Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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genus g . A change of variables permits integration
by quadratures,
q
i
ðtÞ¼
#½
2i1
ð0Þ#½
2i1
ðz
0
D þ 2
ffiffiffiffiffiffiffi
1
p
tUÞ
#½0ð0Þ#½0ðz
0
D þ 2
ffiffiffiffiffiffiffi
1
p
tUÞ
where z
0
, U 2 C
g
are constant vectors, J denotes the
Riemann theta function of X,
k
(k = 1, ...,2g) are
theta characteristics and D is the Riemann constant.
While these are technical objects of classical
Riemann function theory whose detailed definition
is best found in a textbook (see, e.g., Mumford
(1984)), the point here is that the motion is
linearized along the line with direction U, on the
hyperelliptic Jacobian Jac(X), which is a 2
gþ1
:1
cover of the phase space.
A yet deeper fact links the integrable Hamiltonian
motion and the (soliton) PDE, namely the statement
that
P
gþ1
i = 1
(e
i
q
2
i
þ p
2
i
) = u(t
1
, t
3
) solves the KdV
equation, where the variables are renamed as
x = t
1
, t = t
3
to denote two of the g commuting
Hamiltonian flows.
The Neumann system as well allows us to uncover
another deep relation between dynamics and geo-
metry, namely the moduli aspect: on the one hand,
Mumford (1984) used the Neumann system to recover
the equation of the spectral curve from a vanishing
property of theta functions with characteristics,
thereby giving the first characterization of the moduli
subvariety of hyperelliptic curves in terms of thetanulls
(for any genus). On the other hand, Franc¸oise (1987)
explored the relevance of the integrable system to the
Picard–Fuchs equations. The fundamental link is
provided by Arnol’d’s theory, according to which a
set of action-angle variables (q
i
, p
i
), i = 1, ..., n,fora
completely integrable Hamiltonian system can be
calculated in terms of a basis
i
of the first homology
of the fibers, which are n-dimensional tori,
R
j
dq
i
=
ij
; hence, in the case of an algebraically
integrable system such as the Neumann example (or,
in Franc¸oise’s paper, the Kowalevski top), in principle
one can express the (coefficients of the) differential
equations satisfied by the periods in terms of the
commuting Hamiltonians, despite the fact that
periods and Hamiltonians are transcendental func-
tions the ones of the others. A more general family of
period matrices is subject to the Gauss–Manin
connection, and the question of whether its general
abelian variety is Lagrangian with respect to a
holomorphic symplectic structure on the family yields
a cubic condition on the periods (Donagi and Mark-
man 1996).
These are two major applications of PDEs to
algebraic geometry: characterizing subvarieties of
moduli spaces (of curves) and expressing the
Gauss–Manin connection acting on sections of a
Hodge-theoretic bundle over the moduli space in
terms of the evolution equations of a completely
integrable system. In the former case, the flows of
the system act on the theta functions of a (fixed)
curve; in the latter, the Hamiltonians are related,
via the action variables, to computing the mono-
dromy over the branch points of the base of the
system. The generalization of specific (e.g., hyper-
elliptic) cases is very difficult to work out and
remains largely open 40 years after the field of
integrable equations started being actively
investigated.
Before concluding this historical overview, a
beautiful theory that escaped attention is worth
mentioning. In the late nineteenth century, for
example, Baker (1907) constructed the first genus-2
solutions of the KdV equation, although he was
apparently not aware of the equation itself; in the
process, he also defined what is known as the Hirota
bilinear operator, a device introduced by R Hirota
in the 1970s to capture an equivalent version of the
KdV, or the more general Kadomtsev–Petviashvili
(KP) equation,
ðu
t
6uu
x
þ u
xxx
Þ
x
¼ u
yy
Just as the Lax pair allows for a linearization of the
isospectral deformations, Hirota’s bilinear form
reveals the representation-theoretic (and algebro-
geometric) nature of the equations, via the vanishing
of a natural pairing on a pair of solutions, besides
providing a formula for exact solutions; the defini-
tion of the bilinear operation is the following: for
functions F and G,
D
t
n
F G ¼
@
@t
0
n
@
@t
n
Fð
t ÞGðt
0
Þ
t¼t
0
t ¼ðt
1
; t
2
; ...Þ
so that Hirota’s direct method gives the following
solution: set u = 2(@
2
=@x
2
) log F, then
KdV , D
x
D
t
þ D
4
x
F F ¼ 0
KP , D
2
x
D
4
x
þ 3D
2
y
4D
x
D
t
F F
2F
2
¼ 0
Baker was intent on generalizing the properties of
the Weierstrass } function. He focused on genus 2
(and obtained partial results for general genus), in
which case any curve is hyperelliptic,
f :
2
¼
2gþ1
þ a
2g
2g
þþa
0
and used a suitable basis of holomorphic differen-
tials particular to the hyperelliptic case, whose
Integrable Systems and Algebraic Geometry 67
integrals give abelian coordinates z
i
that happen to
be dual to the KdV flows,
!
1
¼
d
2
;!
2
¼
d
2
; ...;!
g
¼
g1
d
2
to characterize the genus-2 theta function by
differential equations (equivalent to the KdV hier-
archy), as well as give the quartic equation for the
Kummer surface in P
3
, namely the 2:1 image of the
Jacobian of the curve mapped by the divisor 2,
that is, by a basis of the space of theta functions
with second-order characteristics, simply as the
determinant of
a
0
1
2
a
1
2}
11
2}
12
1
2
a
1
ða
2
þ 4}
11
Þ
1
2
a
3
þ 2}
12
2}
22
2}
11
1
2
a
3
þ 2}
12
ða
4
þ 4}
22
Þ 2
2}
12
2}
22
20
2
6
6
4
3
7
7
5
where
}
ij
ðzÞ¼
@
2
@z
i
@z
j
log ðzÞ
and the function, defined in analogy to the genus-1
case, is proportional to the Riemann theta function.
To summarize this introduction, the exchange
between algebraic geometry (the classification of
algebraic varieties) and dynamical systems has been
extremely fruitful in either direction: algebraic
geometry surprisingly provides exact solutions to
evolution equations that have special algebraic
symmetries (and arise in nature!), and conversely
those very evolutionary equations yield the structure
of particularly complicated varieties, by characteriz-
ing their (rational) functions.
