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-co...
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2007
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| Цитувати: | Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows / H. Aratyn, J. van de Leur // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1477872019-02-17T01:27:15Z Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows Aratyn, H. van de Leur, J. 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 × 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. 2007 Article Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows / H. Aratyn, J. van de Leur // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 15 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 11E88; 17B67; 22E67; 37K10 http://dspace.nbuv.gov.ua/handle/123456789/147787 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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 × 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. |
| format |
Article |
| author |
Aratyn, H. van de Leur, J. |
| spellingShingle |
Aratyn, H. van de Leur, J. Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Aratyn, H. van de Leur, J. |
| author_sort |
Aratyn, H. |
| title |
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows |
| title_short |
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows |
| title_full |
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows |
| title_fullStr |
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows |
| title_full_unstemmed |
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows |
| title_sort |
clifford algebra derivations of tau-functions for two-dimensional integrable models with positive and negative flows |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/147787 |
| citation_txt |
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows / H. Aratyn, J. van de Leur // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 15 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT aratynh cliffordalgebraderivationsoftaufunctionsfortwodimensionalintegrablemodelswithpositiveandnegativeflows AT vandeleurj cliffordalgebraderivationsoftaufunctionsfortwodimensionalintegrablemodelswithpositiveandnegativeflows |
| first_indexed |
2025-07-11T02:49:25Z |
| last_indexed |
2025-07-11T02:49:25Z |
| _version_ |
1837317148775546880 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 020, 29 pages
Clifford Algebra Derivations of Tau-Functions
for Two-Dimensional Integrable Models
with Positive and Negative Flows?
Henrik ARATYN † and Johan VAN DE LEUR ‡
† Department of Physics, University of Illinois at Chicago,
845 W. Taylor St., Chicago, IL 60607-7059, USA
E-mail: aratyn@uic.edu
‡ Mathematical Institute, University of Utrecht,
P.O. Box 80010, 3508 TA Utrecht, The Netherlands
E-mail: vdleur@math.uu.nl
Received October 11, 2006, in final form January 09, 2007; Published online February 06, 2007
Original article is available at http://www.emis.de/journals/SIGMA/2007/020/
Abstract. 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×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.
Key words: Clifford algebra; tau-functions; Kac–Moody algebras; loop groups; Camassa–
Holm equation; Cecotti–Vafa equations; AKNS hierarchy
2000 Mathematics Subject Classification: 11E88; 17B67; 22E67; 37K10
In memory of Vadim Kuznetsov. One of us (JvdL) f irst met Vadim at a semi-
nar 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 pa-
per 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.
1 Introduction
Tau-functions are the building blocks of integrable models. The Grassmannian techniques have
been shown in the past to be very effective in theory of tau-functions. In this paper we exploit
?This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
mailto:aratyn@uic.edu
mailto:vdleur@math.uu.nl
http://www.emis.de/journals/SIGMA/2007/020/
http://www.emis.de/journals/SIGMA/kuznetsov.html
2 H. Aratyn and J. van de Leur
a Grassmannian approach to constructing tau functions in terms of expectation values of cer-
tain 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
A⊗AΩ = ΩA⊗A, Ω =
∑
l,i
ψ
+(i)
l ⊗ ψ
− (i)
−l
defined on a Clifford algebra with Fermi operators ψ± (i)
l , satisfying the relations
ψ
λ (i)
l ψ
µ (j)
k + ψ
µ (j)
k ψ
λ (i)
l = δi,jδλ,µδl,−k,
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 GLn.
As shown in Section 6 the model obtained in Section 5 embeds both AKNS and the 2-com-
ponent 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-infinite wedge space and Clifford 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 vi1 ∧ vi2 ∧ vi3 · · · , with ij ∈ 1/2 + Z,
where i1 > i2 > i3 > · · · and i`+1 = i` − 1 for ` � 0. Define the wedging and contracting
Clifford Algebra Derivations of Tau-Functions 3
operators ψ+
j and ψ−j (j ∈ Z + 1/2) on F by
ψ+
j (vi1 ∧ vi2 ∧ · · · ) =
{
0 if − j = is for some s,
(−1)svi1 ∧ vi2 · · · ∧ vis ∧ v−j ∧ vis+1 ∧ · · · if is > −j > is+1,
ψ−j (vi1 ∧ vi2 ∧ · · · ) =
{
0 if j 6= is for all s,
(−1)s+1vi1 ∧ vi2 ∧ · · · ∧ vis−1 ∧ vis+1 ∧ · · · if j = is.
These operators satisfy the following relations (i, j ∈ Z + 1/2, λ, µ = +,−):
ψλ
i ψ
µ
j + ψµ
j ψ
λ
i = δλ,−µδi,−j, (2.1)
hence they generate a Clifford algebra, which we denote by C`.
Introduce the following elements of F (m ∈ Z):
|m〉 = vm− 1
2
∧ vm− 3
2
∧ vm− 5
2
∧ · · · .
It is clear that F is an irreducible C`-module such that
ψ±j |0〉 = 0 for j > 0.
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 · · · ∧ vi3 ∧ vi2 ∧ vi1 , where i1 < i2 < i3 < · · · and
i`+1 = i` + 1 for ` � 0. 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.,
(· · · ∧ vi2 ∧ vi1)ψ
+
j =
{
0 if j 6= is for all s,
(−1)s+1 · · · ∧ vis+1 ∧ vis−1 ∧ · · · vi2 ∧ vi1 if is = j,
(· · · ∧ vi2 ∧ vi1)ψ
−
j =
{
0 if − j = is for some s,
(−1)s · · · ∧ vis+1 ∧ vj ∧ vis ∧ · · · vi2 ∧ vi1 if is < −j < is+1.
We introduce the element 〈m| by 〈m| = · · · ∧ vm+ 5
2
∧ vm+ 3
2
∧ vm+ 1
2
, 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
|V (i1, . . . , ik, j1, . . . , j`)〉 = ψ+
i1
ψ+
i2
· · ·ψ+
ik
ψ−j1ψ
−
j2
· · ·ψ−j` |0〉,
〈V (i1, . . . , ik, j1, . . . , j`)| = 〈0|ψ+
−j`
ψ+
−j`−1
· · ·ψ+
−j1
ψ−−ik
ψ−−ik−1
· · ·ψ−−i1
, (2.2)
with i1 < i2 < · · · < ik < 0 and j1 < i2 < · · · < j` < 0 form dual basis of F and F ∗, i.e.,
〈V (r1, . . . , rm, s1, . . . , sq)|V (i1, . . . , ik, j1, . . . , j`)〉 = δ(r1,...,rm),(i1,...,ik)δ(s1,...,sq),(j1,...,j`). (2.3)
We relabel the basis vectors vi 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 = vnk− 1
2
(n−2j+1),
and correspondingly:
ψ
±(j)
k = ψ±
nk± 1
2
(n−2j+1)
.
4 H. Aratyn and J. van de Leur
Notice that with this relabeling we have:
ψ
±(j)
k |0〉 = 0 for k > 0.
Define partial charges and partial energy by
chargej ψ
±(i)
k = ±δij , chargej |0〉 = 0,
energyj ψ
±(i)
k = −δijk, energyj |0〉 = 0.
Total charge and energy is defined as the sum of partial charges, respectively the sum of partial
energy.
Introduce the fermionic fields (0 6= z ∈ C):
ψ±(j)(z) =
∑
k∈Z+1/2
ψ
±(j)
k z−k− 1
2 .
Next, we introduce bosonic fields (1 ≤ i, j ≤ n):
α(ij)(z) ≡
∑
k∈Z
α
(ij)
k z−k−1 =: ψ+(i)(z)ψ−(j)(z) :,
where : : stands for the normal ordered product defined in the usual way (λ, µ = + or −):
: ψλ(i)
k ψ
µ(j)
l :=
{
ψ
λ(i)
k ψ
µ(j)
l if l > 0,
−ψµ(j)
l ψ
λ(i)
k if l < 0.
One checks (using e.g. the Wick formula) that the operators α(ij)
k satisfy the commutation
relations of the affine algebra gln(C)∧ with the central charge 1, i.e.:
[α(ij)
p , α(k`)
q ] = δjkα
(i`)
p+q − δi`α
(kj)
p+q + pδi`δjkδp,−q,
and that
α
(ij)
k |m〉 = 0 if k > 0 or k = 0 and i < j.
The operators α(i)
k ≡ α
(ii)
k satisfy the canonical commutation relation of the associative oscillator
algebra, which we denote by a:
[α(i)
k , α
(j)
` ] = kδijδk,−`,
and one has
α
(i)
k |m〉 = 0 for k > 0, 〈m|α(i)
k = 0 for k < 0.
Note that α(j)
0 is the operator that counts the j-th charge. The j-th energy is counted by the
operator
−
∑
k∈ 1
2
+Z
k : ψ+(j)
k ψ
−(j)
−k : .
The complete energy is counted by the sum over all j of such operators. In (8.1) we will define
another operator L0, which will also count the complete energy. In order to express the fermionic
Clifford Algebra Derivations of Tau-Functions 5
fields ψ±(i)(z) in terms of the bosonic fields α(ii)(z), we need some additional operators Qi,
i = 1, 2, . . . , n, on F . These operators are uniquely defined by the following conditions:
Qi|0〉 = ψ
+(i)
− 1
2
|0〉, Qiψ
±(j)
k = (−1)δij+1ψ
±(j)
k∓δij
Qi. (2.4)
They satisfy the following commutation relations:
QiQj = −QjQi if i 6= j, [α(i)
k , Qj ] = δijδk0Qj .
We shall use below the following notation
|k1, k2, . . . , kn〉 = Qk1
1 Q
k2
2 · · ·Qkn
n |0〉, 〈k1, k2, . . . , kn| = 〈0|Q−kn
n · · ·Q−k2
2 Q−k1
1 ,
such that
〈k1, k2, . . . , kn|k1, k2, . . . , kn〉 = 〈0|0〉 = 1.
