Sympleti Killing spinors
Svatopluk Kr
ysl
Abstrat.Let(M;!)beasympletimanifoldadmittingametapletistruture
(a sympleti analogue of the Riemannianspin struture) and a torsion-free
sympletionnetionr. SympletiKillingspinoreldsforthisstrutureare
setions ofthe sympletispinor bundlesatisfying aertain rstorderpartial
dierential equation andthey arethe mainobjetof thispaper. Wederivea
neessaryonditionwhihhastobesatisedbyasympletiKillingspinoreld.
UsingthisonditiononemayeasilyomputethesympletiKillingspinorelds
for the standardsympleti vetor spaesand the round sphere S 2
equipped
withthevolumeformoftheroundmetri.
Keywords:Fedosov manifolds, sympleti spinors, sympleti Killing spinors,
sympletiDiraoperators,Segal-Shale-Weilrepresentation
Classiation: 58J60,53C07
1. Introdution
Inthis artile weshall study theso alled sympletiKilling spinorelds on
Fedosov manifolds admitting a metapletistruture. A Fedosov manifold is a
strutureonsistingof asympletimanifold(M 2l
;!)andtheso alledFedosov
onnetionon(M;!). AFedosovonnetionr isanaÆneonnetionon(M;!)
suh that it is sympleti, i.e., r! =0, and torsion-free. Let us notie that in
ontrarytotheRiemanniangeometry,aFedosovonnetionisnotunique. Thus,
it seems natural to add the Fedosov onnetion into the studied struture and
obtain the notion of a Fedosov manifold. See, e.g., Tondeur [13℄ for symple-
tionnetionsforpresympletistrutures andGelfand, Retakh, Shubin[3℄for
Fedosovonnetions.
Itisknownthatifl>1,theurvaturetensorofaFedosovonnetiondeom-
posesintotwoinvariantparts,namelyintothesoalledsympletiRiiurvature
and sympletiWeylurvature tensorelds. If l=1, onlythesympletiRii
urvatureours. SeeVaisman[14℄fordetails.
InordertodeneasympletiKillingspinoreld,weshallbrieydesribethe
soalledmetapletistrutures withhelp ofwhihthese elds aredened. Any
sympletigroupSp (2l;R) admitsanon-trivial,i.e.,onneted,two-foldovering,
Theauthorof thisartilewas supported bythe grantGA
CR306-33/80397of the Grant
Agenyof the CzehRepubli. Thework isa partof the researh projetMSM0021620839
thesoalledmetapletigroup,denotedbyMp(2l;R)inthispaper. Ametapleti
strutureoverasympletimanifoldisasympletianalogueoftheRiemannian
spin struture. In partiular, one of its parts is a prinipal Mp(2l;R)-bundle
whihoverstwiethebundleofsympletiframeof(M 2l
;!). Letusdenotethis
prinipalMp (2l;R)-bundle byq: Q!M.
Now, let us say afew words about thesympleti spinor elds. These elds
aresetionsofthesoalledsympletispinorbundleS !M. Thisvetorbundle
isthebundleassoiatedto theprinipalMp(2l;R)-bundle q:Q!M viatheso
alled Segal-Shale-Weil representation. TheSegal-Shale-Weilrepresentationis a
distinguishedrepresentationofthemetapletigroupandplaysasimilarrole in
the quantization of boson partiles asthe spinor representationsof spin groups
play in thequantizationoffermions. See,e.g.,Shale[12℄. TheSegal-Shale-Weil
representation is unitary and does not desend to a representation of the sym-
pletigroup. ThevetorspaeoftheunderlyingHarish-Chandra(g;K)-module
of the Segal-Shale-Weil representation is isomorphi to S
(R l
), the symmetri
powerofaLagrangiansubspaeR l
ofthesympletivetorspaeR 2l
. Thus,the
situationisparalleltotheomplexorthogonalase,wherethespinorrepresenta-
tionanberealizedontheexterior powerofamaximalisotropisubspae. The
Segal-Shale-Weilrepresentation and someof itsanalyti versions are sometimes
alledosillatory representation,metapletirepresentationorsympletispinor
representation. Foradetailedexplanationofthelastname,see,e.g.,Kostant[8℄.
