Dual Baxter equations and quantization of Affine Jacobian

27 Pages·0.21 MB·English

by  Feodor A. Smirnov

LPTHE-00-04
Dual Baxter equations and
quantization of Affine Jacobian.
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F.A. Smirnov0
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LaboratoiredePhysiqueThe´oriqueetHautesEnergies1

1 Universite´PierreetMarieCurie,Tour16,1er e´tage,4placeJussieu
v 75252ParisCedex05,France
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0 Abstract. A quantumintegrablemodelis consideredwhichdescribesa quantization
/ of affinehyper-ellipticJacobian. Thismodelis shown to possess the propertyof du-
h
p ality: a dual model with inverse Planck constant exists such that the eigen-functions
- ofits Hamiltonianscoincidewith theeigen-functionsofHamiltoniansofthe original
h
model. Weexplainthatthisdualitycanbeconsideredasdualitybetweenhomologies
t
a andcohomologiesofquantizedaffinehyper-ellipticJacobian.
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0MembreduCNRS
1Laboratoireassocie´auCNRS.
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1 Introduction.
TherelationbetweentheaffineJacobianandintegrablemodelsiswellknown(cf. [1]).
Inthepaper[2]wehaveshownthatthealgebraoffunctionsontheaffineJacobianis
generatedbyactionofhamiltonianvectorfieldsfromfinitenumberoffunctions. The
latter functions are coefficients of highest non-vanishing cohomologies of the affine
Jacobian.Actually,theideathatsuchdescriptionofthealgebraoffunctionsispossible
appearedfromthepaper[3]whichconsidersthestructureofthealgebraofobservables
forquantumandclassicalTodachain.
Inthepresentpaperwegivequantumversionof[2].Aquantummechanicalmodel
is formulated which gives a quantization of the affine Jacobian. As usual in Quan-
tum Mechanicswe can describe notthe variety itself but the algebra of functionson
it(observables). We needtoshowthatthe quantumalgebraofobservablespossesses
essentialpropertyofcorrespondingclassicalalgebraoffunctions.Inourcasethisprop-
ertyisthepossibilityofcreatingeveryobservablefromafinitenumberofobservables
(cohomologies)byactionofHamiltonians.
In the process of realization of this program we find the Baxter equations which
describethespectrumofthemodel.Ithappensthattheseequationspossesstheproperty
ofduality:thereisdualmodelwithinversePlanckconstantforwhichtheeigen-vectors
are the same. The algebras of observables of two dual models commute. The next
ingredientofourstudyisthemethodofseparationofvariablesdevelopedbySklyanin
[4]. Using thismethodwe presentthematrixelementsofanyobservablein termsof
certainintegrals.
WeshowthattheintegralsinquestionareexpressesintermsofdeformedAbelian
integrals(cf.[3,5]).Theobservablesforbothdualmodelsaredefinedintermsofcoho-
mologies.Themostbeautifulfeatureofourconstructionisthatinthesecohomologies
enter the integrals for matrix elements in such a way that the cohomologies of dual
modelplayroleofhomologiesfororiginaloneandviseaversa.Weconsiderthisrela-
tionbetweenweek-strongdualityinquantumtheorywithdualitybetweenhomologies
andcohomologiesasthemostimportantconclusionofthispaper.
2 Affine Jacobian.
Inthissectionwebrieflysummarizenecessaryfactsconcerningrelationbetweeninte-
grablemodelsandalgebraicgeometryfollowingthepaper[2].Thereasonforrepeating
certainfactsfrom[2]isthatweshallneedtheminslightlydifferentsituation.
Consider2 2matrixwhichdependspolynomiallyontheparameterz:
×
a(z) b(z)
m(z)=
c(z) d(z)
(cid:18) (cid:19)
2
wherethematrixelementsarepolynomialsoftheform:
a(z)=zg+1+a zg+ +a , (1)
1 g+1
···
b(z)=zg+b zg−1+ +b ,
1 g
···
c(z)=c zg+c zg−1 +c ,
2 3 g+2
···
d(z)=d zg−1+d zg−2 +d
2 3 g+1
···
InthetheaffinespaceC4g+2withcoordinatesa , ,a ,b , ,b ,c , ,c ,
1 g+1 1 g 2 g+2
··· ··· ···
d , ,d considerthe(2g+1)-dimensionalaffinevariety definedasquadric
2 g+1
··· M
f(z) a(z)d(z) b(z)c(z)=1 (2)
≡ −
Weconsiderthissimplestsituation,butinprincipleitispossibletoputarbitrarypoly-
nomialofdegree2ginRHS.