One easily checks the following relations:
[α(i)
k , ψ
±(j)
m ] = ±δijψ±(j)
k+m
and
Q±1
i |k1, k2, . . . , kn〉 = (−)k1+k2+···+ki−1 |k1, k2, . . . , ki−1, ki ± 1, ki+1, . . . , kn〉,
〈k1, k2, . . . , kn|Q±1
i = (−)k1+k2+···+ki−1〈k1, k2, . . . , ki−1, ki ∓ 1, ki+1, . . . , kn|.
These formula’s lead to the following vertex operator expression for ψ±(i)(z). Given any sequence
s = (s1, s2, . . . ), define
Γ(j)
± (s) = exp
( ∞∑
k=1
skα
(j)
±k
)
,
then
Theorem 1 ([6, 8]).
ψ±(i)(z) = Q±1
i z±α
(i)
0 exp
(
∓
∑
k<0
1
k
α
(i)
k z−k
)
exp
(
∓
∑
k>0
1
k
α
(i)
k z−k
)
= Q±1
i z±α
(i)
0 Γ(i)
− (±[z])Γ(i)
+ (∓[z−1]),
where [z] =
(
z, z2
2 ,
z3
3 , . . .
)
.
Note,
Γ(j)
+ (s) |k1, k2, . . . , kn〉 = |k1, k2, . . . , kn〉, 〈k1, k2, . . . , kn|Γ(j)
− (s) = 〈k1, k2, . . . , kn|.
Also observe that (Γ(j)
± )∗ = Γ(j)
∓ and
Γ(j)
+ (s) Γ(k)
− (s′) = γ(s, s′)δjk Γ(k)
− (s′) Γ(j)
+ (s),
where
γ(s, s′) = e
∑
n sns′n .
We have
Γ(j)
± (s)ψ+(k)(z) = γ(s, [z±1])δjkψ+(k)(z) Γ(j)
± (s),
Γ(j)
± (s)ψ−(k)(z) = γ(s,−[z±1])δjkψ−(k)(z) Γ(j)
± (s).
Note that
γ(t, [z]) = exp
(∑
n≥1
tn z
n
)
.
6 H. Aratyn and J. van de Leur
3 Tau functions as matrix elements and bilinear identities
Let A be an operator on F such that
[A⊗A,Ω] = 0, Ω =
∑
ψ+
k ⊗ ψ−−k. (3.1)
Note that if A ∈ GL∞ then this A satisfies (3.1). So in the n-component case
Ω =
∑
k,i
ψ
+(i)
k ⊗ ψ
−(i)
−k =
∑
i
Res
z
ψ+(i)(z)⊗ ψ−(i)(z).
Here Resz
∑
i fiz
i = f−1. Define the following functions
τm1,m2,...,mn
k1,k2,...,kn
(t, s) = 〈k1, k2, . . . , kn|Ã |m1,m2, . . . ,mn〉,
à =
n∏
i=1
Γ(i)
+ (t(i))A
n∏
j=1
Γ(j)
− (−s(j)). (3.2)
Instead of τm1,m2,...,mn
k1,k2,...,kn
(t, s) we shall write τβ
α (t, s) for α = (α1, α2, . . . , αn) and a similar expres-
sion for β. We will also use |α| = α1 + α2 + · · · + αn, |α|i = α1 + α2 + · · · + αi, |α|0 = 0 and
εj = (0, . . . , 0, 1, 0, . . . , 0), a 1 on the jth place.
If the operator A satisfies (3.1), then so does Ã. Following Okounkov [14] we calculate in two
ways
〈α|
n∏
i=1
Γ(i)
+ (t(i))⊗ 〈γ|
n∏
k=1
Γ(k)
+ (s(k))(A⊗A) Ω
n∏
j=1
Γ(j)
− (−t′(j))|β〉 ⊗
n∏
`=1
Γ(`)
− (−s′(`))|δ〉
=〈α|
n∏
i=1
Γ(i)
+ (t(i))⊗〈γ|
n∏
k=1
Γ(k)
+ (s(k))Ω(A⊗A)
n∏
j=1
Γ(j)
− (−t′(j))|β〉⊗
n∏
`=1
Γ(`)
− (−s′(`))|δ〉. (3.3)
Clearly
ψ+(m)(z)
n∏
j=1
Γ(j)
− (−t′(j))|β〉 = Qmz
α
(m)
0 Γ(m)
− ([z])Γ(m)
+ (−[z−1])
n∏
j=1
Γ(j)
− (−t′(j))|β〉
= (−)|β|m−1zβmγ([z−1], t′(m))Γ(m)
− ([z])
n∏
j=1
Γ(j)
− (−t′(j))|β + εm〉
= (−)|β|m−1zβmγ([z−1], t′(m))Γ(m)
− (−t′(m) + [z])
n∏
j 6=m
Γ(j)
− (−t′(j))|β + εm〉
and in a similar way
ψ−(m)(z)
n∏
`=1
Γ(`)
− (−s′(1))|δ〉
= (−)|δ|m−1z−δmγ([z−1],−s′(m))Γ(m)
− (−s′(m) − [z])
∏
` 6=m
Γ(`)
− (−s′(`))|δ − εm〉.
Also
〈α|
n∏
i=1
Γ(i)
+ (t(i))ψ+(m)(z) = 〈α|
n∏
i=1
Γ(i)
+ (t(i))Qmz
α
(m)
0 Γ(m)
− ([z])Γ(m)
+ (−[z−1])
Clifford Algebra Derivations of Tau-Functions 7
= (−)|α|m−1zαm−1γ([z], t(m))〈α− εm|
n∏
i=1
Γ(i)
+ (t(i))Γ(m)
+ (−[z−1])
= (−)|α|m−1zαm−1γ([z], t(m))〈α− εm|Γ(m)
+ (t(m) − [z−1])
∏
i6=m
Γ(i)
+ (t(i))
and
〈γ|
n∏
k=1
Γ(k)
+ (s(k))ψ−(m)(z)
= (−)|γ|m−1z−γm−1γ(s(m),−[z])〈γ + εm|Γ(m)
+ (s(m) + [z−1])
∏
k 6=m
Γ(k)
+ (s(k)).
Using this we rewrite (3.3):
Res
z
n∑
m=1
(−)|β+δ|m−1zβm−δmγ([z−1], t′(m) − s′
(m))τβ+εm
α (t(i), t′(j) − δjm[z])
× τ δ−εm
γ (s(k), s′
(`) + δ`m[z]) = Res
z
n∑
m=1
(−)|α+γ|m−1zαm−γm−2γ([z], t(m) − s(m))
× τβ
α−εm
(t(i) − δim[z−1], t′(j))τ δ
γ+εm
(s(k) + δkm[z−1], s′(`)). (3.4)
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:
Res
z
n∑
m=1
(−)|β+δ|m−1zδm−βm−2γ([z], u(m) − v(m))τβ+εm
α (t(i), u(j) − δjm[z−1])
× τ δ−εm
γ (s(k), v(`) + δ`m[z−1]) = Res
z
n∑
m=1
(−)|α+γ|m−1zαm−γm−2γ([z], t(m) − s(m))
× τβ
α−εm
(t(i) − δim[z−1], u(j))τ δ
γ+εm
(s(k) + δkm[z−1], v(`)). (3.5)
From the commutation relations (2.1) one easily deduces the following
Proposition 2. Let λ = +,−, and aj ∈ C, for 1 ≤ j ≤ n.
(a) The operator A =
n∑
j=1
ajψ
λ(j)(z) satisfies (3.1).
(b) Let fj(z) =
∑
i
f i
jz
i, for 1 ≤ j ≤ n, then the operator A =
n∑
j=1
Resz fj(z)ψλ(j)(z) satis-
fies (3.1).
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 = z̄. Then, for each 1 ≤ j, k ≤ n
and λ, ν = +,− the operator A = eB with
B =
∫
ψλ(j)(x)ψν(k)(y) dµ(x, y)
8 H. Aratyn and J. van de Leur
satisfy (3.1). If one chooses j = 1, k = 2 and λ = +, ν = − and defines for the above A
τα1+N,α2−N
α1,α2
(t, u) = 〈α1, α2|Γ(1)
+ (t(1))Γ(2)
+ (t(2))eBΓ(1)
− (−u(1))Γ(2)
− (−u(2))|α1 +N,α2 −N〉
=
1
N !
〈α1, α2|Γ(1)
+ (t(1))Γ(2)
+ (t(2))BN Γ(1)
− (−u(1))Γ(2)
− (−u(2))|α1 +N,α2 −N〉,
then these tau-functions satisfy (3.5).
4 Wave functions and pseudo-differential equations
We will now rewrite the equations (3.5) in another form. Note that for
|β − α| = |δ − γ| = j for fixed j ∈ Z (4.1)
these are the equations of the 2n-component KP hierarchy, see [9]. One obtains equation (66)
of [9] if one chooses t(m+n) = u(m), s(m+n) = v(m) and τβ
α (t) = (i)|α|
2
τ(α,−β)(t), where
τ(α,−β)(t) = τ(α1,α2,...,αn,−β1,−β2,...,−βn)(t)
are the 2n-component KP tau-functions, viz for (α,−β) = ρ and (γ,−δ) = σ they satisfy the
2n-component KP equations:
Res
z
2n∑
m=1
(−)|ρ+σ|m−1zρm−σm−2γ([z], t(m) − s(m))τρ−εm(t(i) − δim[z−1])
× τσ+εm(s(k) + δkm[z−1]) = 0.
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):
P±(s)(α, β, t, u,±z) = (P±(s)(α, β, t, u,±z)k`)1≤k,`≤n,
P±(0)(α, β, t, u,±z)k` = (−)|εk|`−1zδk`−1
τβ
α±(εk−ε`)
(t(i) ∓ δi`[z−1], u(j))
τβ
α (t(i), u(j))
,
P±(1)(α, β, t, u,±z)k` = z−1 τ
β±ε`
α±εk
(t(i), u(j) ∓ δj`[z−1])
τβ
α (t(i), u(j))
,
R±(s)(α, β,±z) = diag(R±(s)(α, β,±z)`),
R±(0)(α, β,±z)` = (−)|α|`−1z±α` ,
R±(1)(α, β,±z)` = (−)|β|`−1z∓β` ,
Q±(t,±z) = diag(γ([z],±t(1)), γ([z],±t(2)), . . . , γ([z],±t(n))). (4.2)
Replace t(a)
1 , s(a)
1 , u(a)
1 , v(a)
1 by t
(a)
1 + x0, s
(a)
1 + y0, u
(a)
1 + x1, v
(a)
1 + y1 then some of the above
functions also depend on x0 and x1 we will add these variables to these functions and write e.g.
