers of the second variable, so 1 We discuss the proof of the above theorem in Appendix B.2.

Birkhoff factorization for the 3 × 3 case
Define centrally extended loop group elements and assume that they have a Birkhoff factorization [28] (see Appendix C.1): an invertible z independent matrix. As in the 2 × 2 case, this assumption is justified precisely when τ ](α,β) v 0 is not zero, see the discussion at the beginning of Section 2.2.
Now we want to display the negative component of π g [k, ](α,β) . As we did in the 2 × 2 case, to calculate this we make some extra structure explicit, see Section 2.2 for the simpler situation.
Let N be the subgroup of elements of GL 3 of the form (3.2). We can think of the coefficients x k , x = c, d, e as coordinates on N , so is the coordinate ring of N .
We first define shifts acting on B: these are multiplicative maps given on generators by (x, y ∈ {c, d, e}) We will often write S ± x for S ±1 x . We also define shift fields. These are multiplicative maps given by We sketch the proof in Appendix C.3. As in the 2 × 2 case, we have now expressed the negative component of the Birkhoff factorization in terms of matrix elements of the centrally extended loop group, so we no longer need the central extension and we will simplify notation by writing g , T i for π(T i ), i = 1, 2 and similarly Q a for π(Q a ), a = 0, 1, 2. In particular, in the rest of this section we write

Matrix Baker functions and connection matrices, 3 × 3 case
Next, we define the Baker functions. These are now elements of the loop group of GL 3 , defined by Since the Baker functions are all invertible, they are related by (right) multiplication by connection matrices belonging to GL 3 . In particular, define The simplest connection matrices are those where (x 0 , x 1 , x 2 ) has two zero components and the other absolute value 1. We therefore define the elementary connection matrices Also define translation matrices Pictorially, fixing k and α, we have Walking around the triangles in this and the similar diagram where , β are fixed we see that we get factorizations of the elementary translation matrices U We will argue that all identities for the connection matrices are the result of those in (3.7).
is a product of the four types of elementary connection matrices (3.5).
Proof . We need to show that we can move from Ψ [k, ](α,β) to Ψ [k , ](α ,β ) just using the elementary connection matrices. First of all, the translation matrices (3.6) are products of elementary connection matrices, see (3.7). We can move from Ψ [k, ](α,β) to Ψ [k , ](α,β) using just U k i+ , i and/or U k i , i+ , keeping (α, β) fixed. Then we use the V (α + ,β) and/or V (α,β + ) to adjust the (α, β) as a product of elementary connection matrices is of course not unique. Each path from Ψ [k, ](α,β) to Ψ [k , ](α ,β ) in the lattice of Baker functions with diagonal or anti-diagonal steps gives a product expression for Γ: the diagonal steps give V factors and the anti-diagonal steps give W factors. Now it should be clear that any two paths from Ψ [k, ](α,β) to Ψ [k , ](α ,β ) can be deformed into each other by moves The moves (3.9) correspond to identities (3.7) and moves (3.8) correspond to similar equations of the form V W = W V (without inverses on W ). These last equations will be equivalent to those in (3.7). So the upshot is that all equations obtained by writing an arbitrary connection matrix Γ as a product of elementary connection matrices follow from (3.7). We will therefore concentrate on (3.7). In particular, we will see that these equations will imply the equations for our τ -functions (1.3).
We first check that our elementary connection matrices and translation matrices do not contain any negative powers of z, which is not obvious from the definitions (3.5) and (3.6).
The following lemma tells us that the elementary connection matrices and translation matrices contain only z k for k ≥ 0.
Similarly, for the translation matrices Proof . From (3.3) it follows that from which the result for the V matrices follows by rearranging factors.
Similarly, using Q −1 1 T 1 = Q −1 0 , Q −1 2 T 2 = Q −1 1 and again (3.3) we find giving the result for the W matrices. Finally the positive expression for U matrices follows from , and rearranging factors.
As in the simpler, 2 × 2 case, this lemma allows us to calculate the connection matrices easily in terms of the τ -functions.
First note that, similarly to the 2 × 2 case, Theorem 3.4 allows us to expand g [k, ](α,β) − and its inverse up to order z −1 as (we suppress the shift (α, β)) where O(z i ) are terms with power of z equal to i or lower, and we define quotients of τ -functions as This formula then gives the following formula for g