Onthequadric letusconsiderthesectionsJ (t)definedbytheequations:
aff
M
a(z)+d(z)=t(z) (3)
wheret(z)isgivenpolynomialoftheform:
t(z)=zg+1+zgt + +t , (4)
1 g+1
···
The notation J (t) stands for affine Jacobi variety. The definition of affine Jacobi
aff
varietyanditsequivalencetoJ (t)describedabovearegivenintheAppendixA.We
aff
includeAppendixAbecausethereisminordifferencewiththesituationconsideredin
[1] and [2]. The variety is foliated into the affine Jacobians J (t). Mechanical
aff
M
modeldescribedbelowprovidesacleverwayofdescribingthisfoliation.
Wewouldliketounderstandthegeometricalmeaningofquantumintegrablemod-
els. The general philosophy teaches that in order to describe the quantization of a
manifoldonehasto deformthe algebraoffunctionsonthismanifoldpreservingcer-
tainessentialpropertiesofthisalgebra. Theclassicalalgebramustallowthe Poisson
structureinorderthatquantizationispossible.
CertainPoissonbracketsforthecoefficientsofmatrixm(z)canbeintroduced.We
donotwritethemdownexplicitly,ifneededtheycanbeobtainedtakingclassicallimit
ofthecommutationrelations(14).Thealgebra
=C[a , ,a ,b , ,b ,c , ,c ,d , ,d ]
1 g+1 1 g 2 g+2 2 g+1
A ··· ··· ··· ···
becomesaPoissonalgebra.ThemostimportantpropertiesofthisPoissonstructureare
b
thefollowing.First,thecoefficientsofthedeterminantf(z)belongtothecenterofthe
Poissonalgebra,so,theequation(2)isconsistentwithPoissonstructure. Second,the
tracet(z)generatescommutativesub-algebra:
t(z),t(z′) =0
{ }
It can be shown, actually, that the coefficientt of the trace belongsto the center,
g+1
it is convenientto put t = ( 1)g+12. The subject of our study is the algebra of
g+1
−
functionsonthe :
M
= A
A f(z)=1, t =( 1)g+12
g+1
{ − }
b
3
onwhichthePoissonstructureiswelldefined.
ThePoissoncommutativealgebrageneratedbythecoefficientst , ,t iscalled
1 g
···
thealgebraofintegralsofmotion.Introducethecommutingvector-fields
∂ g = t ,g , i=1, g
i i
{ } ···
Thevector-fields∂ describemotionalongthesub-varietiesJ (t).
j aff
Onecanthinkofthesevector-fieldsas
∂
∂ =
j
∂τ
j
where τ are ”times” correspondingto the integrals of motion t . Define the ring of
j j
integralsofmotion
=C[t , ,t ] (5)
1 g
T ···
IntroducethespaceofdifferentialformsCk withbasis
xdτ dτ , x
i1 ∧···∧ ik ∈A
andthedifferential
d=∂ dτ
j j
ConsidercorrespondingcohomologiesHk. Inthepaper[2]theargumentsaregivenin
favourofthefollowing
Conjecture1. ThecohomologiesHk arefinite-dimensionaloverthering ,theyare
T
isomorphictothecohomologiesoftheaffinevarietyJ (t)withtingenericposition.
aff
On the algebra and on the spaces Ck one can introduce degree [2]. Take the ba-
A
sis of Hg considered as a vector space over which is composed of homogeneous
T
representatives
Ω =g dτ dτ
α α 1 g
∧···∧
whereαtakesfinitenumberofvalues. Thefactoffoliationof intovarietiesJ (t)
aff
M
correspondstothefollowingstatementconcerningthealgebra [2]
A
Proposition1. Everyelementxof canbepresentedintheform
A
x= p (∂ , ,∂ )g (6)
α 1 g α
···
α
X
wherep (∂ , ,∂ )arepolynomialsof∂ , ,∂ withcoefficientsin .
α 1 g 1 g
··· ··· T
Therepresentation(6)isnotunique,theequations
p (∂ , ,∂ )g =0 (7)
α 1 g α
···
α
X
arecountedbyHg−1[2].
The formula (6) can be useful only if we are able to control the cohomologies.
Concerningthesecohomologiesweadoptseveralconjecturesfollowing[2].
4
3 Conjectured form of cohomologies.
TheaffinevarietyJ (t)allowsthefollowingdescription. Considerthehyper-elliptic
aff
curveX ofgenusg:
w2 t(z)w+1=0 (8)
−
Thiscurvehastwopointsoverthepointz = whichwedenoteby ±.