ThesympletiKillingspinoreldisanon-zerosetionofthesympletispinor
bundleS!Msatisfyingertainlinearrstorderpartialdierentialequationfor-
mulatedbytheonnetionr S
: (M;S) (M;TM)! (M;S),theassoiated
onnetion to the Fedosovonnetion r. This partial dierential equation is a
sympleti analogueof the lassial sympleti Killing spinor equation from at
leasttwoaspets. Oneof them is ratherformal. Namely, thedening equation
for asympleti Killing spinor is of the\same shape" asthat one for aKilling
spinor eld on aRiemannian spin manifold. The seond similarity an be ex-
pressedbyomparingthisequation withthesoalledsympletiDiraequation
andthesympletitwistorequationandwillbedisussedbelowinthispaper. Let
usmentionthatanysympletiKillingspinorelddeterminesauniqueomplex
number,the soalled sympleti Killingspinor number. Letus notie that the
sympletiKillingspinoreldswereonsideredalreadyin aonnetionwiththe
existeneofalinearembeddingofthespetrumofthesoalledsympletiDira
operatorintothespetrumofthesoalledsympletiRarita-Shwingeroperator.
The sympletiKilling spinor elds represent an obstrution for thementioned
embedding. SeeKr ysl[10℄forthisaspet.
Inmanypartiularases,theequationforsympletiKillingspinoreldsseems
toberatherompliated. Ontheotherhand,in manyasesitisknownthatits
solutions are rare. Therefore it is reasonableto look for a neessary ondition
satised by a sympleti Killing spinor eld whih is simpler than the dening
equation itself. Let us notie that similar neessary onditions are known and
parallelmethodswereusedinRiemannianorLorentzianspingeometry. See,e.g.,
Friedrih[2℄.
Inthispaper,weshallprovethatanysympletiKillingspinoreldneessarily
satisesertainzerothorder dierentialequation. Morepreisely, weprovethat
anysympletiKillingspinorisneessarilyasetionofthekernelofasympleti
spinorbundlemorphism. Wederivethisequationbyprolongatingthesympleti
Killingspinorequation. Wemakesuhaprolongationthatenablesustoompare
the result with an appropriate part of the urvature tensor of the assoiated
onnetionr S
atingonsympletispinors. Anexpliitformulaforthis partof
the urvature ation wasalready derived in Kr ysl [11℄. Espeially, it is known
thatthesympletiWeylurvatureofrdoesnotshowupinthispartandthus,
thementionedmorphism depends on thesympletiRiipartof theurvature
of the Fedosov onnetion r only. This will make us able to prove that the
onlysympletiKillingnumberofaFedosovmanifoldofWeyltypeiszero. This
willin turnimplythatanysympletiKillingspinoronthestandardsympleti
vetorspaeofanarbitrarynitedimensionandequippedwiththestandardat
onnetion isonstant. This result anbe obtainedeasily when one knowsthe
prolongatedequation,whereasomputingthesympletiKillingspinorswithout
this knowledge israther ompliated. Thisfat will beillustrated whenwewill
omputethesympletiKillingspinorsonthestandardsympleti2-planeusing
justthedening equationforsympletiKillingspinoreld.
Theaseswhentheprolongatedequationdoesnothelpsoeasilyasinthease
of the Weyl type Fedosov manifolds are the Rii type ones. Nevertheless, we
provethattherearenosympletiKillingspinorsonthe2-sphere,equippedwith
thevolume form of theround metrias the sympletiform and the Riemann-
ian onnetionas theFedosov onnetion. Let us remark that in this ase, the
prolongatedequationhasashapeofastationaryShrodingerequation. Morepre-
isely,ithastheshapeoftheequationfortheeigenvaluesofertainosillator-like
quantum Hamiltoniandetermined ompletely bythe sympletiRiiurvature
tensoroftheFedosovonnetion.
Letusnotiethattherearesomeappliationsofsympletispinorsinphysis
besidesthose in thementionedartileofShale[12℄. Foranappliation in string
theoryphysis,see,e.g.,Green,Hull[4℄.