Isospectral Deformations
The isospectral deformations in question have been
encoded by Lax-pair equations, which take their
name from Peter D Lax, who gave a version of the
KdV equation in such form.
Lax pairs enter in two essentially different ways in
the theory of integrable systems. The evolution
equations take the form: @
t
n
L = [B, L], where
t
1
, t
2
, t
3
, ... is a sequence of commuting time flows,
L is an operator whose coefficients depend on time,
and B is another operator of the same kind; since
heuristically this is the infinitesimal version of the
equation L(t) = U(t)
1
L(0)U(t) (with B = U
1
@
t
U),
the spectrum of L is preserved and provides
conserved quantities; in fact, Moser (1980) specu-
lated that every completely integrable system might
have such a form.
In the form that immediately yields a hierarchy of
PDEs, the (hierarchy of) deformations pertain to a
ring of (formal) pseudodifferential operators, where
the variable x = t
1
is singled out and @ denotes
differentiation with respect to x:
LðtÞ2D¼
X
n
j¼0
u
j
ðxÞ@
j
; u
j
analytic near x ¼ 0
()
P¼
X
n
1
u
j
ðxÞ@
j
()
The multiplication rule that makes P into a ring (in
fact, a C-algebra) is composition:
@ u ¼ u@ þ u
0
@
1
u ¼ u@
1
u
0
@
2
þ u
00
@
3
We normalize L by an automorphism of D
(generated by a change of variable and conjugation
by a function)
L ¼ @
n
þ u
n2
ðxÞ@
n2
þþu
0
ðxÞ
In P any (normalized) L has a unique nth root,
n = ord L, of the form L= @ þ u
1
(x)@
1
þ
u
2
(x)@
2
þ. Finally, the deformation equations,
@
t
n
L ¼ ½ðL
n
Þ
þ
; L
define the KP hierarchy, which takes its name from
the first nontrivial deformation equation, known as
the KP equation encountered above, if we set
x = t
1
, y = t
2
, t = t
3
(notice that this reduces to KdV,
up to rescaling, when the solution is independent
of y). The algebro-geometric solutions are those
with the property that only a finite number of time
evolutions are independent. This turned out to be
equivalent to a classical problem of elementary
differential algebra, known as the Burchnall–
Chaundy problem after the two co-authors who
solved it in the 1920s.
The Burchnall–Chaundy problem: which L(x)’s
have centralizer C
D
(L) that is larger than a
polynomial ring C[L
1
], L
1
2D? The key to the
solution is the following fact (which clearly does
not hold for operators in more than one variable,
or finite-dimensional operators such as matrices):
if ord L > 0andA, B 2Dboth commute with L,
then [A, B] = 0; in particular, C
D
(L)iscommuta-
tive, hence every maximal-commutative subalgebra
of D is a centralizer. It was proved in the early
1900s by I Schur that C
D
(L) = {
P
N
1
c
j
L
j
, c
j
2 C} \
D. It follows that centralizers are rings of affine
curves: their transcendence degree over the field of
coefficients is 1, and Spec C(L) can be regarded as
an affine curve X
0
(with natural compactification
68 Integrable Systems and Algebraic Geometry
X by a smooth point at infinity). Burchnall and
Chaundy proceeded to show that the rings of
operators whose orders are not all multiples of a
fixed integer >1, and having the same spectral
curve X (up to isomorphism), correspond to line
bundles over X (more precisely, rank-1 torsion-
free sheaves); thus, the hierarchy of evolutions
linearizes on Jac X, as indicated by the examples
treated above.
In this setting, it has been very challenging to
generalize the integrable flows, both to the higher-
rank and to the higher-dimensional case. When all
the operators in the commutative ring have order
divisible by an integer r > 1, their common kernel
defines a rank-r vector bundle over the spectral
curve, and although the theory in principle is
similar to the case of line bundles, there are no
explicit formulas for solution. On the other hand,
in order that the spectrum be a variety X of
dimension d > 1 rather than a curve, it is natural to
seek commutative rings of partial differential
operators in d variables; but again, while some
constructions work in principle, explicit formulas
are elusive.
The form in which Lax pairs occur for finite-
dimensional Hamiltonian systems is quite differ-
ent: here what is preserved is the spectrum of a
finite-dimensional linear operator, a matrix. The
first examples, from which the theory took off,
were inspired guesses. The Neumann system
described above fits in the following theory:
Moser (1980) showed that the Neumann system
together with other important classical examples
are special cases of rank-2 perturbations (since
(2 = dimhp, qi)) which preserve the spectrum of a
matrix
L ¼ A þ aq q þ bq p þ cp q þ dp p
where A is a fixed constant matrix which can be
normalized to a diagonal, diag(e
1
, ..., e
gþ1
),
det
ab
cd
6¼ 0
and u v denotes the matrix [u
i
v
j
]. The symplectic
structure is the standard ! =
P
dp
i
^ dq
i
so that a
Hamiltonian H defines a flow
_
q
i
¼
@H
@p
i
;
_
p
i
¼
@H
@q
i
and
@G
@t
¼fH; Gg¼
X
@H
@p
i
@G
@q
i
@G
@p
i
@H
@q
i
The Hamiltonian flow of
H ¼
1
2
ahBq; qiþðb þ cÞhBq; piþdhBp; pi
ad bc
2
X
i6¼j
b
i
b
j
e
i
e
j
ðq
i
p
j
q
j
p
i
Þ
2
!
(where B = diag(b
1
, ..., b
gþ1
) is any fixed diagonal
matrix) is equivalent to the Lax-pair equation
_
L = [M, L], where M is a suitable matrix:
M ¼
1
2
ðb cÞ½b
i
ij
þðad bcÞ
b
i
b
j
e
i
e
j
ðq
i
p
j
q
j
p
i
Þ
The Weinstein–Aronszajn formula
det I
n
X
r
i¼1
i
i
!