P±(0)(α, β, x, t, u,±z) for P±(0)(α, β, t(j)i + δi1x0, u,±z). Introduce for k = 0, 1 the following
differential symbols ∂k = ∂/∂xk ∂′k = ∂/∂yk. Introduce the wave functions, here x is short
hand notation for x = (x0, x1)
Ψ±(0)(α, β, x, t, u, z) = P±(0)(α, β, x, t, u,±z)R±(0)(α, β,±z)Q±(t,±z)e±x0z
= P±(0)(α, β, x, t, u, ∂0)R±(0)(α, β, ∂0)Q±(t, ∂0)e±x0z,
Clifford Algebra Derivations of Tau-Functions 9
Ψ±(1)(α, β, x, t, u, z) = P±(1)(α, β, x, t, u,±z)R±(1)(α, β,±z)Q±(u,±z)e±x1z
= P±(1)(α, β, x, t, u, ∂1)R±(1)(α, β, ∂1)Q±(u, ∂1)e±x1z. (4.3)
Then (3.5) leads to
Res
z
Ψ+(0)(α, β, x, t, u, z)Ψ−(0)(γ, δ, y, s, v, z)T
= Res
z
Ψ+(1)(α, β, x, t, u, z)Ψ−(1)(γ, δ, y, s, v, z)T . (4.4)
One can also deduce the following 6 equations, for a proof see the appendix A:
P+(0)(α, β, x0, x1, t, u, ∂0)−1 = P−(0)(α, β, x0, y1, s, v, ∂0)∗, (4.5)
P+(1)(α, β, x0, x1, t, u, ∂1)−1 =S(∂1)P−(1)
(
α+
n∑
i=1
εi, β+
n∑
i=1
εi, x0, x1, t, u, ∂1
)∗
S(∂1), (4.6)
∂P+(0)(α, β, x0, x1, t, u, ∂0)
∂t
(a)
j
= −(P+(0)(α, β, x0, x1, t, u, ∂0)Eaa∂
j
0P
+(0)(α, β, x0, x1, t, u, ∂0)−1)−
× P+(0)(α, β, x0, x1, t, u, ∂0), (4.7)
∂P+(1)(α, β, x, t, u, ∂1)
∂u
(a)
j
= −(P+(1)(α, β, x, t, u, ∂1)∂
j
1EaaP
+(1)(α, β, x, t, u, ∂1)−1)<1
× P+(1)(α, β, x, t, u, ∂1). (4.8)
And
∞∑
j=1
∂P+(0)(α, β, x, t, u, ∂0)
∂u
(a)
j
z−j−1
= P+(1)(α, β, x, t, u, z)Eaa∂
−1
0 P−(1)(α, β, x, t, u,−z)TP+(0)(α, β, x, t, u, ∂0), (4.9)(
Eaa +
∞∑
j=1
∂
∂t
(a)
j
z−j
)
(P+(1)(α, β, x, t, u, ∂1)) = P+(0)(α, β, x, t, u, z)EaaS(∂1)−1
× P−(0)
(
α+
n∑
i=1
εi, β +
n∑
i=1
εi, x, s, v,−z
)T
S(∂1)P+(1)(α, β, x, t, u, ∂1), (4.10)
where
S(∂) =
n∑
i=1
(−)i+1Eii∂.
Recall that (P (x)∂k)∗ = (−∂)k · P (x)T .
5 First reduction and the generalized AKNS model
Assume from now on that our tau functions satisfy
τ
β−
∑
j
εj
α−
∑
j
εj
= (−)
n−1∑
i=1
|β−α|i
cτβ
α , where 0 6= c ∈ C.
10 H. Aratyn and J. van de Leur
This holds e.g. when
Q1Q2 · · ·QnA = cAQ1Q2 · · ·Qn. (5.1)
Substituting this in (4.2), gives
P±(i)
(
α−
∑
j
εj , β −
∑
j
εj , t, u, ∂i
)
=
n∑
k=1
(−)kEkkP
±(i)(α, β, t, u, ∂i)
n∑
k=1
(−)kEkk. (5.2)
Now from the second equation of (A.3) and (4.7) we deduce that
n∑
j=1
∂P+(0)(α, β, x0, x1, t, u, ∂0)
∂t
(j)
1
= 0. (5.3)
Hence this implies that
[∂0, P
+(0)(α, β, x0, x1, t, u, ∂0)] = 0. (5.4)
In a similar way we deduce from (A.6) and (4.8) that
n∑
j=1
∂P+(1)(α, β, x0, x1, t, u, ∂1)
∂u
(j)
1
= 0. (5.5)
Hence also
[∂1, P
+(1)(α, β, x0, x1, t, u, ∂1)] = 0. (5.6)
Remark. Note that the above does not imply that both
[∂1, P
+(0)(α, β, x0, x1, t, u, ∂0)] = 0 and [∂0, P
+(1)(α, β, x0, x1, t, u, ∂1)] = 0.
Next, using this reduction and (A.4) then (A.9) turns into
∂P+(1)(α, β, x, t, u, ∂1)
∂x0
=
(
P
+(0)
1 (α, β, x, t, u)∂−1
1 − ∂−1
1 P
+(0)
1 (α, β, x, t, u)
)
∂1P
+(1)(α, β, x, t, u, ∂1)
=
∞∑
j=1
(−)j+1∂
jP
+(0)
1 (α, β, x, t, u)
∂xj
1
P+(1)(α, β, x, t, u, ∂1)∂
−j
1 . (5.7)
Now, use (4.6) to see that (A.12) turns into
∂P+(0)(α, β, x, t, u, ∂0)
∂x1
= P
+(1)
1 (α, β, x, t, u)∂−1
0 P
(1)
1 (α, β, x, t, u, )−1P+(0)(k1, k2, x, t, u, ∂0). (5.8)
Taking the coefficient of ∂−1
0 of this equation we thus get
∂P
+(0)
1 (α, β, x, t, u)
∂x1
= I
and hence (5.7) turns into
∂P+(1)(α, β, x, t, u, ∂1)
∂x0
= P+(1)(α, β, x, t, u, ∂1)∂−1
1 .
Clifford Algebra Derivations of Tau-Functions 11
In particular
∂P
+(1)
1 (α, β, x, t, u)
∂x0
= 0.
Substituting this in (5.8) we obtain
∂P+(0)(α, β, x, t, u, ∂0)
∂x1
= P+(0)(α, β, x, t, u, ∂0)∂−1
0 . (5.9)
Note first that from (5.9) one deduces that
∂P+(0)(α, β, x, t, u, ∂0)−1
∂x1
= −P+(0)(α, β, x, t, u, ∂0)−1∂P
+(0)(α, β, x, t, u, ∂0)
∂x1
P+(0)(α, β, x, t, u, ∂0)−1∂−1
0
= −P+(0)(α, β, x, t, u, ∂0)−1P+(0)(α, β, x, t, u, ∂0)∂−1
0 P+(0)(α, β, x, t, u, ∂0)−1
= −P+(0)(α, β, x, t, u, ∂0)−1∂−1
0 .
Clearly also
∂P+(1)(α, β, x, t, u, ∂1)−1
∂x0
= −P+(1)(α, β, x, t, u, ∂1)−1∂−1
1 .
Using (4.5) we rewrite (A.8):
∂P+(1)(α, β, x, t, u, ∂1)
∂t
(a)
j
= Res
z
P+(0)(α, β, x, t, u, z)zj−1Eaa
× ∂−1
1 P+(0)(α, β, x, t, u, z)−1∂1P
+(1)(α, β, x, t, u, ∂1)
= Res
z
P+(0)(α, β, x, t, u, z)zj−1Eaa
×
∞∑
i=0
(−)i∂
iP+(0)(α, β, x, t, u, z)−1
∂xi
1
∂−i
1 P+(1)(α, β, x, t, u, ∂1)
= Res
z
P+(0)(α, β, x, t, u, z)Eaa
× P+(0)(α, β, x, t, u, z)−1
∞∑
i=0
zj−i−1P+(1)(α, β, x, t, u, ∂1)∂−i
1
= Res
z
j∑
i=0
zj−i−1P+(0)(α, β, x, t, u, z)Eaa
× P+(0)(α, β, x, t, u, z)−1P+(1)(α, β, x, t, u, ∂1)∂−i
1 . (5.10)
N.B. This summation starts with 0.
In particular
∂P+(1)(α, β, x, t, u, ∂1)
∂t
(1)
j
+
∂P+(1)(α, β, x, t, u, ∂1)
∂t
(2)
j
= P+(1)(α, β, x, t, u, ∂1)∂
−j
1 . (5.11)
In a similar way we deduce, using (4.6), from (A.10):
∂P+(0)(α, β, x, t, u, ∂0)
∂u
(a)
j
= Res
z
j∑
i=1
zj−i−1P+(1)(α, β, x, t, u, z)Eaa
12 H. Aratyn and J. van de Leur
× P+(1)(α, β, x, t, u, z)−1P+(0)(α, β, x, t, u, ∂0)∂−i
0 . (5.12)
N.B. This summation starts with 1.