Inthe seond setion, someneessarynotions from sympletilinear algebra
and representation theory of redutiveLie groups are explained and the Segal-
Shale-Weil representation and the sympleti Cliord multipliation are intro-
dued. In the third setion, the Fedosov onnetions are introdued and some
properties of theirurvature tensorsating onsympletispinor elds are sum-
marized. In the fourth setion, the sympleti Killing spinors are dened and
sympletiKillingspinors onthestandardsympleti2-planeareomputed. In
this setion, a onnetion of the sympleti Killing spinor elds to the eigen-
funtionsofsympletiDiraandsympletitwistoroperatorsisformulatedand
proved. Further, the mentioned prolongation of the sympleti Killing spinor
equationisderivedandthesympletiKillingspinorelds onthestandardsym-
pletivetorspaesare omputed. Attheend,theaseoftheroundsphereS 2
istreated.
2. Sympletispinors and sympleti spinor valued forms
Let us start realling some notions from sympleti linear algebra. Let us
mentionthatweshalloftenusetheEinsteinsummationonventionwithoutmen-
tioning it expliitly. Let (V;!
0
) be asympleti vetor spae of dimension 2l,
i.e., !
0
is a non-degenerate anti-symmetri bilinear form on thevetorspae V
ofdimension2l. LetLandL 0
betwoLagrangiansubspaes 1
of(V;!
0
)suhthat
LL 0
= V. Let fe
i g
2l
i=1
be an adapted sympleti basis of (V = L L 0
;!
0 ),
i.e., fe
i g
2l
i=1
isasympletibasis and fe
i g
l
i=1
L and fe
i g
2l
i=l+1 L
0
. Beause
thedenitionofasympletibasisisnotunique,weshallxonewhihweshall
usein this text. A basisfe
i g
2l
i=1
of (V;!
0
)is alled sympleti,if!
0 (e
i
;e
j )=1
i 1 i l and j = l+i; !
0 (e
i
;e
j
) = 1 i l+1 i 2l and j = i l
and !
0 (e
i
;e
j
) =0in the remaining ases. Whenevera sympletibasis will be
hosen, we will denote the basis of V
dual to fe
i g
2l
i=1 by f
i
g 2l
i=1
. Further for
i;j =1;:::;2l, we set !
ij := !
0 (e
i
;e
j
) and similarly for other type of tensors.
Fori;j=1;:::;2l,wedene ! ij
bytheequation P
2l
k =1
!
ik
! jk
=Æ i
j .
As in the orthogonal ase, we would like to rise and lowerindies. Beause
thesympletiform !
0
isantisymmetri,weshould bemorearefulin thisase.
ForoordinatesK
ab::::::d rs:::t:::u
of atensorK overV,wedenote theexpression
! i
K
ab::::::d rs:::t
byK
ab:::
i
:::d rs:::t
andK
ab:::
rs:::t:::u
!
ti byK
ab:::
rs:::
i :::u
andsim-
ilarlyforother typesof tensorsand alsoin ageometri setting whenwewill be
onsideringtensoreldsoverasympletimanifold(M;!).
Let us denote the sympleti group Sp(V;!
0
) of (V;!
0
) by G. Beause the
maximalompat subgroupof Gis isomorphito theunitary groupU(l)whih
isofhomotopytypeZ,wehave
1
(G)'Z. Fromthetheoryofoveringspaes,
weknowthatthereexistsuptoanisomorphismauniqueonneteddoubleover
ofG. ThisdoubleoveristhesoalledmetapletigroupMp (V;!
0
)andwillbe
denoted by
~
G in this text. Weshall denote the overing homomorphismby ,
i.e., :
~
G! Gis axed member ofthe isomorphismlass ofall onneted2:1
overings.