¼ det
I
r
½h
i
;
j
i
(where each of the
1
, ...,
r
,
1
, ...,
r
is a
(g þ 1 = n)-vector) gives for the spectral invariants
lðÞ
eðÞ
¼
detð LÞ
detð AÞ
¼ detðI ðð AÞ
1
qÞðaq þ bpÞ
ðð AÞ
1
pÞðcq þ dpÞÞ
¼ detðI
2
W
ðq; pÞÞ
with
W
ðq; pÞ¼
hð AÞ
1
q; qihð AÞ
1
q; pi
hð AÞ
1
q; pihð AÞ
1
p; pi
"#
ab
cd
and det (I W
(q, p)) = 1 tr W
þ det W
= 1
(q, p), defining the rational function
.
Moser also showed that the system is completely
integrable and linearizes on the (generalized) Jaco-
bian of the curve
2
= e
2
()
(x, y). Letting a = 1,
b = c = 1, d = 0 gives the Neumann system.
The dilation q 7!q gives a Lax pair with
a parameter, A 7!A þ
2
q q þ (q p p q),
which makes the spectral curve look more natural.
Indeed,
Remark (Adler and van Moerbeke 1980). The
Neumann flow is equivalent to the Lax pair:
_
L
1
= [M
1
, L
1
], where L
1
= A
2
þ (q p p q) þ
q q and M
1
= A þ q p p q. Moreover, the
Hamiltonians are of Adler–Kostant–Symes (AKS)
type, namely projections (with respect to an ad-
invariant inner product) of gradients of orbit-invariant
functions to half of the splitting of a Lie algebra.
Integrable Systems and Algebraic Geometry 69
Specifically, {
P
N
1
A
j
j
jA
j
2 gl(n, C)} = K N,with
K = {
P
N
0
A
j
j
} and N = {
P
1
1
A
j
j
}; if the inner
product is hA, Bi=
P
iþj = 1
tr A
i
B
j
, the dual of N
can be identified with K = K
?
, and the Hamiltonian
for the Neumann flow can be taken to be
H = (1/2)(L
1
2
)
2
,
3
I
gþ1
under the Lie–Poisson
brackets and suitable reduction. The flows linearize
on the Jacobian of the (hyperelliptic) curve
det(L
1
) = 0.
It is possible to recover the link between the finite
and infinite integrable systems (Neumann and KdV)
mentioned in the introductory overview, if we notice
that squared eigenfunctions for the Lax operator
L = L
2
= @
2
þ u become algebraic on the spectral
curve: Dubrovin et al. (2001) introduced the Baker
function, namely the unique function (x, P) with
the following properties:
(i) For jxj sufficiently small it is meromorphic on X n
{P
1
}, with pole divisor bounded by = P
1
þþ
P
g
, independent of x, such that h
0
( P
1
) = 0,
and near P
1
(x, P)e
xz
= 1 þ O(z
1
) is holo-
morphic, with z chosen to be
1=2
in our case.
(ii) We let be the unique meromorphic differential
with zeros on and a double pole of the
form ( þ holomorphic)dz
1
at P
1
. Note:
(1) that Riemann–Roch show that is unique.
(2) We also get a characterization of the dual
Baker function, defined as (x, P) in the
hyperelliptic case where is the involution
(, ) 7!(, ), as meromorphic on X n {P
1
}
with poles bounded by
0
and behavior e
xz
(1 þ
O(z
1
)) near P
1
, where þ
0
are the 2g zeros of .
(3) Furthermore, =d=W( , ), where W is
the Wronskian (with respect to the variable x).
Then, upon fixing a meromorphic function h,
normalized at P
1
, h =
1=2
þentire, with g þ 1
fixed poles distinct from , we have:
If
j
¼Res
e
j
h; q
j
¼
ffiffiffiffi
j
p
ðx; e
j
Þ; p
j
¼
ffiffiffiffi
j
p
ðx; e
j
Þ;
then
gþ1
j¼1
q
2
j
¼1;
gþ1
j¼1
q
j
p
j
¼0;
gþ1
j¼1
ðe
j
q
2
j
þp
2
j
Þ¼
uðxÞ and fq
j
; p
j
g satisfy the Neumann system:
Indeed, the constraints follow from the ‘‘residue
theorem’’ applied to the differential h (it has
a residue of 1atP
1
); the differential equations
€
q
j
= e
j
q
j
uq
j
follow from the assumption
L = .
The function u = 2
P
gþ1
k = 1
(
P
l6¼k
e
l
)q
2
k
, evolving
under suitable abelian flows, is a solution of the
KdV equation; the ‘‘times’’ of the KdV hierarchy are
linear combinations of the Neumann Hamiltonians;
more precisely, of the invariant vector fields deter-
mined by the tangent directions to the image of X in
JacX, with Abel map normalized at P
1
, at some
point P: D
P
=
P
g
k = 1
(P)
gk
D
k
.
The other way around (Moser–Trubowitz,
McKean–van Moerbeke),
If L = @
2
þ u(x) is a finite-gap operator and
e
1
, ..., e
gþ1
are among the 2g þ 1 edges of the
gaps, there exist constants
1
, ...,
gþ1
so that
the functions p
j
(x) =
ffiffiffiffi
j
p
(x, e
j
) satisfy
P
gþ1
1
p
2
j
(x) 1. Since L
j
= e
j
j
, the p
j
(x) solve the
Neumann system.
The squared eigenfunctions also provide a natural
interpretation for Moser’s Lax pair. If V
is the
kernel of L , then the Baker function (x, P) and
its dual (x, P) give a basis of V
except at the
branch points (e
i
, 0) where = . But then the
normalized basis of V
is related to , by a
constant matrix:
y
0
y
1
¼ C
while
B
¼
0
0
if B is the differential operator of the Burchnall–
Chaundy ring corresponding to multiplication by ,
so that
VU
WV
T
¼ M
B
¼ C
0
0
C
1
By evaluating at x = 0, we find:
C ¼
1
W
0
0
x¼0
with W =
0
0
. Finally, we calculate:
C
0
0
C
1
¼
W
0
þ
0
2
0
0
2 ð
0
þ
0
Þ
so that U() =
0
þ
0
, V() = 2 , W() = 2
0
0
are polynomials like the entries of W
(q, p) e
2
(),
and the fact that UW þV
2
does not depend on x
expresses the fact that W= constant.