Explicit formulae for connection matrices
Lemma 3.7.
Proof . can now easily be calculated.
For the translation matrices, we obtain the following expressions Equivalently, Equivalently,

Difference equations from factorizations
Theorem 3.8. The τ -functions defined by (3.4) satisfy the following system of four equations, referred to as the 3Q-system. For all α, β ∈ Z and k, ≥ 0 Proof . We substitute into the two factorizations (3.7), the results of Lemma 3.7. We find rather complicated rational expressions in the τ -functions. From the two different expressions for U (α,β) and Expressing equations (3.13), (3.14), (3.15) in terms of τ -functions, we get nontrivial equations in 3 components: Following the same procedure, but instead using the two different factorizations of U (α,β) k, + , we get the following relations satisfied by the τ -functions: The two equations In the first term we substitute , which follows from [2] by the change of variables, k → k + 1, → + 1, α → α − 2. The first term then becomes In the second term of (3.16), we use [2] in the form so that the second term becomes −τ After the cancellation of two terms and collecting like terms, we have (3.16) Now observe that the square factor on the right in (3.17) is obtained from the square factor on the left by a shift k → k + 1, and similarly, mutatis mutandis, for the terms in the big parentheses. Therefore, if we know that then (3.17) also tells us that So it suffices to check In the first term, substitute k−1, +1 , which is obtained from [1] by the shift β → β − 1. The first term becomes In the second term we also use [1], in the form After cancellation of two terms and collecting like terms, (3.19) becomes (3.20) As before, (3.20) implies that if

Introduction
In the main text we work with n × n matrices (depending on a spectral parameter z) for n = 2 or 3. In this appendix we will not specify n, as the theory of n-component fermions and the associated semi-infinite wedge space does not significantly depend on n. A convenient reference for background and more details is ten Kroode and van der Leur [31].

A.2 Semi-infinite Wedge space
denote the standard basis of C n . Denote the corresponding elementary matrices by E ab (such that E ab e c = δ bc e a ); they are also indexed by integers 0, 1, . . . , n − 1. We also need the loop space of C n , denoted by with basis e k a = e a z k , for a = 0, . . . , n − 1 and k ∈ Z. Let F (n) be the n-component fermionic Fock space, the semi-infinite wedge space based on H (n) . It is spanned by semi-infinite wedges where the w i satisfy some restrictions that we will presently discuss. Semi-infinite wedges obey the usual rules of exterior algebra, like multilinearity in each factor and antisymmetry under exchange of two factors.
To formulate the restrictions on the w i that can appear in the wedge ω above we introduce the Clifford algebra Cl (n) acting on F (n) : it is generated by exterior and interior products, denoted by e e k a and i e k a , defined as wedging and contracting operators, respectively: It is useful to collect the generators of the Clifford algebra in generating series. Therefore, define fermion fields The fermionic fields satisfy anti-commutation relations 2 where the formal delta distribution is defined by Let v 0 be the vacuum vector Then we define F (n) to be the span of the wedges obtained by acting on the vacuum v 0 by monomials in the wedging and contracting operators. To get a basis for F (n) we specify an ordering on the wedging/contracting operators acting on F (n) .
is a monomial in a ψ ± (k) for k ≤ −1, ordered in increasing order from left to right.
The statement that the elementary wedges form a basis for F (n) follows from the Poincaré-Birkhoff-Witt theorem for the Lie superalgebra underlying the Clifford algebra.
We define a bilinear form, denoted , , on F (n) by declaring the elementary wedges to be orthonormal. We then have The n-component fermionic Fock space F (n) has a grading by the Abelian group Z n , i.e., we have a decomposition F (n) = ⊕ α∈Z n F The grading on F (n) induces a grading on linear maps on F (n) : if L : F (n) → F (n) has the property that there exists a δ ∈ Z n so that, for all ω ∈ Z n , L restricts to a map F (n) ω+δ , then we say that L has degree δ. Then the grading is uniquely determined by declaring wedging operators e a z k ∧ to have degree δ a , and the contracting operators i e a z k to have degree −δ a . The fields ψ ± a (z) have degree ±δ a . The total degree of an element ω of degree α is just the sum of the entries in the degree row vector α.