Now, let us reall some notions from representation theory of redutive Lie
groupswhihweshallneedinthis paper. Letus mentionthat thesenotionsare
rather of tehnial harater for the purpose of ourartile. ForaredutiveLie
groupGin thesenseofVogan[15℄,letR(G)betheategorytheobjetofwhih
areomplete,loallyonvex,Hausdorvetorspaeswithaontinuousationof
Gwhih isadmissible and of nite length; themorphisms areontinuouslinear
G-equivariant maps between the objets. Let us notie that, e.g., nite overs
of the lassial groups are redutive. It is known that any irreduible unitary
representation of a redutive group is in R(G). Let gbe the Lie algebra of G
1
i.e.,maximalisotropiwithrespetto!0,inpartiulardimL=dimL 0
=l
andK beamaximalompatsubgroupof G. It iswellknown thatthere exists
the so alled L 2
-globalization funtor, denoted by L 2
here, from the ategory
of admissible Harish-Chandra modules to the ategory R(G). See Vogan [15℄
for details. Let us notie that this funtor behaves ompatiblywith respet to
Hilbert tensor produts. See, e.g., Vogan [15℄ again. Foran objet E in R(G),
let us denote its underlying Harish-Chandra (g;K)-module by E and when we
will be onsidering only itsg C
-module struture, weshall denote it byE. Ifg C
happensto be asimple omplexLie algebra of rankl, letus denote itsCartan
subalgebra byh C
. The set of roots for (g C
;h C
) is then uniquely determined.
Furtherletushooseaset +
ofpositiverootsanddenotetheorresponding
setoffundamental weightsbyf$
i g
l
i=1
. For2h C
,letusdenotetheirreduible
highestweightmodulewith thehighestweightbyL().
Let us denote by U(W ) the group of unitary operators on a Hilbert spae
Wand letL:Mp(V;!
0
)!U(L 2
(L)) be theSegal-Shale-Weilrepresentationof
themetapletigroup. Itisaninnitedimensional unitaryrepresentationofthe
metapleti group on the omplex valued square Lebesgue integrable funtions
dened on the Lagrangian subspae L . This representation does not desend
to a representation of the sympleti groupSp (V;!
0
). See, e.g., Weil [16℄ and
Kashiwara, Vergne [7℄. For onveniene, let us set S := L 2
(L) and all it the
sympletispinor module and itselementssympleti spinors. It is well known
that as a
~
G -module, S deomposes into the diret sum S = S
+
S of two
irreduiblesubmodules. ThesubmoduleS
+
(S )onsistsofeven(odd)funtions
in L 2
(L). Further, let us notie that in Kr ysl [9℄, a slightly dierent analyti
version(basedonthesoalledminimalglobalizations)ofthisrepresentationwas
used.
Asin theorthogonalase,wemaymultiplyspinorsbyvetors. Themultipli-
ation : : VS! S will be alled sympletiCliord multipliation and it is
denedasfollows. Forf 2Sandi=1;:::;l,weset
(e
i
:f)(x):={x i
f(x);
(e
l+i
:f)(x):=
f
x i
(x); x2L
andextenditlinearlytogetthesympletiCliordmultipliation. Thesympleti
Cliordmultipliation(byaxedvetor)hastobeunderstoodasanunbounded
operator on L 2
(L). See Habermann, Habermann [6℄ for details. Let us also
notie that the sympleti Cliord multipliation orresponds to the so alled
Heisenberganonialquantizationknownfromquantummehanis. (Forbrevity,
weshallwritev:w:s,insteadofv:(w:s),v;w2Vands2S.)
It iseasy tohekthat thesympletiCliord multipliation satisesthe re-
lationdesribedin thefollowing
Lemma1. Forv;w2V ands2S,wehave
v:(w:s) w:(v:s)= {! (v;w)s:
Proof: SeeHabermann,Habermann[6℄.
Letusonsidertherepresentation
:
~
G!Aut (
^
V
S)
ofthemetapletigroup
~
Gon V
V
Sgivenby
(g)(s):=
^r
(g)L(g)s;
where r=0;:::;2l, 2 V
r
V
, s2S and
^r
denotestherth wedgepowerof
the representation
dual to , and extendedlinearly. For deniteness, let us
onsider the vetor spae V
V
S equipped with thetopology of the Hilbert
tensor produt. Beause the L 2
-globalization funtor behaves ompatibly with
respettotheHilberttensorproduts,oneaneasilysee thattherepresentation
belongstothelassR(
~
G ).
Inthe nexttheorem, thespae o sympletivalued exteriortwo-formsis de-
omposedinto irreduiblesummands.