∞ ∞
Consideramatrixm(z)satisfying(2).Takethezerosofb(z):
g
b(z)= (z z )
j
−
j=1
Y
and
w =d(z )
j j
Obviouslyz ,w satisfytheequationofthecurveX (8).Thusm(z)definesapoint
j j
P
(divisor)onthesymmetrizedg-thpowerX[g]ofthecurveX. Thedivisor consists
P
ofthepointsp =(z ,w ) X.
j j j
∈
Oppositelyone can reconstructm(z)starting formthe divisor . Corresponding
P
mapissingular,thesingularitiesbeinglocatedon
D = p =σ(p )forsomei,jorp = ± forsomei (9)
i j i
{P | ∞ }
Whereσishyper-ellipticinvolution.ThusthealternativedescriptionofJ (t)is
aff
J (t)=X[g] D
aff
−
ConsiderthemeromorphicdifferentialsonX withsingularitiesat ±. Wechose
∞
thefollowingbasisofthesedifferentials:
dz
µ (p)=zg+k , g k 0
k
y − ≤ ≤
d dz
µ (p)= y (zk−g−1y) , k 1 (10)
k dz ≥ y ≥
(cid:2) (cid:3)
where p = (z,w), y = 2w t(z), [ ] means that only non-negative degrees of
≥
−
Laurentseriesinthebracketsaretaken.
Theform
µ = µ (p )
k k i
i
X
is viewed as a form on J (t). Iteis easy to see that the forms µ (hence µ ) with
aff k k
k g+1areexact.ConsiderthespaceWm withthebasis:
≥
e
Ω =µ µ
k1,···,km k1 ∧···∧ km
where g k g. Following[2]weadoptthe
− ≤ j ≤ e e
5
Conjecture2. Wehave
Wm
Hm = (11)
σ Wm−2
∧
where
g
σ = µ µ
j −j
∧
j=1
X
Accordingto(9)thesingularitiesofdiffereentialeformsoccureitheratpi =σ(pj)orat
p = ±. Thenon-trivialessenceofConjecture2isthatthefirstkindofsingularities
i
∞
can be eliminated by adding exact forms. There are (g 1)-forms singular at p =
i
−
σ(p )suchthatthesesingularitiesdisappearafterapplyingd. Thisistheoriginofthe
j
spaceσ Wk−2[2].
∧
Considerbrieflythedualpicture. Ontheaffinecurvewithpuncturesat ± there
∞
are 2g + 1 non-trivial cycles δ with k = g, ,g. The cycles δ , k < 0 are
k k
− ···
a-cycles, the cycles δ , k > 0 are b-cycles and δ is the cycle around +. One
k 0
∞
definesthecyclesδ onthesymmetricalpoweroftheaffinecurve. The -operationis
k
∧
introducedforthesecyclesbydualitywithcohomologies.Thenon-trivialconsequence
ofConjecture2 isethateverycycle onJ (t) canbeconstructedbywedgingδ . The
aff k
formuladualto(11)is
W e
H = m (12)
m σ′ W
m−2
∧
whereW isspannedby
m
∆ =δ δ
k1,···,km k1 ∧···∧ km
and
ge e
σ′ = δ δ
j −j
∧
j=1
X
Weneedtofactorizeoverσ′ W becauesethee2-cycleσ′intersectswithD.
m−2
∧
LetusreturntotherelationofHg tothealgebra . Noticethat
A
dτ dτ µ µ Ω
1 g 1 g
∧···∧ ≃ ∧···∧ ≡
Thefunctions
x =Ωe−1Ω e
k1,···,kg k1,···,kg
are symmetricpolynomialsofz , ,z . Recall thatb , ,b arenothingbutele-
1 g 1 g
··· ···
mentarysymmetricpolynomialsofz , ,z .Hencethecoefficientsofcohomologies
1 g
···
havetheform:
g =g (b , ,b )
α α 1 g
···
ThedimensionofHg isdeterminedbyConjecture2:
α=1, , 2g+1 2g+1 ,
··· g − g−2
Theequations(7)areconsequencesofth(cid:0)efollo(cid:1)win(cid:0)gone(cid:1)s
g
∂ Ω−1(µ Ω ) =0, Ω Wg−1 (13)
k −k∧ k1,···,kg−1 ∀ k1,···,kg−1 ∈
k=1
X (cid:0) (cid:1)
