Extended affine Weyl groups and Frobenius manifolds

53 Pages·1998·0.39 MB·English

CompositioMathematica 111: 167–219,1998. 167
c 1998KluwerAcademicPublishers. PrintedintheNetherlands.
(cid:13)
Extended affine Weyl groups and Frobenius
manifolds
BORISDUBROVIN1 andYOUJINZHANG2
1SISSA,ViaBeirut2–4,34014Trieste,Italy,andSteklovMathematicsInstitute,Moscow,
e-mail:[email protected]
2SISSA,ViaBeirut2–4,34014Trieste,Italy,andDepartmentofMathematics,UniversityofScience
andTechnologyofChina,Hefei,
e-mail:[email protected]
Received14May1996;acceptedinfinalform16January1997.
Abstract.WedefinecertainextensionsofaffineWeylgroups(distinctfromtheseconsideredbyK.
Saito[S1]inthetheoryofextendedaffinerootsystems),proveananalogueofChevalleyTheorem
fortheirinvariants,andconstructaFrobeniusstructureontheirorbitspaces.Thisproducessolutions
ofWDVVequationsofassociativitypolynomialin , , ,exp .
1 1 1
F(t ;:::;tn) t ::: tn(cid:0) tn
MathematicsSubjectClassifications(1991).32M10,14B07,20H15.
Keywords:rootsystems,affineWeylgroups,Frobeniusmanifolds,flatcoordinates,WDVVequa-
tions.
0. Introduction
Frobeniusmanifoldisageometricobject(seeprecisedefinitioninSection2below)
designedas acoordinate-freeformulation ofequationsofassociativity,or WDVV
equations (they were invented in the beginning of ’90s by Witten, Dijkgraaf, E.
andH.Verlindeinthesettingoftwodimensionaltopologicalfieldtheory;see[D]
andreferencestherein).In[D]foranarbitrary -dimensionalFrobeniusmanifold
amonodromygroupwasdefined.Itactsin -dinmensionallinearspaceanditisan
extensionofagroupgeneratedbyreflectionns.Lookingatsimpleexamplesitmight
beconjecturedthatforaFrobeniusmanifoldwithgoodanalyticproperties(inthe
senseof[D], AppendixA) the monodromygroup actsdiscretelyin somedomain
of the space.The Frobenius manifold itself can be identified with the orbit space
ofthegroupinthesensetobespecifiedforeachclassofmonodromygroups.
In the present paper we introduce a new class of discrete groups that can be
realized as monodromy groups of Frobenius manifolds (it was shown previously
that any finite Coxeter group can serve as a monodromy group of a polynomial
Frobeniusmanifold,see[D]). WedefinecertainextensionsofaffineWeylgroups,
and construct a Frobenius structure on their orbit spaces. Our groups coincide
with the monodromy groups of the Frobenius manifolds. They are labelled by
pairs where is an irreducible reduced root system, and is a certain
simple(Rro;okt)(showniRnwhiteonTableI,nextpage).OurconstructionkofFrobenius
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168 BORISDUBROVINANDYOUJINZHANG
TableI
*
Al 1r 2r q q q r1 c r1 q q q r1 r 1
k(cid:0) k k+ l(cid:0) l l+
*
Bl 1r 2r q q q r1 r r1 q q q c1 > r 1
k(cid:0) k k+ l(cid:0) l l+
*
Cl 1r 2r q q q r1 r r1 q q q r1 < c 1
k(cid:0) k k+ l(cid:0) l l+
1
l(cid:0)
(cid:8)r
(cid:8)
(cid:8)
Dl 1r 2r q q q r1 r r1 q q q 2cH
k(cid:0) k k+ l(cid:0) HHr * 1
2 l l+
r
6 *
r r c r r
E 1 3 4 5 6 7
2
r
7 *
r r c r r r
E 1 3 4 5 6 7 8
2
r
8 *
r r c r r r r
E 1 3 4 5 6 7 8 9
4 *
r c r r
F 1 2 > 3 4 5
2 *
r c
G 1 < 2 3
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AFFINEWEYLGROUPSANDFROBENIUSMANIFOLDS 169
structureincludes,particularly,aconstructionoftheflatcoordinates inthe
1
appropriateringofinvariantsoftheextendedaffineWeylgroups(flta;t:c:o:o;trdninates
in the ring of polynomial invariants of finite Coxeter groups were discovered by
Saito, Yano, Sekiguchi [SYS, S]). The correspondent solutions of equations of
associativityareweightedhomogeneous(upto aquadraticfunction)polynomials
in , e with all positiveweights ofthe variables.Here 1 is equal
1 1 tn
to tth;e:r:a:n;ktno(cid:0)f the root system . It can be shown (see [D], Appendnix(cid:0)A) that for
3ourconstructionexhaustRsallsuchsolutions.