Theorem2. For 1
2
dim (V)=l>2,thefollowingisomorphism
2
^
V
S
'E
20
E
21
E
22
holds. Forj
2
=0;1;2,theE 2j2
areuniquelydetermined bythe onditionsthat
rst,theyaresubmodules oftheorrespondingtensorprodutsandseond,
E 20
'S 'L($
l 1 3
2
$
l ); E
20
+ 'S
+ 'L(
1
2
$
l );
E 21
'L($
1 1
2
$
l ); E
21
+
'L($
1 +$
l 1 3
2
$
l );
E 2 2
+
'L($
2 1
2
$
l
) and E 2 2
'L($
2 +$
l 1 3
2
$
l ):
Proof: ThistheoremisprovedinKr ysl[10℄orKr ysl[9℄forthesoalledminimal
globalizations. Beause theL 2
-globalizationbehavesompatiblywithrespetto
theonsideredHilberttensorproduttopology,thestatementremainstrue.
Remark. Let us notiethat for l=2, thenumberof irreduible summands in
sympletispinorvaluedtwo-formsisthesameasthatoneforl>2. Inthisase
(l=2), oneonlyhastohangethepresriptionforthehighestweightsdesribed
inthepreeding theorem. Forl=1,thenumberof theirreduiblesummandsis
dierentfrom thatoneforl2. Nevertheless,in thisasethedeompositionis
alsomultipliity-free. SeeKr ysl[9℄fordetails.
InordertomakesomeproofsinthesetiononsympletiKillingspinorelds
F +
:
^
V
S! +1
^
V
S; F +
(s):=
2l
X
i=1
i
^e
i :s;
F :
^
V
S! 1
^
V
S; F (s):=
2l
X
i;j=1
! ij
ei e
j :s;
H :
^
V
S!
^
V
S; H :=fF +
;F g:
Remark. (1) Oneeasilyndsoutthattheoperatorsareindependentofthe
hoieofanadaptedsympletibasisfe
i g
2l
i=1 .
(2) Letus remarkthat theoperatorsF +
;F andH denedheredierfrom
theoperatorsF +
;F ;H denedinKr ysl[9℄. Though,byaonstantreal
multipleonly.
(3) The operators F +
and F are used to provethe Howe orrespondene
forMp(V;!
0
)atingon V
V
Sviatherepresentation. Moreorless,
theortho-sympletisuperLiealgebraosp(1j2)playstheroleofa(super
Lie)algebra,arepresentationofwhihistheappropriateommutant. See
Kr ysl[9℄fordetails.
Inthenextlemmathe
~
G-equivarianeoftheoperatorsF +
;F andHisstated,
somepropertiesofF
arementionedandthevalueofH ondegree-homogeneous
elements is omputed. We shall use this lemma when we will be treating the
sympletiKillingspinorelds inthefourthsetion.
Lemma 3. Let (V = LL 0
;!
0
)be a2l dimensional sympleti vetor spae.
Then
(1) theoperatorsF +
,F +
andH are
~
G -equivariant,
(2) (a) F
jE 11
=0,
(b) F +
jE 00
isanisomorphismontoE 10
,
() (F +
) 2
jS
= {
2
!Id
jS
anditisanisomorphismontoE 20
.
(3) Forr=0;:::;2l,wehave
H
j V
r
V
S
={(r l)Id
j V
r
V
S :
Proof: SeeKr ysl[9℄.
Let us remark that the items 1 and 3 of the preeding lemma follow by a
diretomputation,andtheseonditemfollowsfromtherstitem,deomposition
3. Curvature ofFedosov manifoldsand its ations onsympleti
spinors
After we havenished the \algebraipart" of this paper, let us reall some
basifatsonFedosovmanifolds, theirurvature tensors,metapletistrutures
andtheationoftheurvaturetensoronsympletispinorelds.
LetusstartreallingsomenotionsandresultsrelatedtothesoalledFedosov
manifolds. Let(M 2l
;!)beasympletimanifoldofdimension2l. Anytorsion-
free aÆne onnetion r on M preserving !, i.e., r! = 0, is alled Fedosov
onnetion. Thetriple(M;!;r),whererisaFedosovonnetion,willbealled
Fedosovmanifold. AswehavealreadymentionedintheIntrodution,aFedosov
onnetionforagiven sympletimanifold(M;!)isnotunique. Letus remark
that Fedosovmanifolds areused foraonstrution ofgeometri quantizationof
sympletimanifoldsdueto Fedosov. See,e.g.,Fedosov[1℄.