An object that links the two distinct occurrences
of Lax pairs is Sato’s infinite-dimensional Grass-
mann manifold. One particular model will serve as
illustration, with more general settings covered by
Dickey (2003). Sato defined a one-to-one correspon-
dence between cyclic D-submodules I of P, namely
of the type I = DS (which turns out to be equivalent
to the property: P = IP
(1)
), and subspaces of a
ring of formal power series, which make up an
infinite-dimensional Grassmann manifold, more
70 Integrable Systems and Algebraic Geometry
precisely elements of Gr
;
, the ‘‘big cell.’’ This way,
KP can be viewed as deformation of D modules.
There are two ways to set up the Grassmannian:
(1) more direct as a limit of finite-dimensional
Grassmannians; (2) more intrinsic, using the rings
DP.
1. Let dimV = m þ n = N, Gr(m, V) = {m frames
in V}=GL(m) ,!P( ^
m
V) via
(0)
, ...,
(m1)
7!
(0)
^
^
(m1)
.
If we fix a basis e
0
, ..., e
N1
of V,and
write a frame in coordinates,
(i)
=
0, i
e
0
þþ
N1, i
e
N1
, then
ð0Þ
^^
ðm1Þ
¼
X
0‘
0
<<‘
m1
<N
‘
0
...‘
m1
e
‘
0
^^e
‘
m1
with
‘
0
...‘
m1
¼detð
‘
i
;j
Þ
i;j¼0;...;m1
A point in the ambient P(^
m
V) lies in the embedded
Gr(m,V) , its projective coordinates
‘
0
...‘
m1
(0
‘
i
< N) satisfy the Plu¨ cker relations (PRs):
X
m
i¼0
ð1Þ
i
k
0
...k
m2
‘
i
‘
0
...
^
‘
i
...‘
m
¼ 0
Therefore,
Grðm; VÞ¼ð
f
Grðm; VÞnf0gÞ=GLð1Þ
where
f
Gr(m, V) = {(
Y
)
Y
mN
satisfying the PRs} is a line
bundle over Gr(m, V), Y is a Young diagram con-
sisting of rows
‘
m1
ðm 1Þ
.
.
.
‘
1
1
‘
0
so it is contained in the rectangle
mN
.
For the commutative diagram:
f
Grðm
0
; N
0
Þ
project
!
f
Grðm; NÞ
#
identity # identity
f
Grðm
0
; N
0
Þ
embed
-
f
Grðm; NÞ
the following facts can be checked. Let
m m
0
, n n
0
, N
0
= m
0
þ n
0
:
(i) if (
0
Y
)
Y
m
0
N
0
satisfies PRs, so does its restric-
tion to Y’s within
mN
;
(ii) if (
Y
)
Y
mN
satisfies PRs, so does (
0
Y
)
Y
m
0
N
0
where
0
Y
= 0 unless Y
mN
.
These facts make it possible to define: Gr =
(
f
Grn{0})=GL(1), where
f
Gr = {(
Y
)
Y all Young diagrams
satisfying all PRs}
f
Gr
project
!
f
Grðm; NÞ
"
dense # identity
f
Gr
fin
embed
-
f
Grðm; NÞ
and
f
Gr
fin
¼fðÞ2
f
Gr :
Y
¼ 0 for almost all Yg
¼
[
m;N
f
Grðm; NÞ
The KP time deformations are defined as follows:
Y
ðtÞ:¼
X
allY
0
Y
0
=Y
ðtÞ
Y
0
where
Y
0
=Y
ðtÞ:¼detðp
‘
0
i
‘
j
ðtÞÞ
p
0
ðtÞ¼1; p
n
ðtÞ:¼
X
1
þ2
2
þ3
3
þ¼n
t
1
1
t
2
2
...=ð
1
!
2
!...Þ
Write
Y=;
as
Y
, where
Y
(t)= det(p
‘
i
j
(t)) are
the Schur functions.
To connect with the KP hierarchy, let
w
n
ðx; tÞ :¼ð1Þ
n
n;1
ðx þ tÞ
;
ðx þ tÞ
where x þt=(x þt
1
, t
2
,...), and S:= 1þw
1
(x, t)
@
1
þ. Then L=S@S
1
satisfies the KP hierarchy,
namely @
t
n
S=B
n
SS@
n
, where B
n
:= (S@
n
S
1
)
þ
,()
[@
t
n
B
n
, @
t
k
B
k
]=0() @
t
n
L=[(L
n
)
þ
,L].
Note The Plu¨ cker coordinate
;
(t) =
P
allY
Y
(t)
Y
= (, t) is a generating function for the Plu¨cker
coordinates,
Y
(t) =
Y
(@
t
)
;
(t), where
@
t
:¼
@
@t
1
;
1
2
@
@t
2
;
1
3
@
@t
3
; ...
Now by reducing to Gr(m, N) and checking that
every
Y
(t) satisfies PRs, we have a dynamical
system on
f
Gr.
Conclusion (Sato). Although any f (t) 2 C[[t
1
, t
2
, ...]]
admits a formal expression of the form
P
Y
c
Y
Y
(t),
where the coefficients are
c
Y
¼
Y
ð@
t
Þf ðtÞj
t¼0
it represents the function for some 2
f
Gr () its
coefficients satisfy the following PRs:
X
m
i¼0
ð1Þ
i
k
0
...k
m1
‘
i
@
t
2
‘
o
...
^
‘
i
...‘
m
@
t
2
¼ 0
which is the KP hierarchy in Hirota bilinear form.
Integrable Systems and Algebraic Geometry 71
2. Let
V :¼
P
Px
ffiP
const
¼
X
1<i<<1
a
i
@
i
; a
i
2 C
()
equipped with the induced filtration V
(i)
by order,
induced by
P
ðiÞ
¼
X
1<ki
a
k
@
k
; a
k
2 C
()
and define
Gr ¼fvector subspaces W of V
s.t. dimðW \ V
ð0Þ
Þ¼dim V=ðW þ V
ð0Þ
Þ < 1g
‘‘same size’’ as the reference subspace {
P
0
c
e
:
c
2 C} = V
(0)
.