n6The paper is organized in the following way. In Section 1 we define extended
affineWeylgroupsandproveananalogueofChevalleyTheorem[B]forthem.In
Section2 we construct Frobenius structure on the orbit spaces of our groups and
computeexplicitlyalllow-dimensionalexamplesoftheseFrobeniusmanifolds.In
Section3weshowthatinthecaseoftherootsystemof -type,ourextendedaffine
WeylgroupsdescribemonodromyofrootsoftrigonomeAtricpolynomialsofagiven
bidegree. We discuss topology of the complement to bifurcation variety of such
trigonometricpolynomialsintermsofthecorrespondentFrobeniusmanifolds.
1. ExtendedaffineWeylgroupsandtheirinvariants
Let be an irreducible reduced root systemin -dimensional Euclidean space
withREuclideaninnerproduct . Wefix a basils ofsimple rootVs.
1 2
Let ( ; ) (cid:11) ;(cid:11) ;:::;(cid:11)l
2
1 2
_ (cid:11)j
(cid:11)j = ; j = ; ;:::;l
((cid:11)j;(cid:11)j)
bethecorrespondentcoroots.Allthenumbers : areintegers(these
_
aretheentriesoftheCartanmatrix , Aij = ((cid:11)i2;(cid:11)j) 0 for
_ _
). The Weyl group Ais=a(fiAnijit)e(g(cid:11)rio;u(cid:11)pig)e=nera;t(e(cid:11)di;b(cid:11)yjt)h6e reflectiion6=s
j W = W(R)
1 2
(cid:27) ;(cid:27) ;:::;(cid:27)l
x x x x (1.1)
_
(cid:27)j( ) = (cid:0)((cid:11)j; )(cid:11)j; 2V:
Werecallthattherootsystemisoneofthetype
6 7 8 4 2
(see[B]). Al; Bl; Cl; Dl; E ; E ; E ; F ; G
TheaffineWeylgroup actsinthespace byaffinetransformations
Wa(R) V
x x l Z
_
7!w( )+ 1mj(cid:11)j; w 2W; mj 2 :
j=
X
Soitisisomorphictothesemidirectproductof bythelatticeofcoroots.