A.3 Fermionic translation operators and translation group
Besides the action of fermion operators, a ψ ± (k) , on F (n) , we also have the action of fermionic translation operators Q a : F (n) → F (n) , a = 0, 1, . . . , n − 1, given by The Q a are invertible. Q ±1 a has degree ±δ a . The Q a are unitary for the standard bilinear form of F (n) : The fermionic translation operators belong to the central extension of the loop group GL n (acting on F (n) ). They are lifts of commuting elements of the non-centrally extended loop group, see for instance [7,Proposition 5.3.4]. We have The group generated by Q a , a = 0, 1, . . . , n − 1, contains a subgroup of elements of total degree zero, generated by the translation operators T s = Q s Q −1 s−1 , s = 1, 2, . . . , n − 1, of degree δ s − δ s−1 . Another set of generators for this subgroup is also useful: define

For a = b and for all
3. For all k, ∈ Z we have Proof . Part (1) is clear. Part (2) is a simple induction. For part (3) we get Similarly for part (4) we have Finally, for part (5) we substitute (3)), in the right hand side of part (4). The result then follows from Define the ordered product of k ≥ 0 fermions by The empty product is as usual the identity.
Just as the fermionic translation operators are unitary, so are the translation operators T s and T ab : from (A.8) it follows that A.4 The Lie algebra gl n and fermions The loop algebra gl n is defined as the Lie subalgebra of gl(H (n) ) generated by E ab z k , a, b = 0, 1, . . . , n − 1, k ∈ Z, where E ab z k · e c z m = δ bc e a z k+m .
The loop algebra gl n does not quite act on F (n) . One would like to define the action on F (n) by However, considering the action of E aa z 0 on the vacuum v 0 we would get (since e a z l ∧ i(e a z l ) v 0 = v 0 for l ≥ 0), so that these diagonal elements would have a divergent action. Therefore we introduce a normal ordering on fermion fields [18] by where the creation and annihilation parts of a fermion field are given by Of course, when a = b We define the normal ordering on the components of the fermion fields by Then we have Note that Introduce a generating series of loop algebra elements by The action of this generating series on F (n) can be represented by a normal ordered product of fermion fields Indeed, The series E ab (z 1 ), acting on F (n) , has degree δ a − δ b . Equation (A.13) is the reason we chose to encode the wedging and contracting operators as coefficients of fermion fields according to (A.1).
We also need the commutator of the generating series of Lie algebra elements with fermionic translation operators.

A.5 Root lattice
Recall, see Appendix A.1, the group Z n that gives a grading for fermionic Fock space F (n) . The root lattice A n−1 is a subgroup of Z n . It is generated by We will call elements in A n−1 of the form α = The translation group is also graded by A n−1 : the generator T s = Q s Q −1 s−1 has degree α s . Similarly the Lie algebra generating fields E ab (z) have deg(E aa−1 (z)) = α a .