In particular
n∑
k=1
∂P+(0)(α, β2, x, t, u, ∂0)
∂u
(k)
j
= P+(0)(α, β, x, t, u, ∂0)∂
−j
0 . (5.13)
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
P (0)(α, β, x, t, u, z) = P+(0)(α, β, x0, x1, t, u, z) and
P (1)(α, β, x, t, u, z) = z−1P+(1)(α, β, x0, x1, t, u, z
−1) (5.14)
and thus obtain for fixed α and β (P (j)(x, t, u, z) = P (j)(α, β, x, t, u, z)):
∂P (0)(x, t, u, z)
∂x0
= 0,
∂P (0)(x, t, u, z)
∂x1
= P (0)(x, t, u, z)z−1,
∂P (1)(x, t, u, z)
∂x0
= P (1)(x, t, u, z)z,
∂P (1)(x, t, u, z)
∂x1
= 0,
∂P (0)(x, t, u, z)
∂t
(a)
j
= −(P (0)(x, t, u, z)EaaP
(0)(x, t, u, z)−1zj)−P (0)(x, t, u, z),
∂P (0)(x, t, u, z)
∂u
(a)
j
= (P (1)(x, t, u, z)EaaP
(1)(x, t, u, z)−1z−j)−P (0)(x, t, u, z),
∂P (1)(x, t, u, z)
∂t
(a)
j
= (P (0)(x, t, u, z)EaaP
(0)(x, t, u, z)−1zj)+P (1)(x, t, u, z),
∂P (1)(x, t, u, z)
∂u
(a)
j
= −(P (1)(x, t, u, z)EaaP
(1)(x, t, u, z)−1z−j)+P (1)(x, t, u, z). (5.15)
Which are the generalized AKNS equations (2.5)–(2.10) of [1]. Write
P (0)(x, t, u, z) = I +
∞∑
i=1
Pi(x, t, u)z−i, P (1)(x, t, u, z) =
∞∑
i=0
Mi(x, t, u)zi
and from now on in this section
∂j =
∂
∂t
(j)
1
, ∂−j =
∂
∂u
(j)
1
.
We thus obtain the following equations:
∂jP1 = [P1, Ejj ]P1 + [Ejj , P2], ∂−jP1 = M0EjjM
−1
0 ,
∂jM0 = [P1, Ejj ]M0, ∂jM1 = EjjM0 + [P1, Ejj ]M1,
∂−jM0 = M0[Ejj ,M
−1
0 M1], ∂−jM1 = M0[Ejj ,M
−1
0 M2]. (5.16)
From this one easily deduces the following equations for M0:
∂i(M−1
0 ∂−jM0) = [Ejj ,M
−1
0 EiiM0],
Clifford Algebra Derivations of Tau-Functions 13
∂−i(M0EjjM
−1
0 ) = ∂−j(M0EiiM
−1
0 ),
∂i(M−1
0 EjjM0) = ∂j(M−1
0 EiiM0).
Note that if we define P̄i = M−1
0 Mi, then from (5.16) we get:
∂jP1 = [P1, Ejj ]P1 + [Ejj , P2], ∂−jP1 = M0EjjM
−1
0 , ∂jP̄1 = M−1
0 EjjM0,
∂jP̄1 = [P̄1, Ejj ]P̄1 + [Ejj , P̄2], ∂jM0 = [P1, Ejj ]M0, ∂−jM0 = M0[Ejj , P̄1]. (5.17)
Note that for x0 = x1 = 0 we have
(M0(α, β, t, u))kl =
τβ+ε`
α+εk
(t, u)
τβ
α (t, u)
, (M1(α, β, t, u))kl = −
∂−`(τ
β+ε`
α+εk
(t, u))
τβ
α (t, u)
,
(M2(α, β, t, u))kl =
1
2
(∂2
−` − ∂
u
(`)
2
)(τβ+ε`
α+εk
(t, u))
τβ
α (t, u)
,
(P1(α, β, t, u))kl =
(−)|εk|`−1
τβ
α+εk−ε`
(t, u)
τβ
α (t, u)
if k 6= `,
−∂`(τ
β
α (t, u))
τβ
α (t, u)
if k = `,
(P2(α, β, t, u))kl =
−(−)|εk|`−1
∂`
(
τβ
α+εk−ε`
(t, u)
)
τβ
α (t, u)
if k 6= `,
1
2
(
∂2
` − ∂t`2
)(
τβ
α (t, u)
)
τβ
α (t, u)
if k = `.
(5.18)
6 AKNS and the two-component Camassa–Holm model
We still assume that A satisfies (5.1), hence that we have the reduction of the previous section.
We consider the case n = 2 and define
y =
t
(1)
1 − t
(2)
1
2
, ȳ =
t
(1)
1 + t
(2)
1
2
, s = 2u(1)
1 − 2u(2)
1 , s̄ = 2u(1)
1 + 2u(2)
1 , (6.1)
then
∂
∂y
= ∂1 − ∂2,
∂
∂s
=
1
4
(∂−1 − ∂−2) .
Now Let E = E11 − E22 and define
ψ(z) = P (0)(0, t, u, z)
(
eyz 0
0 e−yz
)
then (5.15) turns into:
∂ψ(z)
∂y
= (zE + [P1, E])ψ(z) =
((
z 0
0 −z
)
+
(
0 q
r 0
))
ψ(z),
∂ψ(z)
∂s
=
z−1
4
M0EM
−1
0 ψ(z) = z−1
(
A B
C −A
)
ψ(z),
14 H. Aratyn and J. van de Leur
where
q = −2(P1)12, r = 2(P1)21, A =
1
4 detM0
((M0)11(M0)22 + (M0)12(M0)21),
B =
−1
2 detM0
(M0)11(M0)12, C =
1
2 detM0
(M0)22(M0)21.
Now (5.16) turns into:
∂P1
∂y
=[P1, E]P1+[E,P2] =
(
0 q
r 0
)
P1+[E,P2],
∂P1
∂s
=
1
4
M0EM
−1
0 =
(
A B
C −A
)
,
∂M0
∂y
= [P1, E]M0 =
(
0 q
r 0
)
M0,
∂M1
∂y
= EM0 + [P1, E]M1 = EM0 +
(
0 q
r 0
)
M1,
∂M0
∂s
=
1
4
M0[E,M−1
0 M1],
∂M1
∂s
=
1
4
M0[E,M−1
0 M2]. (6.2)
We will now describe transition from the matrix equations (6.2) to the 2-component Camassa–
Holm (CH) model. The matrix P1 is parametrized by q = −2(P1)12, r = −2(P1)21 and
Tr (P1E) = −(ln τ0
0 )y, where r and q are variables of the AKNS model and τ0
0 is its tau function.
Furthermore, A2 +BC = 1/16. Calculating Tr ((P1)yE) using the first equation of (6.2) we get
rq = −(ln τ0
0 )yy. (6.3)
Next, from the second equation of (6.2) we find
qs = −2B, rs = 2C (6.4)
and by taking Tr ((P1)sE):
A = −1
2
(ln τ0
0 )ys. (6.5)
By comparing eqs. (6.3) and (6.5) we deduce that Ay = (rq)s/2.
From the third equation of (6.2) we derive:
(P1)sy =
1
4
(
M0EM
−1
0
)
y
=
(
Ay By
Cy −Ay
)
=
[(
0 q
r 0
)
,
(
A B
C −A
)]
or in components:
By = −2qA, Cy = 2rA, Ay = −1
2
(ln τ0
0 )syy = qC − rB (6.6)
and
rsy = 4rA, qsy = 4qA.
In view of the fact that the determinant of the matrix (M0EM
−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:
B = ef
(
A− 1
4
)
, C = −e−f
(
A+
1
4
)
.
Recalling equation (6.4) we easily find
rse
f − qse
−f = 2(Cef +Be−f ) = −1, rse
f + qse
−f = 2(Cef −Be−f ) = −4A. (6.7)
Clifford Algebra Derivations of Tau-Functions 15
Using the first two identities of equation (6.6) and the fact that
Cye
f = fy
(
A+
1
4
)
−Ay, Bye
−f = fy
(
A− 1
4
)
+Ay
we derive expressions for ref ± qe−f as follows:
ref − qe−f =
Cy
2A
ef +
By
2A
e−f =
1
2A
(
fy
(
A+
1
4
)
−Ay + fy
(
A− 1
4
)
+Ay
)
= fy (6.8)
and
ref +qe−f =
Cy
2A
ef−By
2A
e−f =
1
2A
(
fy
(
A+
1
4
)
−Ay−fy
(
A− 1
4
)
−Ay
)
=
fy
4A
−Ay
A
. (6.9)
Taking a derivative of relation ref − qe−f = fy with respect to variable s we find, in view
of (6.7), that:
fys = rse
f − qse
−f + fs(ref + qe−f ) = −1 + fs(ref + qe−f )
or
ref + qe−f = 2g, (6.10)
where we defined:
g =
1
2fs
(1 + fsy). (6.11)
Comparing two expressions (6.9) and (6.10) for the quantity ref + qe−f we find the following
relation
2gA =
fy
4
−Ay. (6.12)
By adding and subtracting (6.10) and (6.8) we get
2ref = 2g + fy, 2qe−f = 2g − fy
or
q = ef
(
g − fy
2
)
, r = e−f
(
g +
fy
2
)
.
Plugging these expressions into the second relation in eq. (6.7) yields:
−4A = rse
f + qse
−f =
(
−fsr + e−f
(
gs +
fsy
2
))
ef +
(
fsq + ef
(
gs −
fsy
2
))
e−f
= −fs(ref − qe−f ) + 2gs = −fsfy + 2gs.
Thus,
A =
1
4
(fsfy − 2gs) =
1
4
(
fsfy −
(
1
fs
)
s
−
(
fsy
fs
)
s
)
. (6.13)
Plugging definition (6.11) of g into relation (6.12) leads to:
A =
1
4
fsfy − fsAy −Afsy =
1
4
fsfy − (fsA)y.
16 H. Aratyn and J. van de Leur
Inserting on the left hand side of the above identity the value of A from equation (6.13) and
multiplying by −4 yields(
1
fs
)
s
= −
(
fsy
fs
)
s
+ 4(fsA)y =
(
−fss
fs
+ 4fsA
)
y
=
(
−fss
fs
+ f2
s fy − fs
(
1 + fsy
fs
)
s
)
y
=
(
f2
s fy − fssy +
fssfsy
fs
)
y
, (6.14)
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:
u = f2
s fy − fssy +
fssfsy
fs
− 1
2
κ, (6.15)
with κ being an integration constant, it holds from relation (6.14) that
uy =
(
1
fs
)
s
. (6.16)
Let us now denote the product f2
s fy by m. Then from relations (6.15) and (6.16) we derive
m = f2
s fy = u+fssy−
fssfsy
fs
+
1
2
κ = u−fs
(
fs
(
1
fs
)
s
)
y
+
1
2
κ = u−fs(fsuy)y+
1
2
κ.