Letusintroducecoordinates inW usingthebasisofcoroots
1 2
x ;x ;:::;xl V
x (1.2)
1 _1 2 _2 _
=x (cid:11) +x (cid:11) +(cid:1)(cid:1)(cid:1)+xl(cid:11)l :
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170 BORISDUBROVINANDYOUJINZHANG
WedefineFourierpolynomialasthefollowingfunctionson
V
x e2 1 1
1 (cid:25)i(m x +(cid:1)(cid:1)(cid:1)+mlxl)
f( ) = 1 Zam ;:::;ml ;
m ;:::;ml2
X
thecoefficientsarearbitrarycomplexnumbersandonlyfinitenumberofthemcould
benonzero.Alternatively,introducingthefundamentalweights
1
! ;:::;!l 2V
_
(!i;(cid:11)j) =(cid:14)ij;
wecanrepresenttheFourierpolynomialasasumovertheweightlattice
x e2 1 1 x
1 (cid:25)i(m ! +(cid:1)(cid:1)(cid:1)+ml!l; )
f( ) = 1 Zam ;:::;ml :
m ;:::;ml2
X
ThustheringofourFourierpolynomialsisidentifiedwiththegroupalgebraofthe
weightlattice[B].WedefinetheoperationofaveragingofaFourierpolynomial
x x x : 1 x (1.3)
f( ) 7! f(cid:22)( ) =SW(f( )) =n(cid:0)f f(w( ));
w2W
X
where # x x . For any x the Fourier polynomial
x nf = fxw 2isWa jfufn(cwti(on))o=n f(in)vgariant withfr(es)pect to the action of the
fa(cid:22)f(fin)e=WeSyWlg(fro(up)) V
x l x
f(cid:22) w( )+ 1mj(cid:11)_j =f(cid:22)( ):
0 1
j=
X
Equival@ently,thisisa -inAvariantFourierpolynomial.
W
THEOREM [B]. The ring of -invariant Fourier polynomials is isomorphic to
thepolynomialringC W ,where x x arethebasic
1 1 1
-invariantFourierp[oyly;n:o:m:;iyall]sdefinedyby= y ( );:::;yl = yl( )
W
e2 x 1 (1.4)
(cid:25)i(!j; )
yj =SW( ); j = ;:::;l:
EXAMPLE 1.1. The Weyl group acts by permutations of the coordinates
onthehyperplane W(Al)
1 1
z ;:::;zl+
0
1 1
z +(cid:1)(cid:1)(cid:1)+zl+ = :
Wechoosethestandardbasisofsimpleroots asin[B,PlanchesI].Then
_
thecoordinates aredefinedby (cid:11)j = (cid:11)j
1
x ;:::;xl
2 (1.5)
1 1 1 1
z =x ; zi =xi(cid:0)xi(cid:0) ; i= ;:::;l; zl+ =(cid:0)xl:
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AFFINEWEYLGROUPSANDFROBENIUSMANIFOLDS 171
Thebasic -invariantFourierpolynomialscoincidewiththeelementarysymmet-
ricfunctioWns
e2 1 e2 1 1 (1.6)
(cid:25)iz (cid:25)izl+
yj =sj( ;:::; ); j = ;:::;l:
We are going nowto definecertain extensionsofthe affineWeylgroupacting
inthe 1 -dimensionalspacewithindefinitemetric.
For(aln+yir)reduciblereducedrootsystem wefixaroot indicatedinTableI.
The Dynkin graphs of R (cid:11)tykpe are shown in the
Table1withonemorevAert(cid:0)exBad(cid:0)dedC(t(cid:0)hisDis(cid:0)indEic(cid:0)ateFdb(cid:0)yGasterisk).Wewillusethis
additionalvertexlateron.ThewhitevertexoftheDynkingraphcorrespondstothe
chosenroot .ObservethattheDynkingraphof :
1
isomitted c(cid:11)onksistsof1,2or3branchesof typeRfokrs=omf(cid:11)e ;.:A:n:o;(cid:11)t^hke;r:o:b:s;e(cid:11)rlvgat(i(cid:11)onk
isthatthe)number Ar r
1
2
((cid:11)k;(cid:11)k)
isanintegerforourchoiceof .