B Expressions for the τ -functions
In this appendix we prove Theorems 2.1 and 3.2. This gives expressions for the τ -functions in terms of coordinates on the lower triangular subgroup N of GL n , for n = 2, 3.
B.1 The case of n = 2, Theorem 2.1 Recall that N ⊂ GL 2 was defined as the subgroup of elements of the form (2.1). The inverse image of N under the projection π : GL 2 → GL 2 can be shown to be isomorphic to the group N × C × of pairs (n, z), with multiplication (n 1 , z 1 ) · (n 2 , z 2 ) = (n 1 n 2 , z 1 z 2 ). In other words, the central extension defining GL 2 is trivial when restricted to π −1 (N ). Denote by N the subgroup of π −1 (N ) corresponding to pairs (n, 1) in N × C × . Then N and N are isomorphic.
The τ -functions for GL 2 are given as matrix elements on F (2) : where the element g (α) C of the lower triangular subgroup N of GL 2 has projection given by (2.3). We write the group element g and C (α) (z) is given by (2.2), and the generating series of loop algebra elements E 10 (z 1 ) by (A.12).
The reader might object that Γ (α) C is an infinite sum of fermion operators each acting on F (2) , and it is not so clear what this sum means. By imposing conditions on the coordinates c k ∈ C we can ensure that Γ (α) C is indeed a map F (2) → F (2) . For our purposes it is easier to think of the c i as formal variables, and interpret Γ T k v 0 has degree k(δ 1 − δ 0 ) in F (2) , and Γ (α) C has degree δ 1 − δ 0 , since E 10 (z 1 ) does. In F (2) homogeneous elements of different degree are orthogonal for , . Hence only the = k term contributes to (B.1) and Recall that C (α) (z) = (−1) α n∈Z c n+α z −n−1 . Associated to the series C (α) (z) is a C-linear map We need multiple copies of the map c (α) acting on series in variables z 1 , z 2 , . . . . We define c (α) i (f (z i )) = c (α) (f (z)), for i = 1, 2, . . . , and impose linearity in the variables z j , j = i, i.e., the condition that (for instance) c and so Here we use the fact that, in the expression (A.13) for E 10 (z 1 ) in fermions, the normal ordering is just the ordinary product of fermion fields. We also use Lemma A.2, Part 2 and the anticommutation relations (A.2). Next we use the factorization Lemmas D.1 and E.1 to calculate the factors involving ψ 0 (z) and ψ 1 (z) separately; we find Here V

(k)
{z i } is the Vandermonde matrix (E.1). This proves the first part of Theorem 2.1, since det V (k) For the second part we need a formula for the square of a Vandermonde determinant. Let the permutation group S k act on C[z 1 , z 2 , . . . , z k ] by permuting the subscripts.
Proof . The right hand side of (B.2) can be written as From this, we see that the σ = e term on the r.h.s. of (B.2) is Now for every permutation σ in S k we have Summing over all permutations, we obtain B.2 Proof of n = 3, Theorem 3.2 Proof . In this proof, for typographical simplicity, we will suppress the shift superscripts (α,β) . We write g = exp(Γ c ) exp(Γ d ) exp(Γ e ), where Γ c = Res z (C(z 1 )E 10 (z 1 )), Γ d = Res z 1 (D(z 1 )E 20 (z 1 )), Γ e = Res z 1 (E(z 1 )E 21 (z 1 )).
(Recall (3.1).) This implies that τ k, is the sum of c nc,n d ,ne = Res x,y,z where n d + n e = , n c + n d = k.
We can factorize this using Appendix D as Since, using (B.2), , the theorem follows from the calculation of correlation functions in Appendix E.

C Birkhoff factorization and matrix elements of semi-infinite wedge space
In this appendix we sketch proofs of Theorems 2.2 and 3.4. First we will discuss a more general statement about the Birkhoff factorization in GL n .

