Cohomologies of Affine Jacobi Varieties and Integrable Systems

33 Pages·0.26 MB·English

by  A. Nakayashiki

LPTHE-0002
Cohomologies of Affine Jacobi Varieties
and Integrable Systems .
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J A. Nakayashiki a and F.A. Smirnov b 0

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1 a Graduate School of Mathematics, Kyushu University
v Ropponmatsu 4-2-1,Fukuoka 810-8560,Japan
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1 b Laboratoire de Physique Th´eorique et Hautes Energies 1
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Universit´e Pierre et Marie Curie, Tour 16 1er ´etage, 4 place Jussieu
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Abstract. WestudytheaffineringoftheaffineJacobivarietyofahyper-elliptic
t
a curve. The matrix construction of the affine hyper-elliptic Jacobi varieties due
m
toMumfordisusedtocalculatethecharacteroftheaffinering. Bydecomposing
: thecharacterwemakeseveralconjecturesonthecohomologygroupsoftheaffine
v
hyper-elliptic Jacobivarieties. In the integrable systemdescribed by the family
i
X
of these affine hyper-elliptic Jacobivarieties, the affine ring is closely related to
r thealgebraoffunctionsonthephasespace,classicalobservables. Weshowthat
a
the affine ring is generated by the highest cohomology group over the action of
the invariant vector fields on the Jacobi variety.
0MembreduCNRS
1Laboratoire associ´e au CNRS.
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1 Introduction.
Ourinitialmotivationisthestudyofintegrablesystems. Consideranintegrable
system with 2n degrees of freedom, by definition it possesses n integrals in
involution. Thelevelsoftheseintegralsaren-dimensionaltori. Thisisageneral
description, but the particular examples of integrable models that we meet in
practice are much more special. Let us explain how they are organized.
The phase space is embedded algebraically into the space RN. The
M
integrals are algebraic functions of coordinates in this space. This situation
allows complexification, the complexified phase space C is an algebraicaffine
varietyembeddedintoCN. ThelevelsofintegralsinthMecomplexifiedcaseallow
the followingbeautiful description. The systems that we consider aresuch that
with every one of them one can identify an algebraiccurve X of genus n whose
moduli are defined by the integrals of motion. On the Jacobian J(X) of this
curve there is a particular divisor D (in this paper we consider the case when
this divisor coincides with the theta divisor, but more complicated situations
arepossible). ThelevelofintegralsisisomorphictotheaffinevarietyJ(X) D.
The real space RN CN intersects with every level of integrals by a com−pact
⊂
real sub-torus of J(X) D.
−
Thisstructureexplainswhythemethodsofalgebraicgeometryaresoimpor-
tant in application to integrable models. Closest to the present paper account
of these methods is given in the Mumfords’s book [1].
Let us describe briefly the results of the present paper. We study the struc-
ture of the ring A of algebraic functions (observables) on the phase space of
certain integrable model. The curve X in our case is hyper-elliptic. As is clear
from the description given above this ring of algebraic functions is, roughly, a
productofthefunctionsofintegralsofmotionbytheaffineringofhyper-elliptic
Jacobian. The commuting vectorfields defined by takingPoissonbracketswith
the integrals of motion are acting on A. We shall show that by the action of
thesevector-fieldstheringAisgeneratedfromfinitenumberoffunctionscorre-
spondingtothehighestnontrivialcohomologygroupoftheaffineJacobian. We
conjecture the formof the cohomologygroupsin everydegreeand demonstrate
the consistence of our conjectures with the structure of the ring A.
Finallywewouldliketosaythatthisrelationtocohomologygroupsbecame
clear analyzing the results of papers [2] and [3] which deal with quantum in-
tegrable models. Very briefly the reason for that is as follows. The quantum
observablesareinone-to-onecorrespondencewiththeclassicalones. Considera
matrixelementofsomeobservablebetweentwoeigen-functionsofHamiltonians.
Aneigen-functionswrittenin“coordinate”representation(for“coordinates”we
takethe anglesonthe torus)mustbe consideredasproportionalto square-root
ofthevolumeformonthe torus. Thematrixelementiswrittenasintegralwith
respectto“coordinates”,theproductoftwoeigen-functionsgivesavolumeform
onthe torus,andthe operatoritself canbe considered,atleastsemi-classically,
as a multiplier in front of this volume form, i.e. as coefficient of some differen-
tial topform on the torus which is the same as the form of one-halfof maximal
dimension on the phase space. Further, those operators which correspond to
2
“exact form” have vanishing matrix elements. This is how the relation to the
cohomologies appears.