The correspondence between such a W and a
cyclic submodule of P is given as follows:
I 7!W ¼ S
1
V
ð0Þ
¼fv 2 V : Iv V
ð0Þ
g
W 7!I ¼fA 2P: AW V
ð0Þ
g
Generic points of particular interest in construct-
ing KP solutions make up the ‘‘big cell’’:
Gr
;
open dense
Gr () V ¼ W V
ð0Þ
()
;
6¼ 0 and a function can
be defined as above
In standard basis of V, e
i
:= @
i1
mod Px, i 2 Z,
the action
xe
i
¼ði þ 1Þe
iþ1
@e
i
¼ e
i1
:
gives V a P-module structure. Let be the shift
operator: @e
i
= e
i1
; then
ðtÞ¼e
ðt
1
þt
2
2
þÞ
so, this linearizes the flows!
This survey would not be complete without an
example of the formula that links the and the
theta function; more general statements and groups of
symmetries can be found in Dickey (2003). A solution
of the KP hierarchy can be expressed in terms of the
function
W
associated with an element W of Gr(H), in
the model Gr(H), where H = L
2
(S
1
), H = H
þ
H
with standard basis H
þ
= h1, z, z
2
, ...i, H
=
hz
1
, z
2
, ...i and p
the projections,
W
(g) = det (
g
p
þ
g
1
(p
þ
j
W
)
1
), where g = e
t
i
z
i
. The associated
Baker function
W
(g, z ) is a function of the form
W
ðg; zÞ¼gðzÞ 1 þ
X
1
i¼1
a
i
z
i
!
with a
i
2 C[[t
1
, t
2
, ...]] for each i, such that the map
z 7!
W
(g, z ) is an element of g
1
W.If = 1 þ
P
1
i = 1
a
i
z
i
, then L = @
1
is a solution of the KP
hierarchy.
Moreover,
g
1
W
ðg;Þ¼
W
t
1
W
ððt
ÞÞ
This is the analog of the expression for the Baker
function in terms of the theta function, when W
corresponds to an element of the Jacobian of the
spectral curve via the Krichever map
ðx; PÞ¼exp x
Z
xa
#ðUx þ AðPÞAðDÞÞ#ðAðDÞþÞ
#ðAðPÞAðDÞÞ#ðUx AðDÞÞ
where P 2 , A() is the Abel map, the Riemann
constant, U 2 C
g
a suitable vector, D a generic
divisor of points P
1
, ..., P
g
2 , a differential of
the second kind, and a a constant depending on the
curve. For the KdV solutions, the condition on W 2
Gr
;
is that z
2
W W and the solution is
u
W
ðx; t
2
; t
3
; ...Þ¼2@ log
W
ðx; t
2
; t
3
; ...Þ
In the Grassmannian formulation, the Hirota
bilinear operator mentioned in the introductory
overview makes its third and most general appear-
ance (we regard Baker’s and Hirota’s definitions as
the first two – the one based on a residue formula in
algebraic geometry, the other on the vanishing of a
differential form):
Definitions
(i) In P, it is possible to conjugate any
L= @ þ u
1
(x)@
1
þ into @ by a K = 1 þ
v
1
(x)@
1
þ,determineduptoelements
of C[@] = C
D
(@): K
1
LK = @.
(ii) We define a formal Baker function for L as the
element of the module M (the free, rank-1
P-module = space of formal expressions f = e
xz
~
f
where
~
f =
P
N
1
f
j
(x)z
j
, with generator e
xz
)such
that L = z ;sothat = Ke
xz
for K as in (i).
(iii) We say that the formal adjoint A
y
of a (formal
pseudo) differential operator A =
P
N
j = 1
u
j
(x)@
j
is A
y
=
P
N
j = 1
(@)
j
u
j
(x), and that the dual
Baker function
y
to = Ke
t
j
z
j
is the Baker
function of (L
y
); the operator which corresponds
to K in (i) is (K
y
)
1
, that is, (K
y
)
1
L
y
K
y
= @.
Then, the KP hierarchy is equivalent to the
following formula:
Res
z
ðt
0
; zÞ
y
ðt; zÞ¼0
72 Integrable Systems and Algebraic Geometry
Moreover, as proved in Dickey (2003),if
1
and
2
are formal power series of the form
1
= Ke
t
i
z
i
,
2
= Je
t
i
z
i
, for K, J 2 1 þP
(1)
, satis-
fying the condition
Res
z
@
1
i
1
@
2
i
2
@
m
i
m
¼ 0
then there exists an operator L satisfying the Lax
equations, whose wave function and adjoint wave
function are
1
,
2
, respectively.
To conclude this overview of Lax equations, we
point out that they can be viewed as zero-curvature
condition for a (formal) connection (on the trivial
bundle over the formal deformation space whose
fiber is P), rephrasing the fact that the time flows
commute and hence define time deformations; such
formulation can be found in Mulase (1984).
Symplectic Reduction and r Matrices
While the Lax-pair presentation provides natural
spectral invariants, the group/representation-
theoretic nature of integrability (sometimes referred
to as hidden symmetries) is best seen in the context
of Marsden–Weinstein reduction. We perform it in
the example of a generalization of Moser’s rank-2
perturbation; we extract the basic construction from
Adams et al. (1988). A more comprehensive treat-
ment can be found in Babelon et al. (2003).
Definition We let M
n, r
denote the space of n r
complex matrices, with n r and give M = M
n, r
M
n, r
the symplectic structure !(F, G) = tr(dF ^ dG
T
)
for F, G 2 M. A rank-r perturbation of a fixed n n
matrix A is L = A þ FG
T
.
Definition We split the formal loop algebra
g
gl(r) =
g
gl(r)
þ
g
gl(r)
where
g
gl(r)
þ
consists of r r
matricial polynomials in and
g
gl(r)
of strictly
negative formal power series. Under the pairing
hX(), Y()i= tr(X()Y())
(where the subscript
means the coefficient of
1
), the dual of
g
gl(r)
þ
is
identified with
g
gl(r)
, which therefore admits a Lie–
Poisson structure.