C.1 Birkhoff factorization and n-component fermions
Recall that most elements γ ∈ GL n have a Gauss factorization: with γ − = 1 n×n + strictly lower triangular, γ 0+ upper triangular (and invertible). Only the γ for which the principal minors vanish don't have a Gauss factorization.
A similar story works for the loop group GL n of GL n . Let GL n− be the subgroup of GL n of elements g − = 1 + O z −1 , and let GL n0+ be the subgroup of elements g 0+ = A + O(z), where A is invertible (and independent of z). Then most elements in GL n have a Birkhoff factorization The existence of such a factorization is controlled by the non-vanishing of a fermion matrix element in the semi-infinite wedge space F (n) . We will express g − in terms of such fermion matrix elements.. Recall that, just as in the case of the loop algebra gl n , the loop group GL n does not actually act on F (n) . We instead have a central extension (cf. [28]) and an action of GL n on F (n) . The inverse images π −1 GL n− and π −1 GL n0+ can be shown to be isomorphic to product groups GL n− × C × and GL n0+ × C × , respectively, i.e., the central extension defining GL n is trivial over the two inverse images. Denote by GL n− the subgroup of π −1 GL n− corresponding to pairs (g, 1), and let GL n0+ denote the full preimage of GL n0+ . Then the intersection of GL n− and GL n0+ will be the element 1 ∈ GL n ; the image of C × belongs to GL n0+ . Most elementsĝ ∈ GL n will have a (unique) Birkhoff factorization, If v 0 is the vacuum (A.3) of F (n) , the τ -function is defined as the matrix element The elementĝ (and also g = π(ĝ)) has a Birkhoff factorization as long as τ (ĝ) is not zero.
To calculate the negative component of g in the factorization (C.1), we choose a liftĝ of g, i.e., π(ĝ) = g, and study the action ofĝ on F (n) .
We havê This is explained in the case n = 2 in [6] (cf. [27,32]). Hence (assumingĝ has a Birkhoff factorization, or τ (ĝ) = 0) Now write g − in terms of matrix elements On For the E ab z −k−1 appearing in g − (i.e., those with k ≥ 0) the normal ordering can be omitted, see (A.11). Now to find g (k) ab we calculate where the omitted terms are quadratic and higher in the E ab . We see that g (k) ab appears as the coefficient of many elementary wedges a ψ + ab we just pick one of these elementary wedges, say the l = 0 term, and use orthogonality of elementary wedges to find (k ≥ 0) using (A.4) and (A.5). Now observe that This allows us to calculate g (k) ab for k < 0 in the same way as for k ≥ 0, see (C.3). These remarks prove the following theorem.
Theorem C.1. Let g ∈ GL n admit a Birkhoff factorization g = g − g 0+ . Then andĝ ∈ GL n is any lift of g, so that π(ĝ) = g. The τ -function is given by (C.2).

C.2 The 2 × 2-case; proof of Theorem 2.2
We now specialize n and g in Theorem C.1: in this subsection we set n = 2 and where π(g C gives a well defined operator on F (2) , and the corresponding τ -function will be a complex number. If the τ -function is not zero, g will have a Birkhoff factorization. However, we prefer to think of g [k](α) as a map F (2) → F (2) [[c i ]], so that the τ -function is also a formal series in C[[c i ]], which is not zero, and so the "formal group element" g in this case will always have a Birkhoff factorization.
In this case our calculation of the Birkhoff factorization in Theorem C.1 gives us where the τ -function is defined in (2.6). We will proceed to rewrite g [k](α) ab (z). First of all, we will expand g (α) C in fermion operators. Note that g (α) and C (α) (z) is given by (2.2), and the generating series E 10 (z 1 ) by (A.12). (See the beginning of Appendix B.1 for an interpretation of g (α) C and Γ (α) C as operators on F (2) .) This means that g / !, both acting on H (2) and on F (2) . Hence Next, we use the standard grading on F (2) . Note that Γ Hence, by orthogonality of terms of different degree in F (2) , we find that the only non-zero contribution to the sum (C.5) arises when = k + a − b and From now on, we will often use the abbreviation = k + a − b in formulas for g [k](α) ab (z). Next, we need to commute T −k through ψ − a (z).
Proof . By (A.6), (A.7) we have By unitarity of the translation operators, (A.10), this implies that Next we want to apply the factorization Lemma D.1. We need to write Using Lemma E.3 to calculate the factors involving ψ 0 (z) and ψ 1 (z) separately we find Comparing this with the expression for the τ -function in Theorem 2.1 and the definition of the shift fields (2.7) gives Theorem 2.2.
C.3 Birkhoff factorization in the 3 × 3 case, proof of Theorem 3.4 The proof of Theorem 3.4 is similar to that of Theorem 2.2 sketched in the previous subsection. We leave the details to the reader.