6
4 Quantization of affine Jacobian.
Let us consider a quanization of algebra . The parameter of deformation (Planck
A
constant)isdenotedbyγ,weshallalsouse
q =eiγ
Considerthe2 2matrixm(z)withnoncommutingentries. Supposethatthedepen-
×
dence on the spectral parameter z is exactly the same as in classical case (1). The
variablesa ,b ,c ,d aresubjecttocommutationrelationswhicharesummarizedas
j j j j
follows:
r (z ,z )m (z )k (z )s m (z )k (z )=
21 1 2 1 1 12 1 12 2 2 21 2
=m (z )k (z )s m (z )k (z )r (z ,z ) (14)
2 2 21 2 21 1 1 12 1 12 1 2
where usual conventionsare used: the equation (14) is written in the tensor product
C2 C2, a = a I, a = I 2, a = Pa P where P is the operation of
1 2 21 12
perm⊗uattions.TheC⊗-numbermatri⊗cesr,k,sare:
z qz z +qz
r (z ,z )= 1− 2 (I I)+ 1 2 (σ3 σ3)+
12 1 2
1 q ⊗ 1+q ⊗
−
+2(z σ− σ++z σ+ σ−),
1 2
⊗ ⊗
k (z)=I (I σ3)+ q−σ3 +z(q2 1)σ− (I +σ3),
12
⊗ − − ⊗
s =I I (q q−1)σ(cid:16)− σ+ (cid:17) (15)
12
⊗ − − ⊗
These commutation relations are important because they respect the form of matrix
m(z)prescribedby(1),weshallexplainhowtheyarerelatedtomoreusualr-matrix
relationsinthenextsection.
Definethepolynomials:
t(z)=qa(z)+q2d(z) z(q2 1)b(z)
− −
f(z)=qd(z)t(zq−2) q2d(z)d(zq−2) qb(z)c(zq−2) (16)
− −
The algebra (q) is generated by a , ,a , b , ,b , c , ,c ,
1 g+1 1 g 2 g+2
A ··· ··· ···
d , ,d ,Thepolynomialf(z)belongstothecenterof (q). Thecoefficientsof
2 g+1
··· A
t(z)arecommbuting,actually,t belongstothecenterof (q). Wedefine:
g+1
A
b
(q) b
(q)= A
A f(z)=1, t =( 1)g+12
g+1
{ − }
b
The non-commutativealgebra (q) defines a quantizationof the algebra of function
A
onthequadric . However,wecannotdefinedirectlythequantizationofthealgebra
M
offunctionsontheaffineJacobianbecausethecoefficientsoft(z)arenotinthecenter
of (q). Whatwe candois todescribethequantumversionofProposition1 andof
A
descriptionofcohomologies.Theexpositionwillbemoredetailedthanintheclassical
case.
7
Likeinthepaper[3]weacceptthefollowing
Conjecture3. Thealgebra (q)isspannedaslinearspacebyelementsoftheform:
A
x=p (t , ,t )g(b , ,b )p (t , ,t ) (17)
L 1 g 1 g R 1 g
··· ··· ···
wherep (t , ,t ), g(b , ,b ), p (t , ,t )arepolynomials.
L 1 g 1 g R 1 g
··· ··· ···
Wewerenotabletoprovethisstatement,however,sincethealgebra (q)isgradedwe
A
cancheckitdegreebydegree.Thishasbeendoneuptodegree8. Noticethesimilarity
between the representation (17) and the representation for spin operators proved in
[6]. Conjecture 3 implies that certain generalization of the results of [6] is possible.
In fact the formula (17) is similar to the formula (6): we can either symmetrize or
anti-symmetrize t in (17) which correspondsin classics to multipilation by t or to
j j
applying∂ . Inordertohavecompleteagreementwithclasicalcasewehavetoshow
j
thatonlyfinitelymanydifferentpolynomialsg(b , b )(cohomologies)createentier
1 g
···
algebra (q).
A
Noticethatthecommutationrelations(14)implyinparticularthat
[b(z),b(z′)]=0
whichmeansthatwehavethecommutativefamilyofoperatorsz definedby
j
b(z)= (z z )
j
−
Y
So, every polynomialsg(b , ,b ) can be considered as symmetric polynomialof
1 g
···
z andviceversa.
j
Itisveryconvenienttouse thefollowingformaldefinitions. Considerthering
T
definedin(5).By kwedenotethespaceofanti-symmetricpolynomialsofkvariables
V
suchthattheirdegreeswithrespecttoeveryvariableisnotlessthan1withcoefficients
in . Inotherwords k isthespacespannedbythepolynomials:
T ⊗T V
p h p p (t , ,t )h(z , ,z )p (t′, ,t′)
L· · R ≡ L 1 ··· g 1 ··· k R 1 ··· g
wherehisanti-symmetric,vanishingwhenoneofz vanishes. Thefollowingopera-
j
tionscanbedefined.