Weconstructagroup k
(k)
W =W (R)
actingin
f f
R
V =V (cid:8)
generatedbythetransformations
e
x x l Z (1.7a)
1 _ 1
x=( ;xl+ )7! w( )+ 1mj(cid:11)j; xl+ ; w 2W; mj 2 ;
0 1
j=
and X
@ A
x x 1 (1.7b)
1 1
x=( ;xl+ )7!( +!k; xl+ (cid:0) ):
DEFINITION. istheringofall -invariantFourierpolynomialsof
(k)
1 A=Atha(tRa)reboundedintheWlimit
1 1
x ;:::;xl;( =f)xl+
x x0 1 0 1 f (1.8)
= (cid:0)i!k(cid:28); xl+ =xl+ +i(cid:28); (cid:28) !+1;
for any 0 x0 0 , here is the determinant of the Cartan matrix of the
1
rootsystxem= ,(se;exTla+bl)e2. f A
Weput R
1 (1.8)
dj =(!j;!k); j = ;:::;l;
these are certain positive rational numbers that can be found in Table II. All the
numbers areintegers.Indeed,theyaretheelementsofthe thcolumnofthe
matrix f1(cid:1)tdimj es 1 . k
(cid:0) 2
A ((cid:11)k;(cid:11)k)
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172 BORISDUBROVINANDYOUJINZHANG
TableII.
1
R d ;:::;dl f dk
1 1
Al di= ik((ll(cid:0)l(cid:0)+ki11++1)); 61i6k l+1 k(l(cid:0)l+k1+1)
( l+ ; k+ 6i6l
1 1
i; 6i6l(cid:0) 2 1
Bl di= l(cid:0)21 l(cid:0)
( ; i=l
2
Cl di=i l
1 2
i; 6i6l(cid:0) 4 2
Dl di= l(cid:0)22 1 l(cid:0)
( ; i=l(cid:0) ;l
2,3,4,6,4,2 3 6
6
E
4,6,8,12,9,6,3 2 12
7
E
10,15,20,30,24,18,12,6 1 30
8
E
3,6,4,2 1 6
4
F
3,6 1 6
2
G
LEMMA1.1.TheFourierpolynomials
e2 1 x 1
y~j(x) = (cid:25)idjxl+ yj( ); j = ;:::;l; (1.10)
2
1 (cid:25)ixl+1
y~l+ (x) = e ;
are -invariant.
PWroof.Weshowthatall are -invariant(invarianceof is obvi-
1 1
ous).Itsufficestoprovethay~t ;:::;y~l W y~l+
f
x e2 x f (1.11)
(cid:25)idj
yj( +!k) = yj( ):
Wecanrepresent x as
yj( )
x 1 e2 x (1.12)
(cid:0) (cid:25)i(w(!j); )
yj( ) =nj ;
w2W
X
where for 2 x . Accordingto [B, VI, Sect. 1.6, Prop. 18] for
(cid:25)i(!j; )
any nj = nf f = e
w 2W
l (1.13)
w(!j) =!j (cid:0) 1mi(cid:11)i;
i=
X
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AFFINEWEYLGROUPSANDFROBENIUSMANIFOLDS 173
forsomenon-negativeintegers .So
1
m ;:::;ml
l
(w(!j);!k) =(!j;!k)(cid:0) 1mi((cid:11)i;!k)= dj (cid:0)mw;
i=
X
foraninteger
1 (1.14)
2
mw = mk ((cid:11)k;(cid:11)k);
thisleadsto(1.11),andweprovedthelemma.
2
Letusprovenowboundednessofthefunctions inthelimit(1.8).
1
y~ ;:::;y~l
LEMMA1.2.Inthelimit
x x0 (1.15)
= (cid:0)i!k(cid:28); (cid:28) ! +1;
thefunctions x x havetheexpansion
1
y ( );:::;yl( )
x e2 0 x0 e 2 1 (1.16)
(cid:25)dj(cid:28) (cid:0) (cid:25)(cid:28)
yj( ) = [yj( )+O( )]; j = ;:::;l;
where
0 x0 1 e2 x0 (1.17)
(cid:0) (cid:25)i(w(!j); )
yj( ) =nj :
wX2W 0
(w(!j)(cid:0)!j;!k)=
Proof.Fromtherepresentation(1.13)weseethattheexponential
exp2 x inthelimit(1.15)behavesas
[ (cid:25)i(w(!j); )]
e2 e2 x0
(cid:25)(dj(cid:0)mw)(cid:28) (cid:25)i(w(!j); )
;
where the non-negative integer is defined in (1.14). Thus the leading contri-
bution in the asymptotic behavimouwr of the sum (1.12) comes from those
satisfying 0.Lemmaisproved. w 2 W
mk = 2
COROLLARY1.1.Thefunctions belongto .