Taking a derivative of m = f2
s fy with respect to s yields
ms = 2fyfsfss+f2
s fsy = 2m
fss
fs
+f2
s fsy =−2mfs
(
1
fs
)
s
+f2
s fsy =−2mfsuy+f2
s fsy. (6.17)
In terms of the basic quantities u and ρ = fs of the two-component Camassa–Holm model
equations (6.16) and (6.17) take the following form
ρs = −ρ2uy, (6.18)
ms = −2mρuy + ρ2ρy (6.19)
for
m = u− ρ(ρuy)y +
1
2
κ. (6.20)
Performing an inverse reciprocal transformation (y, s) 7→ (x, t) defined by relations:
Fx = ρFy, Ft = Fs − ρ uFy
for an arbitrary function F , we find that equations (6.18), (6.19) and (6.20) become
ρt = −(uρ)x, (6.21)
mt = −2mux −mxu+ ρρx, (6.22)
m = u− uxx +
1
2
κ (6.23)
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]).
Clifford Algebra Derivations of Tau-Functions 17
The relation (6.14) is equivalent to the following condition
fss
2f3
s
+ fsyfy +
1
2
fsfyy −
fssyy
2fs
+
fssyfsy
2f2
s
+
fssfsyy
2f2
s
−
fssf
2
sy
2f3
s
= 0,
which first appeared in [4].
Comparing equations (6.13) and (6.15) we find that
4fsA = u+
1
2
κ+
fss
fs
= u− ux +
1
2
κ
where
4fsA = − rss
2rs
(
A+
1
4
)
+
qss
2qs
(
A− 1
4
)
= − rssqs
2
(
A− 1
4
) +
qssrs
2
(
A+ 1
4
)
or
4fsA = −2fs(ln τ0
0 )sy = −2(ln τ0
0 )sx.
These relations give u, fs in terms of the AKNS quantities r, q, τ0
0 .
7 A second reduction and the Cecotti–Vafa equations
In order to obtain the Cecotti–Vafa equations we define an automorphism of order 4 on the
Clifford algebra by
ω(ψ±(j)
k ) = (−)k+ 1
2 iψ
∓(j)
k
then ω(ψ±(j)(z)) = iψ∓(j)(−z) and
ω(α(j)
m ) = −(−)mα(j)
m , ω(Q±1
j ) = iQ∓1
j (−)α
(j)
0 .
For the derivation of the last equation see [11]. Next, let
ω(|0〉) = |0〉, and ω(〈0|) = 〈0|,
then this induces also an automorphism on the representation spaces F and F ∗. It is straight-
forward to check that
ω(|α〉) = (−)
n∑
j=1
1
2
|αj |(|αj |−1)
i
n∑
j=1
|αj |
| − α〉,
ω(〈α|) = (−)
n∑
j=1
1
2
|αj |(|αj |+1)
i
n∑
j=1
|αj |
〈−α|.
Since
ω(ψλ(j)
k ψ
−λ(j)
−k ) = ψ
−λ(j)
k ψ
λ(j)
−k
one easily deduces that from (2.2) and (2.3) that
〈ω(〈V (r1, . . . , rm, s1, . . . , sq)|)|ω(|V (i1, . . . , ik, j1, . . . , j`)〉)〉
= 〈V (r1, . . . , rm, s1, . . . , sq)|V (i1, . . . , ik, j1, . . . , j`)〉. (7.1)
18 H. Aratyn and J. van de Leur
Assume for the second reduction that our A, which commutes with Ω, also satisfies
ω(A) = A, (7.2)
then one deduces from (7.1) that
ω(τβ
α (t, u)) = τβ
α (t, u).
On the other hand if we use the definition (3.2) for the tau-functions one deduces
ω(τβ
α (t(j)m , u(q)
p ))=(−)
n∑
j=1
1
2
|αj |(|αj |+1)+ 1
2
|βj |(|βj |−1)
i
n∑
j=1
|αj |+|βj |
τ−β
−α (−(−)mt(j)m ,−(−)pu(q)
p ).
From now on lets write t̃(j)m = −(−)mt
(j)
m , then combining the above two equations, one has
when (7.2) holds:
τβ
α (t, u) = (−)
n∑
j=1
1
2
|αj |(|αj |+1)+ 1
2
|βj |(|βj |−1)
i
n∑
j=1
|αj |+|βj |
τ−β
−α (t̃, ũ).
In particular
τ0
0 (t, u) = τ0
0 (t̃, ũ), τ0
εk−ε`
(t, u) = −τ0
ε`−εk
(t̃, ũ), τ ε`
εk
(t, u) = τ−εk
−ε`
(t̃, ũ).
Now substituting this in (4.3) for α = β = 0 we obtain that
ψ+(0)(0, 0, x, t, u, z) = ψ−(0)(0, 0, x, t̃, ũ,−z),
ψ+(1)(0, 0, x, t, u, z) = −ψ−(1)(0, 0, x, t̃, ũ,−z).
Assume from now on that A satisfies (5.1) and (7.2), viz. that both reductions hold, then
from (4.5), (4.6), (5.2) we deduce that
P+(0)(0, 0, x, t, u, z)−1 = P+(0)(0, 0, x, t̃, ũ,−z)T ,
P+(1)(0, 0, x, t, u, z)−1 = −P+(1)(0, 0, x, t̃, ũ,−z)T z2,
then using the definition (5.14) one finally obtains
P (0)(0, 0, x, t, u, z)−1 = P (0)(0, 0, x, t̃, ũ,−z)T ,
P (1)(0, 0, x, t, u, z)−1 = P (1)(0, 0, x, t̃, ũ,−z)T .
Now putting all t(j)2m = u
(j)
2m = 0, this gives the following equations for P1, M0, M1 and P̄1:
P T
1 = P1, MT
0 = M−1
0 , MT
1 = MT
0 M1M
T
0 , P̄ T
1 = P̄1. (7.3)
Now denote
P1 = (βij)1≤i,j≤n, M0 = (mij)1≤i,j≤n P̄1 = (β̄ij)1≤i,j≤n,
then the equations (7.3) and (5.17) give the following system of Cecotti–Vafa equations (see
e.g. [1, 3, 5]):
βij = βji, β̄ij = β̄ji, ∂jβik = βijβjk, ∂−j β̄ik = β̄ij β̄jk, i, j, k distinct,
n∑
j=1
∂jβik =
n∑
j=1
∂j β̄ik =
n∑
j=1
∂−jβik =
n∑
j=1
∂−j β̄ik = 0, i 6= k,
∂−jβik = mijmkj , ∂j β̄ik = mjimjk i 6= k,
n∑
j=1
∂jmik =
n∑
j=1
∂−jmik = 0, ∂jmik = βijmjk, ∂−jmik = mij β̄jk. (7.4)
Clifford Algebra Derivations of Tau-Functions 19
8 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 L0 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.
L0 =
n∑
j=1
1
2
(α(j)
0 )2 +
∞∑
k=1
α
(j)
−kα
(j)
k . (8.1)
It is straightforward to check that
[L0, α
(j)
k ] = −kα(j)
k
and
〈β|L0 =
1
2
n∑
j=1
|βj |2〈β|, L0|β〉 =
1
2
n∑
j=1
|βj |2|β〉.
Moreover, one also has[
L0, ψ
±(j)
k
]
= −kψ
±(j)
k .
Assume from now on that our operator A that commutes with Ω is homogeneous of degree p
with respect to L0, i.e.,
[L0, A] = pA.
We then calculate
〈α|L0Ã|β〉 = 〈α|L0
n∏
j=1
Γ(j)
+ (t(j))A
n∏
k=1
Γ(k)
− (−u(k))|β〉.
It is straightforward to see that this is equal to
1
2
n∑
j=1
|αj |2〈α|L0Ã|β〉.
On the other hand using
[L0,Γ
(j)
+ (t(j))] = −
∞∑
k=1
kt
(j)
k α
(j)
k Γ(j)
+ (t(j)) = −
∞∑
k=1
kt
(j)
k
∂
∂t
(j)
k
Γ(j)
+ (t(j))
and
[L0,Γ
(j)
− (−u(j)] =
∞∑
k=1
ku
(j)
k
∂
∂u
(j)
k
Γ(j)
− (−u(j))
we also have:
〈α|L0Ã|β〉 =
(
p+
1
2
n∑
j=1
|βj |2 +
n∑
j=1
∞∑
k=1
ku
(j)
k
∂
∂u
(j)
k
− kt
(j)
k
∂
∂t
(j)
k
)
〈α|Ã|β〉.
20 H. Aratyn and J. van de Leur
Now let
E =
∞∑
k=1
kt
(j)
k
∂
∂t
(j)
k
− ku
(j)
k
∂
∂u
(j)
k
,
then
Eτβ
α (t, u) =
(
p+
1
2
n∑
j=1
|βj |2 − |αj |2
)
τβ
α (t, u),
in particular
Eτ0
0 (t, u) = pτ0
0 (t, u), Eτ0
εk−ε`
(t, u) = (p− 1)τ0
εk−ε`
(t, u), Eτ ε`
εk
(t, u) = pτ ε`
εk
(t, u).
Assume now that we also have imposed the first and second reduction, then one has
EP1 = −P1, EP0 = 0, EM1 = M1
and thus also
EP̄1 = P̄1.
Putting all t(j)m = u
(j)
m = 0 for all m > 1 and all 1 ≤ j ≤ n one has that the βij , mij and β̄ij not
only satisfy (7.4), but also
n∑
j=1
(tj∂j − uj∂−j)βij = −βij ,
n∑
j=1
(tj∂j − uj∂−j)mij = 0,
n∑
j=1
(tj∂j − uj∂−j)β̄ij = β̄ij , (8.2)
for tj = t
(j)
1 and uj = 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.
9 Explicit construction of solutions in the AKNS case
We will construct an operator A that satisfies (3.1) and (5.1) in the Camassa–Holm case, i.e.,
the case that n = 2. Now using Proposition 2, we see that the element
Ak = (a(1)
1 ψλ1(1)(z1) + a
(2)
1 ψλ1(2)(z1))(a
(1)
2 ψλ2(1)(z2) + a
(2)
2 ψλ2(2)(z2)) · · ·
· · · (a(1)
k ψλk(1)(zk) + a
(2)
k ψλk(2)(zk)),
satisfies condition (3.1). Using (2.4) and the definition of the fermionic fields, we see that
Q1Q2Ak = zλ1
1 zλ2
2 · · · zλk
k AkQ1Q2.