1. Multiplicationbyt andt′.
j j
2. Operation : k l k+l whichisdefinedasfollows:
∧ V ⊗V →V
(p h p ) (p′ h′ p′ )=p p′ (h h′) p p′
L· · R ∧ L· · R L L· ∧ · R R
where
(h h′)(z , ,z )=
1 k+l
∧ ···
1
= ( 1)π h(z , ,z )h′(z , ,z )
k!l! − π(1) ··· π(k) π(k+1) ··· π(k+l)
π∈XSk+l
Wehaveamap
g χ (q)
V −→A
8
definedonthebasiselementsas
h(z , ,z )
χ(p h p )=p (t , ,t ) 1 ··· g p (t , ,t )
L· · R L 1 ··· g z (z z ) R 1 ··· g
i i<j i− j
andcontinuedlinearly. TheConjecture3QstatesQthatthismapissurjective. Wewantto
describethekernelofthemapχ.
First, consider the space 1. The elements of this space are polynomials of one
V
variable z with coefficients in . In Appendix B we describe certain basis in
T ⊗ T
1 considered as a linear space over . The basis in question consists of the
V T ⊗ T
polynomials: s with k g such that the degree of s with respect to z equals
k k
≥ −
g+k+1. Thekernelofχisthejointofthreesub-spaces,letusdescribethem.
1. Fork g+1wehave:
≥
χ s g−1 =0 (18)
k
∧V
(cid:0) (cid:1)
2. Considerc 2definedas
∈V
g
c= s s
j −j
∧
j=1
X
wehave
χ c g−2 =0 (19)
∧V
(cid:0) (cid:1)
3. Considerd 1definedas
∈V
d=(t t′)s
j − j −j
wehave
χ d k−1 =0 (20)
∧V
Theconstructionofthespace (cid:0) (cid:1)
g
V (q)
Ker(χ) ≃A
is in complete correspondence with the classiccal case. In classics we start with all
the 1-forms µ . Imposing (18) corresponds to throwing away the exact forms and
k
working with µ for k = g ,g only. Imposing (19) correspondsto factorizing
k
− ···
overσ Wk−2 inclassics. Finally,(20)correspondstotheequation(13).
e
∧
Theoriginoftheequations(18),(19),(20)willbeexplainedintheSection9.There
e
shouldbepurelyalgebraicmethodofproovingtheseequations,butwedonotknowit.
Itisimportantto mentionthatacceptingConjecture3 weare forcedto concludethat
thekernelofχiscompletelydescribedbytheequations(18),(19),(20).Thisisproved
bycalculationofcharacterssimilarlytothatof[3].
9
5 The realization of A(q).
Wewanttodescribearealizationofthealgebra (q)inaspaceoffunctions.Consider
A
thequantummechanicalsystemdescribedbytheoperatorsx withj =1, ,2g+2
j
···
andy(zeromode). Theoperatorsx andy areself-adjoint,theysatisfythecommuta-
j
tionrelations:
x x =q2x x k <l,
k l l k
yx =q2x y k
k k
∀
Thehamiltonianofthesystemis
2g+2
h=q−1 x x−1
k k−1
k=1
X
where
x qyx
2g+3 1
≡
PhysicallythismodeldefinesthesimplestlatticeregularizationofthechiralBosefield
withmodifiedenergy-momentumtensor.
It is usefulto double the numberof degreesof freedom. Consider the algebra A
generatedbytwooperatorsuandvsatisfyingthecommutationrelations:
uv =qvu
TakethealgebraA⊗(2g+2) theoperatorsu , v (j = 1, ,2g+2)aredefinedasu
j j
···
andvactinginj-thtensorcomponent.Theoriginaloperatorsx areexpressedinterms
i
ofu ,v asfollows:
i i
k−1 2g+2
x =v u−2, y = u
k k j j
j=1 j=1
Y Y
Considerthe“monodromymatrix”
a(z) b(z)
m(z)= =l (z) l (z) (21)
2g+2 1
c(z) d(z)! ···
e e
wherethel-operatorfsare e e
1 zu qvu
l(z)= − (22)
√z zv−1u−1 0
(cid:18) (cid:19)
This is a particular case of more genaral l-operator l(z,κ) in which the last matrix
elementisnot0butκzu,themodelcorrespondingtothelatterl-operatorisasubject
ofstudyinaseriesofpapers[7].
Thematrixelementsofthematrixm(z)satisfythecommutationrelations:
r (z ,z )m (z )m (z )=m (z )m (z )r (z ,z ) (23)
12 1 2 1 1 2 2 2 2 1 1 12 1 2
f f f f
10