1 1
Proof.From(1.16)itfollowsthy~at(x);:::;y~l+ (x) A
0 0 e2 0 1 0 x0 1
(cid:25)idjxl+
y~j(x) !y~j(x ) = yj( ); j = ;:::;l;
0
1
y~l+ !
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174 BORISDUBROVINANDYOUJINZHANG
in the limit (1.8), where the functions 0 x0 are defined in (1.17). Corollary is
proved. yj( )
2
Themainresultofthissectionis
THEOREM1.1.Thering isisomorphictotheringofpolynomialsof .
1 1
Proof.WewillshowthAatanyelement ofthering canberey~pr;e:s:e:n;tey~dl+as
apolynomialof .Thiswillbefe(nxo)ughsincethAefunctions
1 1 1 1
arealgebraicallyy~in;d:e:p:e;ny~dl+ent. y~ ;:::;y~l+
Fromtheinvariancew.r.t. iteasilyfollows(usingTheorem[B])thatany
canberepresentedasapolynWomialof 1.Weneefd(xto)
1 1 (cid:0)1
showthatin therearenonegativey~p(oxw)e;r:s:o:;fy~l(x);y~l+ (x);y~l+
f(x) f
e2 1
1 (cid:25)ixl+
y~l+ (x) = :
Let’sassume
n 1 1
f(x)= y~l+ Pn(y~ (x);:::;y~l(x))
n>(cid:0)N
X
andthepolynomial doesnotvanishidenticallyforcertain
1
positiveinteger .FPro(cid:0)mN(Cy~or(oxl)la;r:y::1;.y1~lw(xe)o)btainthatinthelimit(1.8)thefunction
behavesasN
f(x)
e2 2 0 1 0 0 0 0 e 2
(cid:25)N(cid:28)(cid:0) (cid:25)iNxl+ 1 (cid:0) (cid:25)(cid:28)
f(x)= [P(cid:0)N(y~ (x );:::;y~l(x ))+O( )];
where
0 0 e2 0 1 0 x0 1
(cid:25)idjxl+
y~j(x )= yj( ); j = ;:::;l
andthefunctions 0 x0 aredefinedin(1.17).
Toobtainafunycjti(on)boundedfor itisnecessarytohave
(cid:28) !+1
0 0 0 0 0
1
P(cid:0)N(y~ (x );:::;y~l(x )) (cid:17) ;
for any 0 x0 0 . We show now that this is impossible due to algebraic
1
independxenc=eo(f th;xelf+un)ctions 0 0. It is sufficient to prove algebraic inde-
1
pendenceofthefunctions 0 xy~ ;:::;0y~lx .
1
y ( );:::;yl( )
MAINLEMMA.TheFourierpolynomials 0 x 0 x arealgebraicallyinde-
1
pendent. y ( );:::;yl( )
We will prove that these functions are functionally independent, i.e., that the
Jacobian
0 x
det @yj( )
@xi !
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AFFINEWEYLGROUPSANDFROBENIUSMANIFOLDS 175
does not vanish identically. At this end we derive explicit formulae for these
functionsandthenprovenonvanishingoftheJacobian.
Consider
1 2
( ) ( )
Rk =R [R [:::;
hereanysubsystem 1 isarootsystemofthetype forsome (seeTableI).
( )
Let beafundaRmen;t:a:l:weightorthogonalto A1 r. r
( )
!i RnR
LEMMA1.3.Letforsome
w 2W(R)
l
w(!i)=!i(cid:0) 1cm(cid:11)m;
m=
X
suchthat 0,then 0onlyif 1 .