Thepaper,aftertheintroduction,consistsoffivesectionsandsixappendices
which contain technical details and some proofs.
In section 2 we recall the standard construction of the Jacobi variety which
is valid for any Riemann surface.
An algebraic construction of the affine Jacobi variety J(X) Θ of a hyper-
−
elliptic curve X is reviewed in section 3 following the book [1]vol.II. This con-
struction is specific to hyper-elliptic curves or more generally spectral curves
[5].
In section 4 we study the affine ring of J(X) Θ using the description in
−
section 3. The main ingredient here is the character of the affine ring. To be
precise we consider the ring A corresponding to the most degenerate curve
0
y2 =z2g+1. The ring A and the affine ring A of J(X) Θ for a non-singular
f
−
X can be studied using A . It is important that A is a graded ring and the
0 0
character ch(A ) is defined. We calculate it by determining explicitly a C-
0
basis of A . The relation between A and A for a non-singular X is given in
0 0 f
Appendix E.
AsetofcommutingvectorfieldsactingonAisintroducedinsection5. This
action descends to the quotients A and A . The action of the vector fields
0 f
coincides with the action of invariant vector fields on J(X). With the help
of these vector fields we define the de Rham type complexes (C∗,d), (C∗,d),
0
(C∗,d) with the coefficients in A, A , A respectively. The complex (C∗,d)
f 0 f f
is nothing but the algebraic de Rham complex of J(X) Θ whose cohomology
−
groupsareknowntobeisomorphictothesingularcohomologygroupsofJ(X)
−
Θ. What is interested for us is the cohomology groups of (C∗,d). We calculate
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theq-Eulercharacteristicof(C∗,d)andshowthatitcoincideswiththequotient
0
ofch(A )bythecharacterch( )ofthespaceofcommutingvectorfields. Then,
0
D
by the Euler-Poincar´eprinciple,ch(A )/ch( ) is found to be expressibleas the
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D
alternatingsumofthecharactersofcohomologygroupsof(C∗,d). Decomposing
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independently theexplicitformulaofch(A )intothe alternatingsum,wemake
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conjectures on the cohomology groups of (C∗,d) which are formulated in the
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next section.
In section six and in Appendix B we study the singular homology and co-
homology groups of J(X) Θ. The Riemann bilinear relation plays an impor-
−
tant role here. We formulate conjectures on the cohomology groups of (C∗,d),
(C∗,d), (C∗,d).
0 f
Acknowledgements. This work was begun during the visit of one of the
authors (A.N.) to LPTHE of Universit´e Paris VI and VII in 1998-1999. We
express our sincere gratitude to this institution for generous hospitality. A.N
thanks K. Cho for helpful discussions.
3
2 Hyper-elliptic curves and their Jacobians.
Consider the hyper-elliptic curve X of genus g described by the equation:
y2 =f(z),
where
f(z)=z2g+1+f z2g+ +f . (1)
1 2g+1
···
The hyper-elliptic involution σ is defined by
σ(z,y)=(z, y).
−
The Riemannsurface X canbe realizedastwo-sheetedcoveringofthe z-sphere
with the quadratic branch points which are zeros of the polynomial f(z) and
.
∞
A basis of holomorphic differentials is given by:
dz
µ =zg−j , j =1, ,g.
j
y ···
ChooseacanonicalhomologybasisofX: α , ,α ,β , ,β . Thebasis
1 g 1 g
··· ···
of normalized differentials is defined as
g
ω = (M−1) µ ,
i ij j
j=1
X
where the matrix M consists of α-periods of holomorphic differentials µ :
i
M = µ , i,j =1 ,g. (2)
ij i
···
Z
αj
The period matrix
B = ω
ij j
Z
βi
defines a point B in the Siegel upper half space:
B =B , Im(B)>0.
ij ji
The Jacobi variety of X is a g-dimensional complex torus:
Cg
J(X)= .