Thus Ak also satisfies (5.1). Since we want τ0
0 6= 0, we take A2k and will assume that
λ1 = λ2 = · · · = λk = + and λk+1 = λk+2 = · · · = λ2k = −.
Clifford Algebra Derivations of Tau-Functions 21
We now want to calculate τ0
εi−εj
and τ
εj
εi , for 1 ≤ i, j ≤ 2 in order to get some solutions related
to the Camassa–Holm equation. Let us start with τ0
0 :
τ0
0 (t, u) = 〈0|
2∏
i=1
Γ(i)
+ (t(i))A2k
2∏
j=1
Γ(j)
− (−u(j))|0〉
=
∑
`1+···`k=`k+1+···`2k
a
(`1)
1 a
(`2)
2 · · · a(`2k)
2k 〈0|
2∏
i=1
Γ(i)
+ (t(i))ψ+(`1)(z1)ψ+(`2)(z2) · · ·
· · ·ψ+(`k)(zk)ψ−(`k+1)(zk+1) · · ·ψ−(`2k)(z2k)
2∏
j=1
Γ(j)
− (−u(j))|0〉
= γ(t(1),−u(1))γ(t(2),−u(2))
∑
`1+···`k=`k+1+···`2k
k∏
i=1
a
(`i)
i a
(`i+k)
i+k γ(t(`i), [zi])
× γ(−t(`k+i), [zk+i])γ(u(`i), [z−1
i ])γ(−u(`k+i), [z−1
k+i])
× 〈0|ψ+(`1)(z1)ψ+(`2)(z2) · · · · · ·ψ+(`k)(zk)ψ−(`k+1)(zk+1) · · ·ψ−(`2k)(z2k)|0〉.
So we have to calculate explicitly:
〈0|ψ+(`1)(z1)ψ+(`2)(z2) · · ·ψ+(`k)(zk)ψ−(`k+1)(zk+1) · · ·ψ−(`2k)(z2k)|0〉
= σ(`1, `2, . . . `2k)
∏
1≤i<j≤k
(zi − zj)
δ`i,`j (zk+i − zk+j)
δ`k+i,`k+j
∏
1≤i,j≤k
(zi − zk+j)
δ`i,`k+j
,
where
σ(`1, `2, . . . , `2k) = 〈0|Q`1Q`2 · · ·Q`k
Q−1
`k+1
· · ·Q−1
`2k
|0〉.
Note that the above expression takes only values −1, 0 or 1. Thus
τ0
0 (t, u) = γ(t(1),−u(1))γ(t(2),−u(2))
×
∑
`1+···`k=`k+1+···`2k
σ(`1, `2, . . . , `2k)
∏
1≤i<j≤k
(zi − zj)
δ`i,`j (zk+i − zk+j)
δ`k+i,`k+j
∏
1≤i,j≤k
(zi − zk+j)
δ`i,`k+j
×
k∏
i=1
a
(`i)
i a
(`i+k)
i+k γ(t(`i), [zi])γ(−t(`k+i), [zk+i])γ(u(`i), [z−1
i ])γ(−u(`k+i), [z−1
k+i]).
Now τ0
εi−εj
for i 6= j has the same expression, except that we have to take the summation over
j + `1 + `2 + · · ·+ `k = i+ `k+1 + `k+2 + · · ·+ `2k
and replace the σ by
(−)iσ(j, `1, `2, . . . , `2k, i).
For τ εj
εi , where i can be equal to j, we find
τ
εj
εi (t, u) = (−)1−δijγ(t(1),−u(1))γ(t(2),−u(2))
∑
j+`1+···`k=i+`k+1+···`2k
σ(j, `1, `2, . . . , `2k, i)
22 H. Aratyn and J. van de Leur
× z
δj`1
1 z
δj`2
2 · · · zδj`k
k z
−δj`k+1
k+1 · · · z−δj`2k
2k
∏
1≤i<j≤k
(zi − zj)
δ`i,`j (zk+i − zk+j)
δ`k+i,`k+j
∏
1≤i,j≤k
(zi − zk+j)
δ`i,`k+j
×
k∏
i=1
a
(`i)
i a
(`i+k)
i+k γ(t(`i), [zi])γ(−t(`k+i), [zk+i])γ(u(`i), [z−1
i ])γ(−u(`k+i), [z−1
k+i]).
We now give the simplest of such expressions. We take k = 1 and put all t(i)j = u
(i)
j = 0 for
j > 2, then
τ0
0 (t, u) =
T (t, u)
z1 − z2
(T11(t, u) + T22(t, u)), τ0
εi−εj
(t, u) = (−)i+1T (t, u)Tij(t, u),
τ
εj
εi (t, u) = z−1
2 T (t, u)Tij(t, u), for i 6= j,
τ εi
εi
(t, u) =
T (t, u)
z1 − z2
((
z1
z2
)δ1i
T11(t, u) +
(
z1
z2
)δ2i
T22(t, u)
)
,
where
T (t, u) = e−t
(1)
1 u
(1)
1 −t
(2)
1 u
(2)
1 −2t
(1)
2 u
(2)
1 −2t
(2)
2 u
(2)
2 ,
Tij(t, u) = Tij(t, u, z1, z2)
= a
(i)
1 a
(j)
2 et
(i)
1 z1+u
(i)
1 z−1
1 +t
(i)
2 z2
1+u
(i)
2 z−2
1 −(t
(j)
1 z2+u
(j)
1 z−1
2 +t
(j)
2 z2
2+u
(j)
2 z−2
2 . (9.1)
In order to determine M0, M1 and P1, see (5.18), we also calculate
∂jτ
0
0 (t, u) = −u(j)
1
T (t, u)
z1 − z2
(T11(t, u) + T22(t, u)) + T (t, u)Tjj(t, u),
∂−jτ
εj
εi (t, u) = −(z−1
2 t
(j)
1 + z−2
2 )T (t, u)Tij(t, u), for i 6= j,
∂−iτ
εi
εi
(t, u) = −t(i)1
T (t, u)
z1 − z2
((
z1
z2
)δ1i
T11(t, u) +
(
z1
z2
)δ2i
T22(t, u)
)
− z−2
2 T (t, u)Tii(t, u).
Now make the change of variables (6.1) and choose t(i)2 = u
(i)
2 = 0. One thus obtains
t
(1)
1 = ȳ + y, t
(2)
1 = ȳ − y, u
(1)
1 =
s̄+ s
4
, u
(2)
1 =
s̄− s
4
(9.2)
and
T (y, s) = e−
1
4
(ȳ+y)(s̄+s)− 1
4
(ȳ−y)(s̄−s),
Tij(y, s, z1, z2)=Tij(y, s)=a(i)
1 a
(j)
2 e(ȳ−(−)iy)z1+
1
4
(s̄−(−)is)z−1
1 −((ȳ−(−)jy)z2+ 1
4
(s̄−(−)js)z−1
2 ). (9.3)
Note that in the calculation of M0, M1 and P1, see (5.18), T (y, s) drops out. Thus
M0 =
(
δij +
(z1z−1
2 − 1)Tij(y, s)
T11(y, s) + T22(y, s)
)
1≤i,j≤2
,
M1 =
(
δij(ȳ − (−)jy) +
(z1z−1
2 − 1)(z−1
2 + ȳ − (−)jy)Tij(y, s)
T11(y, s) + T22(y, s)
)
1≤i,j≤2
,
P1 =
(
δij
s̄− (−)js
4
− (z1 − z2)Tij(y, s)
T11(y, s) + T22(y, s)
)
1≤i,j≤2
.
Clifford Algebra Derivations of Tau-Functions 23
One has detM0 = z1/z2 and M0, M1 and P1 satisfy (6.2). The above solutions should in
principle be closely related to the ones obtained in [2] using vertex method. Finally, we give the
Camassa–Holm function f for this case:
f(y, s) = log
a
(1)
1 a
(1)
2 z1e
s
2z1
+2yz1 + a
(2)
1 a
(2)
2 z2e
s
2z2
+2yz2
(z2 − z1)a
(2)
1 a
(1)
2
.
We give expression for tau functions for k = 2. Again we set t(i)j = u
(i)
j = 0 for i, j > 2 and
use T (t, u) and Tij(t, u, zk, zl) with i, j = 1, 2, (k, `) = (1, 3), (k, `) = (2, 4). For brevity we will
denote
T = T (t, u), T k,`
ij = Tij(t, u, zk, z`).
Then, the τ0
0 function is given by
τ0
0 = T
[
(z1 − z2)(z3 − z4)
(z1 − z3)(z2 − z3)(z1 − z4)(z2 − z4)
(T 1,3
11 T
2,4
11 + T 1,3
22 T
2,4
22 )
− 1
(z1 − z3)(z2 − z4)
(T 1,3
11 T
2,4
22 + T 1,3
22 T
2,4
11 ) +
1
(z1 − z4)(z2 − z3)
(T 1,3
12 T
2,4
21 + T 1,3
21 T
2,4
12 )
]
.
In the following formulas for remaining tau functions we choose indices i, j = 1, 2 so that i 6= j.