( )
Proof.cIkn= cm1 6=there exist(cid:11)sm 2sRuch that it maps all positive roots of
( ) 0
1 intoWneg(RatkivneRroo)ts.Clearly, wpreservesall 1 ,and .
( ) 0 ( ) 0
RWkenrRepresent w (cid:11)m 2R w (!i) =!i
1 2
( ) ( )
w(!i)=!i(cid:0)(cid:11) (cid:0)(cid:11) (cid:0):::;
where ’saresumofsomepositiverootsof .Then
(i) (i)
(cid:11) R
1 2 1
0 ( ) 0 ( ) ( )
w w(!i) =!i(cid:0)(cid:11) (cid:0)w ((cid:11) +(cid:1)(cid:1)(cid:1)) =!i(cid:0)(cid:11) + 1 c~m(cid:11)m;
(cid:11)m2RknR()
X
forsomenonnegativeintegers ,andnotalloftheseintegersvanishifthereexists
certain 1 suchthat c~m 0.Thiscontradictstonegativityof .
( ) 0
Lemma(cid:11)imsp2=roRved. cm 6= w w(!i)(cid:0)!i
2
A similar statement holds true for other components 2 (if any) of .
( )
R ;::: Rk
LEMMA1.4.Ifforsome
w 2W(R)
w(!k) =!k(cid:0) cm(cid:11)m;
m6=k
X
thenall 0.
Proofc.mTh=ereexists suchthatitmapsanypositiverootsof into
0
negativeones,andpreswerv2esW(.RSko) Rk
!k
sumof somepositiveroots
0
w w(!k) =!k + ;
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176 BORISDUBROVINANDYOUJINZHANG
whichleadstotheresultofthelemma.
2
LEMMA 1.5. Under the assumption of Lemma 1 3 there exists 1
( )
suchthat : w~ 2 W(R )
w~(!i)=w(!i)=!i(cid:0) 1 cm(cid:11)m:
(cid:11)m2R( )
X
Proof.Wewilluseinductiononthelengthof . Ifthelengthof equalsone,
then the lemma holds true obviously. We now aswsume that the lemmwa holds true
when the length of is less than . Let has the reduced expression ,
1
thenitfollowsfromwLemma1.3thpat w (cid:27)i :::(cid:27)ip
(1.18)
1
(cid:27)i :::(cid:27)ip(!i)=!i(cid:0) 1 cm(cid:11)m:
(cid:11)m2R( )
X
Ifall 0,thenwecanput 1,otherwisewerewrite(1.18)intheform
cm = w~ =
1 1 1 1
!i =(cid:27)ip(cid:27)ip(cid:0) :::(cid:27)i (!i)(cid:0) 1 cm(cid:27)ip(cid:27)ip(cid:0) :::(cid:27)i ((cid:11)m):
(cid:11)m2R()
X
Weput
l
1 1
(cid:27)ip(cid:27)ip(cid:0) :::(cid:27)i (!i) =!i(cid:0) 1bm(cid:11)m;
m=
X
for some nonnegative integers . We claim now that there exists a root
1
1 suchthat 0inb(;1:.1::8;)balnd
1 ( ) 1
(cid:11)m 2R cm 6=
1
(cid:0) 1 1 1 1
w ((cid:11)m )=(cid:27)ip(cid:27)ip(cid:0) :::(cid:27)i ((cid:11)m )
isanegativeroot.Indeed,otherwisetheroot
l
1 1
1bm(cid:11)m+ 1 cm(cid:27)ip(cid:27)ip(cid:0) :::(cid:27)i (cid:11)m
m= (cid:11)m2R( )
X X
could not be equal to zero since ’s are nonnegative integers. We use now the
followingproposition. cm
PROPOSITION [H, page 50]. Let be some simple roots of not
1
necessarilydistinct .If (cid:11)j ;:::;(cid:11)jt R (
)
1 1
(cid:27)j (cid:1)(cid:1)(cid:1)(cid:27)jt(cid:0) ((cid:11)jt)
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