Zg +BZg
The Riemann theta function associated with J(X) is defined by
θ(ζ)= exp 2πi 1 tmBm+tmζ ,
2
m∈Zg
X (cid:0) (cid:1)
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where ζ Cg. The theta function satisfies
∈
θ(ζ+m+Bn)=exp 2πi 1 tnBn tnζ θ(ζ),
−2 −
for m,n Zg. (cid:0) (cid:1)
∈
Consider the symmetric product of X, the quotient of the product space by
the action of the symmetric group:
X(n)=Xn/S .
n
The Abel transformation defines the map
a
X(g) J(X)
−→
explicitly given by
g pk
w = ω +∆,
j j
Xk=1∞Z
wherep , ,p arepointsofX,∆istheRiemanncharacteristiccorresponding
1 g
···
tothechoiceof forthereferencepoint. Inthepresentcase∆isahalf-period
∞
because is a branch point [1].
∞
The divisor Θ is the (g 1)-dimensional subvariety of J(X) defined by
−
Θ= w θ(w)=0 . (3)
{ | }
The mainsubject of our study is the ring A ofmeromorphic functions onJ(X)
with singularities only on Θ. The simple way to describe this ring is provided
by theta functions:
∞
Θ (w)
k
A= , (4)
θ(w)k
k[=0(cid:16) (cid:17)
where Θ is the space of theta functions of order k i.e. the space of regular
k
functions on Cg satisfying
θ (w+m+Bn)=exp 2kπi 1 tnBn tnw θ (w).
k −2 − k
There are kg linearly independent theta f(cid:0)unctions of order(cid:1)k.
Let us discuss the geometric meaning of the ring A. It is well known that
with the help of theta functions one can embed the complex torus J(X) into
the complex projective space as a non singular algebraic subvariety. It can be
done, for example, using theta functions of third order:
1. 3g theta functions of third order define an embedding of J(X) into the
complex projective space P3g−1,
2. a set of homogeneous algebraic equations for these theta functions can be
written, which allows to describe this embedding as algebraic one.
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Now consider the functions
Θ (w)
3
.
θ(w)3
Obviously, with the help of these functions, we can embed the non-compact
variety J(X) Θ into the complex affine space C3g−1. Denote the coordinates
−
inthis space by x , ,x , the affine ringofJ(X) Θ is defined asthe ring
1 3g−1
··· −
C [x1, ,x3g−1]/(gα),
···
where (g ) is the ideal generated by the polynomials g such that g = 0
α α α
{ } { }
defines the embedding. It is known that the affine ring is the characteristic of
thenon-compactvarietyJ(X) Θindependentofaparticularembeddingofthis
−
variety into affine space. Obviously the ring A defined above is isomorphic to
theaffinering. WeremarkthattheaboveargumentontheembeddingJ(X) Θ
−
intoanaffinespaceis validif(J(X),Θ)isreplacedby anyprincipallypolarized
abelian variety.
Consider X(g)which is mapped to J(X) by the Abel map a. The Riemann
theorem says that
g−1 pj
θ(w)=0 iff w = ω+∆
Xj=1∞Z
whichallowstodescribeΘintermsofthesymmetricproduct. Oneeasilyargues
that the preimage of Θ under the Abel map is described as
D :=D D ,
∞ 0
∪
where
D = (p , ,p ) X(g) p = for some i , (5)
∞ 1 g i
{ ··· ∈ | ∞ }
D = (p , ,p ) X(g) p =σ(p ) for some i=j .
0 1 g i j
{ ··· ∈ | 6 }
The Abel map is not one-to-one, and the compact varieties J(X) and X(g)
are not isomorphic. However, the affine varieties J(X) Θ and X(g) D are
− −
isomorphic since the Abel map
a
X(g) D J(X) Θ
− −→ −
isanisomorphism. InwhatfollowsweshallstudytheaffinevarietyX(g) D
− ≃
J(X) Θ.
−
3 Affine model of hyperelliptic Jacobian.
Consider a traceless 2 2 matrix
×
a(z) b(z)
m(z)= ,
c(z) a(z)
(cid:18) − (cid:19)
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where the matrix elements are polynomials of the form:
a(z)=a zg−1+a zg−2+ +a , (6)
23 25 ··· g+12
b(z)=zg+b zg−1+ +b ,
1 g
···
c(z)=zg+1+c zg+c zg−1+ +c .