Then
τεi−εj = (−1)i+1T
[
(z1 − z2)
(z1 − z3)(z2 − z3)
T 1,3
ii T 2,4
ij − (z1 − z2)
(z1 − z4)(z2 − z4)
T 1,3
ij T 2,4
ii
+
(z3 − z4)
(z2 − z3)(z2 − z4)
T 1,3
ij T 2,4
jj − (z3 − z4)
(z1 − z3)(z1 − z4)
T 1,3
jj T
2,4
ij
]
,
τ εi
εj
= −T
[
z1(z3 − z4)
z3z4(z1 − z3)(z1 − z4)
T 1,3
ii T 2,4
ji − z2(z3 − z4)
z3z4(z2 − z3)(z2 − z4)
T 1,3
ji T
2,4
ii
+
(z1 − z2)
z3(z1 − z4)(z2 − z4)
T 1,3
ji T
2,4
jj − (z1 − z2)
z4(z1 − z3)(z2 − z3)
T 1,3
jj T
2,4
ji
]
,
τ εi
εi
= T
[
(z1 − z2)(z3 − z4)
(z1 − z3)(z1 − z4)(z2 − z3)(z2 − z4)
(
z1z2
z3z4
T 1,3
ii T 2,4
ii + T 1,3
jj T
2,4
jj
)
− 1
(z1 − z3)(z2 − z4)
(
z1
z3
T 1,3
ii T 2,4
jj +
z2
z4
T 1,3
jj T
2,4
ii
)
+
1
(z1 − z4)(z2 − z3)
(
z1
z4
T 1,3
ij T 2,4
ji +
z2
z3
T 1,3
ji T
2,4
ij
)]
.
If we make the change of variables (9.2) then the formula’s (9.1) change into (9.3) and the same
formula’s hold for the tau functions.
Another way to construct solutions is as follows. We take an arbitrary element of the loop
algebra of sl2(C). Such an element is given by
g = Res
z
a(z)(ψ+(1)(z)ψ−(1)(z)− ψ+(2)(z)ψ−(2)(z)) + b(z)ψ+(1)(z)ψ−(2)(z)
+ c(z)ψ+(2)(z)ψ−(2)(z),
where a(z), b(z) and c(z) are arbitrary functions. Then the element A = eg commutes with the
action of Ω and satisfies (5.1). Hence the corresponding tau-functions will satisfy the equations
of the AKNS hierarchy.
24 H. Aratyn and J. van de Leur
10 Explicit construction of solutions in the Cecotti–Vafa case
We will construct an operator A that satisfies (3.1), (5.1) and (7.2). We generalize the construc-
tion of the previous section. Using Proposition 2, we see that the element
Ak =
(
n∑
j1=1
a
(j1)
1 ψ+(j1)(z1)
)(
n∑
j1=1
a
(j1)
1 ψ−(j1)(−z1)
)(
n∑
j1=1
a
(j2)
2 ψ+(j2)(z2)
)
×
(
n∑
j1=1
a
(j2)
2 ψ−(j2)(−z2)
)
· · ·
(
n∑
jk=1
a
(jk)
k ψ+(jk)(zk)
)(
n∑
jk=1
a
(jk)
k ψ−(jk)(−zk)
)
,
satisfies condition (3.1). Moreover, if we assume that
n∑
j=1
(a(j)
` )2 = 0 for all 1 ≤ ` ≤ k, (10.1)
then (
n∑
j=1
a
(j)
` ψ+(j)(z)
)(
n∑
k=1
a
(k)
` ψ−(k)(y)
)
= −
(
n∑
k=1
a
(k)
` ψ−(k)(y)
)(
n∑
j=1
a
(j)
` ψ+(j)(z)
)
.
Thus
ω
((
n∑
j=1
a
(j)
` ψ+(j)(z)
)(
n∑
k=1
a
(k)
` ψ−(k)(−z)
))
= −
(
n∑
j=1
a
(j)
` ψ−(j)(−z)
)(
n∑
k=1
a
(k)
` ψ+(k)(z)
)
=
(
n∑
k=1
a
(k)
` ψ+(k)(z)
)(
n∑
j=1
a
(j)
` ψ−(j)(−z)
)
and hence Ak also satisfies (7.2). Let us calculate for k = 1 the corresponding tau functions.
One finds
τ0
0 =
T
2z
n∑
j=1
Tjj(z), τ0
εi−εj
= (−)|εj |iT (z)Tij(z), τ
εj
εi = −T
z
Tij(z) for i 6= j,
τ εi
εi
=
T
2z
n∑
j=1
(−)δijTjj(z),
where
T =
n∏
i=1
γ(t(i),−u(i)),
Tij(zk) = a
(i)
k a
(j)
k γ(t(i), [zk])γ(−t(j), [−zk])γ(u(i), [z−1
k ])γ(−u(j), [−z−1
k )]).
If we only keep t(i)1 and u(i)
1 and put all higher times equal to zero, we find that
βij = zmij = z2β̄ij , for i 6= j
and that
mij = δij −
2a(i)
1 a
(j)
1 e(t
(i)
1 +t
(j)
1 )z+(u
(i)
1 +u
(j)
1 )z−1
n∑
i=1
(a(i)
1 )2e2t
(i)
1 z+2u
(i)
1 z−1
,
where (10.1) still holds. These βij , mij and β̄ij satisfy (7.4).
Clifford Algebra Derivations of Tau-Functions 25
For the case that k = 2 we describe the tau functions
τ0
0 = T
(
(z1 − z2)2
4z1z2(z1 + z2)2
n∑
i=1
Tii(z1)Tii(z2) +
∑
i6=j
1
4z1z2
Tii(z1)Tjj(z2)
−
∑
i6=j
1
(z1 + z2)2
Tij(z1)Tji(z2)
)
,
τ εk
εk
= T
(
(z1 − z2)2
4z1z2(z1 + z2)2
n∑
i=1
Tii(z1)Tii(z2) +
∑
i6=j
(−)δik+δjk
1
4z1z2
Tii(z1)Tjj(z2)
−
∑
i6=j
(
−z1
z2
)δik
(
−z2
z1
)δjk 1
(z1 + z2)2
Tij(z1)Tji(z2)
)
,
τ0
εi−εj
= T
(
z2 − z1
2(z1 + z2)
(
1
z2
Tij(z1)(Tii(z2) + Tjj(z2))−
1
z1
(Tjj(z1) + Tii(z1))Tij(z2)
)
+
∑
k 6=i,j
(
1
z1 + z2
(Tik(z1)Tkj(z2) + Tik(z2)Tkj(z1))−
1
2z2
Tij(z1)Tkk(z2)
− 1
2z1
Tkk(z1)Tij(z2)
))
,
τ
εj
εi = T
(
z2 − z1
2z1z2(z1 + z2)
(Tij(z1)(Tii(z2)− Tjj(z2)) + (Tjj(z1)− Tii(z1))Tij(z2))
+
∑
k 6=i,j
(
1
z1 + z2
(
1
z2
Tik(z1)Tkj(z2) +
1
z1
Tik(z2)Tkj(z1)
)
− 1
2z1z2
(Tij(z1)Tkk(z2)− Tkk(z1)Tij(z2))
))
.
A Appendix
In this appendix we want to proof the equations (4.5)–(4.10). For the proof of these equations
we need the following lemma:
Lemma 1.
ReszP (xi, t, ∂i)exiz(Q(x′i, t
′, ∂′i)e
−x′iz)T =
∑
j
Rj(xi, t)Sj(x′i, t
′)
if and only if(
P (xi, t, ∂i)Q(xi, t
′, ∂i)∗
)
− =
∑
j
Rj(xi, t)∂−1
i Sj(xi, t
′)
This Lemma is a consequence of Lemma 4.1 of [9]. We rewrite (4.4):
Res
z
P+(0)(α, β, x, t, u, z)R+(0)(α, β, z)Q+(t, z)ex0z(P−(0)(γ, δ, y, s, v,−z)
×R−(0)(γ, δ,−z)Q−(s,−z)e−y0z)T = Res
z
P+(1)(α, β, x, t, u, z)R+(1)(α, β, z)
×Q+(u, z)ex1z(P−(1)(γ, δ, y, s, v,−z)R−(1)(γ, δ,−z)Q−(v,−z)e−y1z)T . (A.1)
First, applying the Lemma to (A.1) gives
(P+(0)(α, β, x0, x1, t, u, ∂0)R+(0)(α, β, ∂0)Q+(t, ∂0)(P−(0)(γ, δ, x0, y1, s, v, ∂0)
26 H. Aratyn and J. van de Leur
×R−(0)(γ, δ, ∂0)Q−(s, ∂0))∗)− = Res
z
P+(1)(α, β, x0, x1, t, u, z)R+(1)(α, β, z)Q+(u, z)
× ex1z∂−1
0 (P−(1)(γ, δ, x0, y1, s, v,−z)R−(1)(γ, δ,−z)Q−(v,−z)e−y1z)T .
Now putting s = t, u = v, x1 = y1, we obtain
(P+(0)(α, β, x0, x1, t, u, ∂0)P−(0)(α, β, x0, y1, s, v, ∂0)∗)− = 0, (A.2)(
P+(0)(α, β, x0, x1, t, u, ∂0)S(∂0)P−(0)(α−
n∑
j=1
εj , β −
n∑
j=1
εj , x0, y1, s, v, ∂0)∗
)
−
= 0. (A.3)
Since
P±(0)(α, β, x0, x1, t, u, ∂0) = I +
∞∑
j=1
P
±(0)
j (α, β, x0, x1, t, u)∂
−j
0
P±(1)(α, β, x0, x1, t, u, ∂1) =
∞∑
j=1
P
±(1)
j (α, β, x0, x1, t, u)∂
−j
1 ,
this implies (4.5), one thus also has:
P
−(0)
1 (α, β, x0, x1, s, v)T = P
+(0)
1 (α, β, x0, x1, s, v). (A.4)
If we apply the Lemma in the other way we obtain
(P+(1)(α, β, x0, x1, t, u, ∂1)R+(1)(α, β, ∂1)Q+(u, ∂1)(P−(1)(γ, δ, y0, x1, s, v, ∂1)
×R−(1)(γ, δ, ∂1)Q−(v, ∂1))∗)− = Res
z
P+(0)(α, β, x0, x1, t, u, z)R+(0)(α, β, z)
×Q+(t, z)ex0z∂−1
1 (P−(0)(γ, δ, y0, x1, s, v,−z)R−(0)(γ, δ,−z)Q−(s,−z)e−y0z)T . (A.5)
Now taking s = t, u = v and y0 = x0, we find
(P+(1)(α, β, x0, x1, t, u, ∂1)R+(1)(α− γ, β − δ, ∂1)P−(1)(γ, δ, x0, x1, t, u, ∂1)∗)−
=Res
z
P+(0)(α, β, x0, x1, t, u, z)R+(0)(α−γ, β−δ, z)∂−1
1 P−(0)(γ, δ, x0, x1, t, u,−z)T. (A.6)
Now choose γ − α =
n∑
i=1
εi, then
R+(0)(α− γ, β − δ, z) =
n∑
i=1
(−)i+1z−1Eii = S(z)−1.