In sketch, we consider an action on M whose
moment map lands in
g
gl(r)
; we check that the
AKS flows on
g
gl(r)
correspond to isospectral
deformations of L = A þ FG
T
for flows on M
A
;
finally, we perform a Marsden–Weinstein reduction
for an (equivariant) GL(r) action to obtain a
completely integrable system on a symplectic leaf,
whose flows are linear on the Jacobian of the
spectral curve. We recall very briefly the general
definitions.
Moment Map
1. A smooth group action of G on a symplectic
manifold (M, !) is said to be Hamiltonian if there
exists a ‘‘moment map’’ J : M ! g
such that the
Hamiltonian vector field associated with J and a
fixed element 2 g is the same as the infinitesi-
mal action associated with . However, an
infinitesimal definition is given because in the
formal setup the group of a Lie algebra is often
delicate to define. We recall that:
2. The Lie–Poisson structure of g
is defined by
f; g
g
ðÞ¼<;½dðÞ; d ðÞ >
for ; 2C
1
ðg
Þ;2 g
where d : g
! g
(which in our situation will
always be identified with g) is defined by
<dðÞ;>¼
d
dt
ð þ tÞ
t¼0
;;2 g
Now we say that J : M ! g
is a moment map if
3. its linear dual j : g !C
1
(M) is a Lie-algebra
homomorphism; or if
4. it is a Poisson map with respect to the Lie–Poisson
structure: , 2C
1
(g
) ) {J
, J
} = J
{, }
g
.
In case we do have a Hamiltonian G-action, then
the subspace C
1
G
(M)ofG-invariant functions is a Lie
subalgebra of C
1
(M). If G acts freely and properly on
M,thenM/G is a manifold with a Poisson structure
inherited from the one on M through the identifica-
tion C
1
(M=G) ffiC
1
G
(M). The symplectic leaves of
M/G have the form M
= J
1
()=G
= J
1
(O
)=G,
where 2 g
, G
is the isotropy group of in G and
O
is the G-orbit through . The reduced manifold
M
has a natural symplectic structure !
such that
i
! =
!
,wherei : J
1
() ! M is inclusion and
: J
1
() ! M
is the natural projection taking
points to their G
-orbits.
This class of examples can be treated with the
technique of a (classical) r-matrix, as follows. Given a
linear map R : g ! g, the alternating bilinear form
[X, Y]
R
= (1/2)([RX, Y] þ [X, RY]) satisfies the Jacobi
identity , certain quadratic conditions on R are
satisfied. Assuming they are, for all pairs of invariant
functions I, J on g
,wehave{I, J }
R
= 0(where{,}
R
is the attendant (Lie–Poisson) structure). Indeed,
{I, J}
R
() = h[dI(), dJ()]
R
, i= (1/2)h[RdI(),dJ()],
iþ(1/2)h[dI(), RdJ()], i, but, for example,
h[RdI(), dJ()], i= hRdI(), ad
dJ()()i= 0.
Remark As is clear from the proof above, our
definition of invariant need only be infinitesimal,
that is, f 2 I(g
) iff <,[df (), X]> = 0 8 2 g
,
X 2 g. Of course, when we have a corresponding Lie
group the invariants are the functions which are
Integrable Systems and Algebraic Geometry 73
invariant under the natural action, such as the
symmetric functions of the eigenvalues of a matrix.
AKS Flows
For a splitting g = K N, as given above, with
g
= N
K
, an example of r-matrix is given by
R(X) = X
þ
X
(where þ, denote projection to
K, N): the Jacobi identity is straightforward to
check. As a consequence, invariants on g
are in
involution with respect to { , }
R
and these are called
AKS flows, after work done independently by AKS:
_
X = [df (
~
X)
þ
, X] = [df (
~
X)
, X], given here for the
special case in which we can identify K with K
and
~
X is the element in K
that corresponds to X 2 K.
We now proceed to the appropriate moment maps.
We generalize the constant matrix A introduced
above (isospectral deformations) by allowing multiple
eigenvalues
i
of multiplicities n
i
r, n
1
þþ
n
k
= n, so that det (A I) =
Q
k
i = 1
(
i
)
n
i
.Let
a() =
Q
k
i = 1
(
i
). We split an n r matrix F
into k blocks F
i
accordingly.
Definition/statement
(i) J
n
r
(F, G)(x
1
, ..., x
n
) =
P
n
j = 1
tr(F
j
X
j
G
T
j
) is the
moment map of the action [(g
1
, ..., g
n
)
(F, G)]
i
= (F
i
g
1
i
, G
i
g
T
i
), where g
i
2 GL(r)so
that under standard identifications J
n
r
(F , G) =
(G
T
1
F
1
, ..., G
T
n
F
n
) and restricting the action to
the diagonal subgroup {(g, ..., g)}, J
r
(F , G) =
G
T
F.
(ii) For X() 2
g
gl(r)
þ
we define (X()) = (X(
1
), ...,
X(
r
)) and obtain the exact sequence
0 ! aðÞ
g
glðrÞ
þ
!
g
glðrÞ
þ
!
g
n
r
! 0
By dualizing, and identifying g
n
r
to its dual by
using the trace componentwise, we get
ðY
1
; ...; Y
n
Þ¼
X
k
i¼1
Y
i
i
and finally check that
~
J
r
=
J
n
r
is a moment
map. By combining (i) and (ii), we get a
moment map
~
J
r
ðF; GÞ¼
X
k
i¼1
G
T
i
F
i
i
¼ G
T
ðA Þ
1
F
which becomes injective on M=H, where M is
a suitable open submanifold of M and
H = GL(n
1
) GL(n
k
) acts blockwise by
(h
i
F
i
, h
1T
i
G
i
).
(iii) We also notice that the ‘‘Moser space’’ M
A
=
{A þ FG
T
jF, G 2M}ofrank-r perturbations can
be identified with the orbit space M=G
r
, G
r
=
GL(r) acting as in (i).