D Factorization and reduction to one-component fermions
Often we want to calculate a matrix element in F (n) of fermion fields of the form where P is some polynomial in the fermion fields ψ ± a (z a ), a = 0, 1, . . . , n − 1. By linearity, we can reduce to the case where P = M is a monomial, and then we can rearrange the factors in the monomial as in Definition A.1: where M ± a is a monomial in a single type of fermions, ordered according to the subscript of the arguments of the fields: This defines the ordered product of fermion fields. We calculate such matrix elements using the following factorization lemma. Let us first introduce some notation. Let F = F (1) be 1-component fermionic Fock space, with vacuum v (1) 0 , and with bilinear form , F . The fermionic translation operators Q, Q −1 , defined as in Section A.3, act on F . Let N a = N a ψ ± a (z a i ) , a = 0, . . . , n − 1, be monomials in fermion fields ψ + a (z a i ), ψ − a (z a j ) of just type a (acting on F (n) ), and let N a be the corresponding monomial in 1-component fermion fields ψ ± (z a i ) (acting on F ). So for example if .

Proof .
A basis for F is given by elementary wedges (see Definition A.1) ω k , labeled by pairs of sequences k = (k + , k − ), where each sequence k ± is of length l ± (in general l + = l − ) and is of the form k ± = k ± 1 < k ± 2 < · · · < k ± l ± ≤ −1 . Roughly speaking ω k is obtained from the vacuum v then adding factors z k + j (using the wedging operators z k + j ∧ ). Since the order in which we apply these operations matters, we must be more precise. So define Here, we still have k ± 1 < k ± 2 < · · · < k ± s ≤ −1. Similarly, elementary wedges in F (n) are labelled by n-tuples (k n−1 , k n−2 , . . . , k 0 ), and are defined by For the duration of the proof, we will write v (n) 0 for the vacuum in F (n) . Now define a multilinear map from the n-fold product of F with itself to n-component fermionic Fock space F (n) : (ω k n−1 , ω k n−2 , . . . , ω k 0 ) → ω (k n−1 ,k n−2 ,...,k 0 ) .
By the universal property of the tensor product, this induces a unique map This map is an isomorphism of vector spaces, and is in fact an isometry, if we define a bilinear form on F ⊗ F ⊗ · · · ⊗ F by declaring the basis {ω k n−1 ⊗ ω k n−2 ⊗ · · · ⊗ ω k 0 } to be orthonormal. For this bilinear form on F ⊗n we have (given ω a ,ω a ∈ F ) ω n−1 ⊗ ω n−2 ⊗ · · · ⊗ ω 0 ,ω n−1 ⊗ω n−2 ⊗ · · · ⊗ω 0 F ⊗n = ω a ,ω a F . (D.1) Now one checks that The lemma then follows from (D.1) and the fact thatφ is an isometry.
So the correlation functions we want to calculate reduce to products of correlation functions on one-component semi-infinite wedge space F . In Appendix E we review some formulas for one component fermions.