Since we have assumed that (4.1) holds for j = 0
|δ − β| = |γ − α| = n.
Now choose δ − β =
n∑
i=1
εi, then (A.5) turns into
P+(1)(α, β, x0, x1, t, u, ∂1)S(∂1)
× P−(1)
(
α+
n∑
i=1
εi, β +
n∑
i=1
εi, x0, x1, t, u, ∂1
)∗
= S(∂1)−1,
thus we obtain (4.6).
Clifford Algebra Derivations of Tau-Functions 27
Differentiate the bilinear identity (A.1) to t(a)
j , one obtains
Res
z
(
∂P+(0)(α, β, x, t, u, z)
∂t
(a)
j
+ P+(0)(α, β, x, t, u, z)zjEaa
)
R+(0)(α, β, z)
×Q+(t, z)ex0z(P−(0)(γ, δ, y, s, v,−z)R−(0)(γ, δ,−z)Q−(s,−z)e−y0z)T
= Res
z
∂P+(1)(α, β, x, t, u, z)
∂t
(a)
j
R+(1)(α, β, z)Q+(u, z)ex1z
× (P−(1)(γ, δ, y, s, v,−z)R−(1)(γ, δ,−z)Q−(v,−z)e−y1z)T . (A.7)
Now using the Lemma and choosing α = γ, β = δ, s = t, u = v and x1 = y1 this gives the
familiar Sato–Wilson equations (4.7). Taking j = 1 one obtains
n∑
i=1
∂P+(0)(α, β, x0, x1, t, u, ∂0)
∂t
(i)
1
= [∂0, P
+(0)(α, β, x0, x1, t, u, ∂0)].
Return to (A.7) and use the Lemma in the opposite way and choosing, γ − α = δ − β =
n∑
i=1
εi,
s = t, u = v and x0 = y0 this gives:
Res
z
P+(0)(α, β, x, t, u, z)zjEaaS(z)−1∂−1
1 P−(0)
(
α+
n∑
i=1
εi, β +
n∑
i=1
εi, x, s, v,−z
)T
=
∂P+(1)(α, β, x, t, u, ∂1)
∂t
(a)
j
S(∂1)P−(1)
(
α+
n∑
i=1
εi, β +
n∑
i=1
εi, y, s, v,−z
)∗
using (4.6) one deduces:
∂P+(1)(α, β, x, t, u, ∂1)
∂t
(a)
j
= Res
z
zj−1P+(0)(α, β, x, t, u, z)EaaS(∂1)−1
× P−(0)(α+
n∑
i=1
εi, β +
n∑
i=1
εi, x, s, v,−z)TS(∂1)P+(1)(α, β, x, t, u, ∂1). (A.8)
Note that we can rewrite (A.8) as (4.10). Since we have replaced in the wave matrices t(a)
1 , by
t
(a)
1 + x0, (A.8) for j = 1 implies
∂P+(1)(α, β, x, t, u, ∂1)
∂x0
= Res
z
P+(0)(α, β, x, t, u, z)S(∂1)−1
× P−(0)(α+
n∑
i=1
εi, β +
n∑
i=1
εi, x, s, v,−z)TS(∂1)P+(1)(α, β, x, t, u, ∂1). (A.9)
Differentiating the bilinear identity to u(a)
j gives that
Res
z
∂P+(0)(α, β, x, t, u, z)
∂u
(a)
j
R+(0)(α, β, z)Q+(t, z)ex0z
× (P−(0)(γ, δ, y, s, v,−z)R−(0)(γ, δ,−z)Q−(s,−z)e−y0z)T
= Res
z
(
∂P+(1)(α, β, x, t, u, z)
∂u
(a)
j
+ P+(1)(α, β, x, t, u, z)zjEaa
)
R+(1)(α, β, z)
28 H. Aratyn and J. van de Leur
×Q+(u, z)ex1z(P−(1)(γ, δ, y, s, v,−z)R−(1)(γ, δ,−z)Q−(v,−z)e−y1z)T .
Using the Lemma and next choosing, α = γ, β = δ, s = t, u = v and x1 = y1 we obtain
∂P+(0)(α, β, x, t, u, ∂0)
∂u
(a)
j
P+(0)(α, β, x, t, u, ∂0)−1
= Res
z
zjP+(1)(α, β, x, t, u, z)Eaa∂
−1
0 P−(1)(α, β, x, t, u,−z)T . (A.10)
In particular taking j = 1 one obtains
∂P+(0)(α, β, x, t, u, ∂0)
∂u
(a)
1
P+(0)(α, β, x, t, u, ∂0)−1
= − P
+(1)
1 (α, β, x, t, u)Eaa∂
−1
0 P
−(1)
1 (α, β, x, t, u, )T . (A.11)
Since we have replaced in the wave matrices u(a)
1 , by u(a)
1 + x1, (A.11) implies
∂P+(0)(α, β, x, t, u, ∂0)
∂x1
= −P+(1)
1 (α, β, x, t, u)∂−1
0 P
−(1)
1 (α, β, x, t, u, )TP+(0)(α, β, x, t, u, ∂0) (A.12)
which is equivalent to (4.9).
Using the Lemma in the other way one deduces for γ − α = δ − β =
n∑
i=1
εi, that
((
∂P+(1)(α, β, x, t, u, ∂1)
∂u
(a)
j
+ P+(1)(α, β, x, t, u, ∂1)∂
j
1Eaa
)
S(∂1)
× P−(1)
(
α+
n∑
i=1
εi, β +
n∑
i=1
εi, x, t, u, ∂1
)∗)
−
= 0.
Now, using (4.6) one deduces the Sato–Wilson equations (4.8). A special case of (4.8), viz j = 1
states that
n∑
i=1
∂P+(1)(α, β, x, t, u, ∂1)
∂u
(i)
1
= [∂1, P
+(1)(α, β, x, t, u, ∂1)].
Acknowledgements
This work has been partially supported by the European Union through the FP6 Marie Curie
RTN ENIGMA (Contract number MRTN-CT-2004-5652) and the European Science Foundation
Program MISGAM.
References
[1] Aratyn H., Gomes J.F., Zimerman A.H., Integrable hierarchy for multidimensional Toda equations and
topological–anti-topological fusion, J. Geom. Phys. 46 (2003), 21–47, Erratum, J. Geom. Phys. 46 (2003),
201, hep-th/0107056.
[2] Aratyn H., Gomes J.F., Zimerman A.H., On negative flows of the AKNS hierarchy and a class of defor-
mations of bihamiltonian structure of hydrodynamic type, J. Phys. A: Math. Gen. 39 (2006), 1099–1114,
nlin.SI/0507062.
[3] Cecotti S., Vafa C., Topological–anti-topological fusion, Nuclear Phys. B 367 (1991), 359–461.
http://arxiv.org/abs/hep-th/0107056
http://arxiv.org/abs/nlin.SI/0507062
Clifford Algebra Derivations of Tau-Functions 29
[4] Chen M., Liu S-Q., Zhang Y., A 2-component generalization of the Camassa–Holm equation and its solutions,
Lett. Math. Phys. 75 (2006), 1–15, nlin.SI/0501028.
[5] Dubrovin B., Geometry and integrability of topological-antitopological fusion, Comm. Math. Phys. 152
(1993), 539–564, hep-th/9206037.
[6] Date E., Jimbo M., Kashiwara M., Miwa T. Operator approach to the Kadomtsev–Petviashvili equation.
Transformation groups for soliton equations. III, J. Phys. Soc. Japan 50 (1981), 3806–3812.
[7] Harnad J., Orlov A.Yu., Fermionic construction of partition functions for two-matrix models and perturba-
tive Schur function expansions, J. Phys. A: Math. Gen. 39 (2006), 8783–8809, math-ph/0512056.
[8] Jimbo M., Miwa M., Miwa T., Solitons and infinite dimensional Lie algebras, Publ. RIMS, Kyoto Univ. 19
(1983), 943–1001.
[9] Kac V.G., van de Leur J.W., The n-component KP hierarchy and representation theory, J. Math. Phys. 44
(2003), 3245–3293, hep-th/9308137.
[10] ten Kroode F., van de Leur J., Bosonic and fermionic realizations of the affine algebra ĝln, Comm. Math.
Phys. 137 (1991), 67–107.
[11] van de Leur J., Twisted GLn loop group orbit and solutions of the WDVV equations, Int. Math. Res. Not.
11 (2001), 551–574, nlin.SI/0004021.
[12] van de Leur J.W., Martini R., The construction of Frobenius manifolds from KP tau-functions, Comm.
Math. Phys. 205 (1999), 587–616, solv-int/9808008.
[13] Liu S-Q., Zhang Y., Deformations of semisimple bihamiltonian structures of hydrodynamic type, J. Geom.
Phys. 54 (2005), 427–453, math.DG/0405146.
[14] Okounkov A., Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), 57–81,
math.RT/9907127.
[15] Ueno K., Takasaki K., Toda lattice hierarchy, Adv. Stud. Pure Math. 4 (1984), 1–95.
http://arxiv.org/abs/nlin.SI/0501028
http://arxiv.org/abs/hep-th/9206037
http://arxiv.org/abs/math-ph/0512056
http://arxiv.org/abs/hep-th/9308137
http://arxiv.org/abs/nlin.SI/0004021
http://arxiv.org/abs/solv-int/9808008
http://arxiv.org/abs/math.DG/0405146
http://arxiv.org/abs/math.RT/9907127
1 Introduction
2 Semi-infinite wedge space and Clifford algebra
3 Tau functions as matrix elements and bilinear identities
4 Wave functions and pseudo-differential equations
5 First reduction and the generalized AKNS model
6 AKNS and the two-component Camassa-Holm model
7 A second reduction and the Cecotti-Vafa equations
8 Homogeneity
9 Explicit construction of solutions in the AKNS case
10 Explicit construction of solutions in the Cecotti-Vafa case
A Appendix
References
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