To finish, we turn on the obvious AKS flows on
g
gl(r)
: the key observation is that they are isospec-
tral for the rank-r perturbation A þ FG
T
: we see that
the Poisson-commutative ring F
þ
of projected
invariants defines, by composition with
~
J
r
,a
Poisson-commutative ring F of isospectral flows on
M
n, r
M
n, r
.
Hitchin Systems
The Hitchin system, introduced in the late 1980s,
20 years later still encompasses the most general class
of ‘‘algebraically completely integrable’’ systems, which
we now discuss. In its most basic form, the concept of
‘‘algebraic completely integrable’’ (ACI) Hamiltonian
system, is an extra condition on the integrability of
classical mechanics, in the following sense.
A Hamiltonian system with n degrees of freedom,
that is, defined on a symplectic manifold M of (real)
dimension 2n is (Arnol’d–Liouville) completely
integrable if it admits n functions in involution
whose differentials are linearly independent (possi-
bly, generically on M). When M is a component of
the set of real points of an algebraic variety M
C
and
the symplectic form ! and Hamiltonian function H
are rational without poles on M, the concept of
algebraic complete integrability can be introduced.
For this condition to hold, we require that the vector
fields corresponding to the Hamiltonians in involu-
tion still have no poles on a compactification of the
fibers on M
C
.
Nonexample (Mumford 1984, x4). Consider
M ¼ R
2
;!¼ dx ^ dy; H ¼ x
4
þ y
4
Here a compactification of the fiber, the affine
curve x
4
þ y
4
= c, is the projective curve X
4
þ
Y
4
= cZ
4
, which is smooth (provided c 6¼ 0) and
has four points at infinity. The vector field X
H
defined by H, X
H
c! = dH, is tangent to the fiber
in the affine plane, but has a pole at infinity as can
be checked by a change of coordinates; 4 is the
lowest exponent for which this simple nonexample
works!
Note In the algebraically completely integrable
situation, the fibers are abelian varieties or exten-
sions of such by C
k
for some power k. This gives
rise to the issues of variations of periods over the
base mentioned in the introductory overview.
The Neumann system is ACI, with integral tori
given by the Jacobians of the spectral curves:
:
2
¼ gðÞ¼
Y
2gþ1
1
ð e
i
Þ¼UW þ V
2
74 Integrable Systems and Algebraic Geometry
where
L ¼ eðÞ
X
q
i
p
i
e
i
X
p
2
i
e
i
1 þ
X
q
2
i
e
i
X
q
i
p
i
e
i
2
6
6
6
4
3
7
7
7
5
¼
VU
W V
"#
; eðÞ¼
Y
gþ1
1
ð e
i
Þ
U ¼
Y
g
i¼1
ð
i
Þ; ð
1
; ...;
g
Þ ‘‘elliptical spherical
coordinates’’
¼ 1;
U
V þ
eigenvector: L ¼
divisor:
X
g
i¼1
ð
i
; Vð
i
ÞÞ
Hitchin (1982) devised a geometrical model of the
spectral curve, a compact algebraic curve contained in
the surface T
P
1
, and its line bundles. He also provided
subsequently (1987) a dramatic generalization.
Hitchin’s construction, in the Neumann-system
example, highlights the following objects:
L 2 H
0
(P
1
, End(E) O(g þ 1)), E rank (r = )2
bundle over P
1
;
T = total space of the line bundle O(g þ1) over P
1
;
= tautological section: P
1
! T,where
~
L
~
I 2
H
0
(T,End(
~
E)
~
O(g þ 1)) (tildes denote pullback);
: det (
~
L
~
I) = 0. The line bundle (eigenvec-
tors) is defined as the kernel of
~
L
~
I; and
the moduli space of spectral curves is a linear
system on the surface T. Fixing {e
1
, ..., e
gþ1
}in
the above example gives constraints that define it
as subsystem of a complete linear system, as well
as providing a Poisson structure on the whole
((2g 1) þ g)-dimensional manifold (base =
curves, fiber = Jacobians) which reduces to the
standard
P
dp
i
^ dq
i
. Equivalent to choosing a
section s 2 H
0
(P
1
, O(g 1) K
1
P
1
),
# r : 1
P
1
s $ðe
1
; ...; e
gþ1
Þ
E ! E O
P
1
ðg þ 1Þ$L
ð : 1Þ2P
1
Generalizations
P
1
!
Riemann surface X of genus g > 1;
E stable rank-r vector bundle over X. To give
a concrete example, we will take r = 2 and fix
det E ¼O
X
:
Hitchin’s Abelianization Program
Fact (Hitchin). Every such bundle E over X can be
realized as the direct image of a line bundle over a
spectral curve
r :I
!
X.
We introduce the moduli space M=
SU
X
(2, O
X
) = S-equivalence classes of E’s, E semi-
stable rank-2 bundle over X, det E = O
X
. The
dimension of M is 3g 3.
Hitchin (1987) proved that T
M is ACI (gener-
ically, there exist 3g 3 regular functions in involu-
tion with respect to the standard symplectic
structure, with invariant manifolds isomorphic to
Prym , where =spectral curve).
To recognize the analog of the features high-
lighted above, we recall that Kodaira–Spencer
deformation theory gives the following description
of the cotangent bundle: since a rank-r vector bundle
over X is determined by a 1-cocycle with values in
GL(r, O
X
), a first-order deformation of E is given by
a 1-cocycle with values in the associated bundle of
Lie algebras, hence by a class in H
1
(X, End(E)), so
the cotangent bundle has Serre-dual fiber
H
0
(X, E E
K).
Hitchin map (E, ) 2T
M (Higgs field, trace zero,
2 H
0
(X, End
0
(E) K)):
H: 7! det (more generally for any r 2,
tr ^
i
2 H
0
(X, K
i
)) i = 2, ..., r;
7! defines Prym ,
2
= det 2 H
0
(X, K
2
)
defines .
Explicit Hamiltonians for the Hitchin System
The cases in which X is genus 0 and 1 were solved
explicitly by Nekrasov (1996) using explicit parame-
trizations of the moduli spaces; this includes the case of
insertions (singular curves), yielding (elliptic) Gaudin
models. We report the solution for the genus-2 case
(van Geemen and Previato 1996).