1 2 g+1
···
Laterweshallsetb =c =1. ConsidertheaffinespaceC3g+1 withcoordinates
0 0
a , ,a , b , ,b , c , ,c . Fix the determinant of m(z):
3 g+1 1 g 1 g+1
2 ··· 2 ··· ···
a2(z)+b(z)c(z)=f(z), (7)
where the polynomial f(z) is the same as used above (1). Comparing each
coefficient of zi (i = 0,1, ,2g) of (7) one gets 2g+1 different equations. In
factthe equations(7) defi·n·e·g-dimensionalsub-varietyofC3g+1. This algebraic
variety is isomorphic to J(X) Θ as shown in the book [1]. We shall briefly
−
recall the proof.
Consider a matrix m(z) satisfying (7). Take the zeros of b(z):
g
b(z)= (z z )
j
−
j=1
Y
and set
y =a(z ).
j j
Obviously z ,y satisfy the equation
j j
y2 =f(z ),
j j
which defines the curve X. So, we have constructed a point of X(g) for every
m(z) which satisfies the equations (7). Conversely, for a point (p , ,p ) of
1 g
···
X(g), construct the matrix m(z) as
g g
z z
k
b(z)= (z z ), a(z)= y − ,
j j
− z z
j=1 j=1 k6=j(cid:18) j − k(cid:19)
Y X Y
a(z)2+f(z)
c(z)= − ,
b(z)
where z = z(p ) is the z-coordinate of p . Considering the function b(z) as a
j j j
function on X(g) one finds that it has singularities when one of z equals .
j
∞
The function a(z) is singular at z = and also at the points where z = z
j i j
∞
but y = y . This is exactly the description of the variety D. The functions
i j
−
a(z)andc(z)do notaddnew singularities. Thus we havethe embedding ofthe
affine variety X(g) D into the affine space:
−
X(g) D ֒ C3g+1.
− →
Therefore we can profit from the wonderful property of the hyper-elliptic Jaco-
bian: it allows an affine embedding into a space of very small dimension equal
to 3g+1 (compare with 3g 1 which we have for any Abelian variety). Actu-
ally,the spaceC3g+1 occurs−foliatedwithgenericleavesisomorphictotheaffine
Jacobians.
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4 Properties of affine ring.
Consider the free polynomial ring A:
A=C [a , ,a ,b , ,b ,c , ,c ].
32 ··· g+12 1 ··· g 1 ··· g+1
Onthe ring A one cannaturally introduce a grading. Prescribethe degreej to
any of generatorsa , b , c and extend this definition to all monomials in A by
j j j
deg(xy)=deg(x)+deg(y).
Everymonomialoftheringhaspositivedegree(exceptfor1whosedegreeequals
0). Thus, as a linear space, A splits into
A= A(p),
2pM∈Z+
where A(j) is the subspace of degree j and Z = 0,1,2, . Define the
+
{ ···}
character of A by
ch(A)= qp dim(A(p)).
2pX∈Z+
Since the ring A is freely generated by a , b , c one easily finds
p p p
1
ch(A)= 2 , (8)
g+ 1 ! [g]! [g+1]!
2 (cid:2) (cid:3)
where, for k Z , (cid:2) (cid:3)
+
∈
[k]=1 qk, [k]!=[1] [k], k+ 1 != 1 3 k+ 1 .
− ··· 2 2 2 ··· 2
This important formula allows to control(cid:2)the siz(cid:3)e of(cid:2)th(cid:3)e(cid:2)rin(cid:3)g A(cid:2). (cid:3)
The relation of the ring A to the affine ring A is obvious. The latter is the
quotientof A by the idealgeneratedby the relations det(m(z))=f(z) where
−
the coefficients of f are considered fixed constants.
From the point of view of integrable models, it is more natural to see
f , ,f as variables than complex numbers. If we assign degree j to the
1 2g+1
···
variablesf ,alltheequationsin(7)arehomogeneous. Considerthe polynomial
j
ring
F=C [f , ,f ].
1 2g+1
···
The ring F is graded and its character is
1
ch(F)= .
[2g+1]!
The ring F acts on A, that is, f(z) acts by the multiplication of the left
handsideof(7). ConsiderthespaceA whichconsistsofF-equivalenceclasses:
0
2g+1
A =A / (F×A), F× = Ff .
0 i
i=1
X
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Since F×A is a homogeneous ideal of A, A is a graded vector space:
0
(p)
A = A .