E One-component fermion correlation functions
In this appendix we collect some results on one-component fermions. In other words, we are dealing with the fermionic Fock space F = F (1) , based on H = H (1) . The results in this Appendix should be known, for instance Lemma E.2 can be found (without proof) in [4], but we could not find references with complete proofs of the facts we need.
The whole discussion of Appendix A transfers to the present one-component context. For typographical convenience we will write ψ ± (z) for ψ ± 0 (z) and similarly we write Q ±1 for Q ±1 0 .
The remaining contributions for this choice of σ come from pulling out coefficients of products of wedging and contracting operators from ψ + (w 1 ) · · · ψ + (w m )ψ − (y σ(1) ) · · · ψ − (y σ(m) ) whose actions cancel with each other. We again pull out these terms such a way that no additional sign changes occur: We count only contributions coming from terms in ψ + (w m )ψ − (y σ(1) ) that cancel with each other, terms in ψ + (w m−1 )ψ − (y σ (2) ) that cancel with each other, . . . and terms in ψ + (w 1 )ψ − (y σ(m) ) that cancel with each other. We claim that we can count each pair, ψ + (w m−i )ψ − (y σ(i+1) ), 0 ≤ i ≤ m − 1, as contributing . We know we are not omitting any nontrivial terms in doing this, since any y σ(i+1) w +1 m−i with < 0 corresponds to ψ + ( ) ψ − (− −1) and ψ − (− −1) kills the vacuum or any vector obtained by acting by contracting operators on the vacuum. We must therefore only prove that we are not including any extra nontrivial terms. Towards this end, consider some monomial, (−1) |σ| y 0 σ(n) y σ(n−1) · · · y n−m−1 σ(m+1) corresponding to a product of wedging operators acting on the vacuum vector which give 0. Since all of the wedging operators, ψ − ( ) , are such that < 0, the only way this is possible is if two of the wedging operators are the same. But this means that two of the y i s in the above expression are being raised to the same power. Define a new element, γ ∈ S n by composing σ with the transposition that interchanges these two y i s. The sign of this new element is −(−1) |σ| . So there is a monomial in the expansion of (−1) |γ| y 0 γ(n) y γ(n−1) · · · y n−m−1 , which cancels with the above monomial.
Using Leibniz's formula to expand this as a determinant and then computing the determinant, we find that this is exactly equal to The proof in the case that m ≥ n is similar. Here, we need to argue that Q m−n v 0 , ψ + (w 1 ) · · · ψ + (w m )ψ − (y 1 ) · · · ψ − (y n )v 0 .
In Lemma E.2 we see that this particular matrix element of fermion fields is the expansion of a rational function in the variables appearing in the fermion fields. This is not an accident, but is a basic property of vertex algebras, see [13,14,18], referred as rationality of vertex operators. Indeed, the one-component fermionic Fock space F (1) is an example of a vertex algebra, and the fermionic fields are vertex operators for this vertex algebra structure. Another basic property of vertex algebras is called commutativity; roughly speaking it says that if we permute the vertex operators in a matrix element of a product of vertex operators the answer is again an expansion of the same rational function, but in a different region, up to a sign. See also [8].
For instance we will also need the following matrix elements. .
One could derive this lemma from Lemma E.2 by commutativity of vertex operators, using the general theory of vertex algebras. For the convenience of the reader we give an elementary proof of this lemma, just using the commutation relations of fermion fields.
Consider the following rational function The partial fraction expansion 3 of R is Instead of R we can also consider the rational function Recall the formal series δ(z, w) = 1 z − w + 1 w − z .
Here (and from now on) we use the convention that we expand in the second variable, so that for example 1 z−w = ∞ k=0 w k z k+1 . In particular we will think of R as a series in positive powers of z and R as a series in positive powers of the w i s. Then we have the following identity: or, writing out the definitions, multiplying by (−1) m+1 and rearranging terms: .

(E.2)
Now, after this preparation, we turn to the matrix element, call it A, that we actually want to compute. By the fermion field commutation relation (A.2) and the previous Lemma E. 2  Hence, using (E.2), , which is what we wanted to show.