Remark The map H projectivizes,
H : PH
0
ðX; End
0
ðEÞKÞ!PH
0
ðX; K
2
Þ
detðcÞ¼c
2
det
Coordinates on T
M can be given as follows :
Pic
g1
X = canonical theta divisor
:M!j2j¼P
2
g
1
E 7!D
E
¼f 2 Pic
g1
X : h
0
ðE Þ > 0g
X hyperelliptic ) is 2:1 except for g = 2 (every
point of M is fixed under the hyperelliptic
Integrable Systems and Algebraic Geometry 75
involution), where MffiP
3
. For a vector space V
the Euler sequence gives
PT
PV ffi I ¼fðx; hÞ2PV PV
: x 2 hg
In our case,
PV PV
¼j2jj2
0
j
Define six polynomial functions H
i
on P
3
P
3
by the requirement: for generic q 2 P
3
,(H
i
= 0) \
PT
q
P
3
= ‘
i
[ ‘
0
i
, the six pairs of bitangents to K\
PT
q
P
3
, where K is the Kummer surface (the
remaining 16 bitangents are cut out by the tropes.)
Recall that the Grassmannian of lines in
P
3
, Gr(2, 4), is defined by an equation
P
6
1
X
2
i
= 0
in Klein’s coordinates
ðX
1
: ... : X
6
Þ2P
5
X
1
¼ p
01
þ p
23
; X
2
¼ iðp
01
p
23
Þ
X
3
¼ iðp
02
p
13
Þ; X
4
¼ p
02
þ p
13
X
5
¼ p
03
þ p
12
; X
6
¼ iðp
03
p
12
Þ
where p
ij
= Z
i
W
j
W
j
Z
i
are Plu¨ cker’s coordinates
on the line
hðZ
0
:... : Z
3
ÞðW
0
: ... : W
3
Þi P
3
Using coordinates on the incidence variety I given
by the sections
i
of the bundle projection PT
P
3
!
P
3
,
i
: P
3
! PT
P
3
= I P
3
P
3
, q 7!(q,
i
(q)) =
(q, X
i
(q, )), explicitly given, for q = (x : y : z : t), by
1
¼ðy : x : t : zÞ;
2
¼ðy : x : t : zÞ
3
¼ðz : t : x : yÞ;
4
¼ðz : t : x : yÞ
5
¼ðt : z : y : xÞ;
6
¼ðt : z : y : xÞ
x
j
¼ X
j
ðh
i
ðqÞ; piÞ
Fact For a point q 2 P
3
, p 2 PT
q
P
3
, p 62
i
(q), the
ith Klein coordinate of the line h
i
(q), pi is zero and
p 2 ‘
i
[ ‘
0
i
, H
i
ðp; qÞ¼
X
j6¼i
x
2
j
i
j
¼ 0
with x
j
= X
j
(h
i
(q), pi).
Conclusion In an affine patch C
3
C
3
3 (q, p) =
((x : y : z : 1), (u : v : w : (xu þ yv þ zw)))
H
a
i
ðp; qÞ¼
X
j6¼i
X
j
ð
i
ðqÞ; pÞ
2
i
j
give six Hitchin Hamiltonians, any three of which
are generically independent. The H
a
i
have degree 4
in x, y, z and are homogeneous of degree 2 in
u, v, w; they Poisson-commute with respect to
dx ^ du þ dy ^ dv þ dz ^ dw.
Example An example is constituted by
2
¼ð
2
1Þð
2
4Þð
2
9Þ
ððx : y : z : 1Þ; ðu : v : w : ðxu þ yv þ zwÞÞÞ
2 A
3
A
3
H
1
¼uvð70xy 32x
3
y 18xy
3
10z 32x
2
z
þ18y
2
zÞþv
2
ð9 30y
2
16x
2
y
2
9y
4
32xy
2
16z
2
Þþu
2
ð16 40x
2
16x
4
9x
2
y
2
þ18xyz
9z
2
Þþvwð18x þ10xy
2
þ10yz 32x
2
yz
18y
3
z 32xz
2
Þþuwð32y þ10x
2
y 10xz
32x
2
z 18xy
2
z þ18yz
2
Þþw
2
ð9x
2
16y
2
þ10xyz 16x
2
z
2
9y
2
z
2
Þ
The concept of reduction and r-matrix have been
generalized to Hitchin systems. Notably, Hitchin later
showed that the Hamiltonians of the system appear as
symbols of a heat operator that corresponds to a
projectively flat connection, the quantization of the
moduli space of bundles, obtained by changing the
complex structure of the Riemann surface X.
Other Aspects
Special Functions
Special functions have also been traditionally signifi-
cant in both algebraic geometry and integrable
systems. Within the examples presented, elliptic
functions gave rise to surprisingly sophisticated the-
ories. The 1-wave solution encountered in the intro-
duction, u = 2} þ const. in the limit when one or both
periods of the Weierstrass function go to zero,
becomes exponential or rational, respectively. The
higher-genus analogs give rise to solitons, or rational
solutions. On the other hand, the KP solutions which
are doubly periodic in the x variable (‘‘elliptic
solitons’’) were classified by Krichever (cf. Dubrovin
et al. (2001)), as forming an ACI Hamiltonian system
(‘‘elliptic Calogero–Moser’’), which, 25 years later, is
still generating important work, with Hamiltonian
H ¼
X
n
i¼1
p
2
1
þ
1
2
X
i6¼j
}ðq
i
q
j
Þ
(where } is the Weierstrass function of a lattice L
with associated elliptic curve X = C=L, q 2 X the
origin) and u = 2
P
n
i = 1
}(x x
i
(t
2
, t
3
, ...)) is a solu-
tion of the KP hierarchy for suitable time flows t
j
of
the system (t
1
= x) and KP Baker function
ðx; Þ¼
ð xÞ
ðÞðxÞ
e
ððÞxÞ
The associated spectral curves have been classified in
moduli by Treibich and Verdier (cf. Treibich
(2001)); Krichever produced a two-field model as
76 Integrable Systems and Algebraic Geometry