0 0
2pM∈Z+
One can consider the space A as a subspace of A taking a set of homo-
0
geneous representativesof the equivalence classes (being homogeneous they are
automatically of smallest possible degree). Consider any homogeneous x A.
∈
One can write x as
2g+1
x=x(0)+ f x ,
i i
i=1
X
where x(0) A and x is a homogeneous element in A satisfying degx =
0 i i
∈
degx i. Since the degreeofx is less thanthe degreeofx, repeating the same
i
−
procedure for x one arrives, by finite number of steps, at
i
x= h x(0), (9)
j j
X
wherex(0) A ,h Fandthe summationisfinite. ThereisanF-linearmap:
j ∈ 0 i ∈
m
F CA0 A,
⊗ −→
which corresponds to multiplying the elements of A by elements from F and
0
takinglinearcombinations. TheabovereasoningshowsthatIm(m)=A. Hence
ch(A)
ch(A ) . (10)
0
≥ ch(F)
The equality takes place iff Ker(m) = 0. We shall see that this is indeed the
case. Informally the equality Ker(m) = 0 is a manifestation of the fact that
the space C3g+1 is foliated into g-dimensional sub-varieties, the coordinates f
j
describe transverse direction. The pure algebraic proof of this fact is given by
the following proposition.
Proposition 1. The set of elements
g g
uij ulk ,
1+j g+1+k
2 2
j=1 k=1
Y Y
where
a , p=half-integer
u = p
p b , p=integer,
p
(cid:26)
is a basis of A as avector space, where i , ,i are non-negativeintegers and
0 1 g
···
l , ,l are 0 or 1.
1 g
···
The proof of Proposition 1 is given in Appendix A. Proposition 1 shows that
ch(A0)= g 1+1j g 1+qg+12+k = g+ 112 ![2[gg]!+[1g]!+1]! = cchh((AF)), (11)
jY=1 2 kY=1(cid:16) (cid:17) (cid:2)2(cid:3)
(cid:2) (cid:3) (cid:2) (cid:3)
9
which means that Ker(m)=0. We summarize this in the following:
Proposition 2. As an F module, A is a free module, A F CA0. In other
≃ ⊗
words every element x A can be uniquely presented as a finite sum:
∈
(0)
x= h x ,
j j
X
where x(0) is a basis of the C-vector space A and h F.
{ j } 0 j ∈
5 Poisson structure and cohomology groups.
The affine model of hyper-elliptic Jacobian is interesting for its application to
integrablemodels. TheringAthatweintroducedintheprevioussectioncanbe
suppliedwithPoissonstructure. Thisfactisalsoimportantbecauseintroducing
the Poisson structure is the first step towards the quantization. The Poisson
structure in question is described in r-matrix formalism as follows:
m(z ) I,I m(z ) =[r(z ,z ),m(z ) I] [r(z ,z ),I m(z )]. (12)
1 2 1 2 1 2 1 2
{ ⊗ ⊗ } ⊗ − ⊗
The r-matrix acting in C2 C2 is
⊗
z
r(z ,z )= 2 1σ3 σ3+σ+ σ−+σ− σ+ +z σ− σ−,
1 2 z z 2 ⊗ ⊗ ⊗ 2 ⊗
1 2
−
(cid:0) (cid:1)
where σ3, σ± are Pauli matrices.
Thevariablesz , ,z (zerosofb(z))andy =a(z )havedynamicalmean-
1 g j j
···
ing of separated variables [4]. The Poisson brackets (12) imply the following
Poissonbrackets for the separated variables:
z ,y =δ z .
i j i,j i
{ }
The determinant f(z) of the matrix m(z) generates Poisson commutative
subalgebra:
f(z ),f(z ) =0.
1 2
{ }
It can be shown that the coefficients f ,f , ,f and f belongs to the
1 2 g 2g+1
···
center of Poisson algebra. The Poisson commutative coefficients f , ,f
g+1 2g
···
are the integrals of motion. Introduce the commuting vector-fields
D h= f ,h , i=1, g.
i g+i
{ } ···
For completeness let us describe explicitely the action of these vector-fields on
m(z). Define
g
D(z)= zj−1D .
g+1−j
j=1
X
Then the Poisson brackets (12) imply:
1
D(z )m(z )= [m(z ),m(z )] [σ−m(z )σ−,m(z )].
1 2 1 2 1 2
z z −
1 2
−
10