Toxournotation,letusreallthelassialdenition oftheurvaturetensor
R r
oftheonnetionr,weshallbeusinghere. Weset
R r
(X;Y)Z:=r
X r
Y
Z r
Y r
X
Z r
[X ;Y℄ Z
forX;Y;Z 2X(M).
Letushoosealoaladaptedsympletiframefe
i g
2l
i=1
onaxedopensubset
U M. Byaloaladaptedsympletiframefe
i g
2l
i=1
overU,wemeansuhaloal
frame that for eah m 2U thebasis f(e
i )
m g
2l
i=1
is an adapted sympletibasis
of(T
m M;!
m
). Wheneverasympletiframeishosen,wedenoteitsdualframe
byf i
g 2l
i=1
. Althoughsomeoftheformulasbelowholdonlyloally, ontaininga
loaladaptedsympletiframe,wewillnotmentionthisrestritionexpliitly.
From the sympleti urvature tensor eld R r
, we anbuild thesympleti
Riiurvaturetensoreld r
dened bythelassialformula
r
(X;Y):=Tr(V 7!R r
(V;X)Y)
for eah X;Y 2 X(M) (the variable V denotes a vetoreld on M). Forthe
hosenframeandi;j =1;:::;2l,wedene
ij :=
r
(e
i
;e
j ):
LetusdenetheextendedRiitensoreldbytheequation
e
(X;Y;Z ;U):=e
ijk n X
i
Y j
Z k
U n
; X;Y;Z ;U 2X(M);
wherefori;j;k;n=1;:::;2l,
2(l+1)e
ijk n :=!
in
jk
!
ik
jn +!
jn
ik
!
jk
in +2
ij
!
k n :
AFedosovmanifold (M;!;r) isalled ofWeyltype,if =0. Letus notie,
that it is alled of Rii type, if R = .e In Vaisman [14℄, one an nd more
informationontheSp (2l;R)-invariantdeompositionofthe urvature tensorsof
Now,letusdesribethegeometristruturewithhelpofwhihthesympleti
Killingspinoreldsaredened. Thisstruture, alledmetapleti,is asymple-
tianalogueof thenotion ofa spinstruture in theRiemannian geometry. For
asympletimanifold(M 2l
;!)ofdimension2l,letusdenotethebundleofsym-
pletiframeinTMbyP andthefoot-pointprojetionofP ontoM byp. Thus
(p:P!M;G),whereG'Sp(2l;R),isaprinipalG-bundleoverM. AsinSub-
setion2,let:
~
G!Gbeamemberoftheisomorphismlassofthenon-trivial
two-foldoveringsofthesympletigroupG. Inpartiular,
~
G'Mp(2l;R). Fur-
ther, let us onsider aprinipal
~
G-bundle (q : Q ! M;
~
G )over thesympleti
manifold (M;!). We all a pair (Q;) metapleti struture if : Q ! P is
asurjetivebundle homomorphism overthe identity onM and if the following
diagram,
Q
~
G
//
Q
q
M
PG
//
P p
>>
}
}
}
}
}
}
}
}
withthehorizontalarrowsbeingrespetiveationsofthedisplayedgroups,om-
mutes. See, e.g., Habermann, Habermann [6℄ and Kostant [8℄ for details on
metapletistrutures. Letus only remark that typialexamples ofsympleti
manifoldsadmitting ametapleti strutureare otangentbundlesof orientable
manifolds (phase spaes), Calabi-Yau manifolds and omplex projetive spaes
CP 2k +1
,k2N
0 .
Let us denote the vetor bundle assoiated to the introdued prinipal
~
G -
bundle(q : Q!M;
~
G) viathe representationating onSbyS, and allthis
assoiated vetorbundle sympleti spinorbundle. Thus, wehaveS =Q
S.
The setions 2 (M;S) will be alled sympleti spinor elds. Further for
j
2
= 0;1;2, we dene the assoiated vetor bundles E 2j
2
by the presription
E 2j
2
:= Q
E
2j
2
. Further, we dene E r
:= (M;Q
V
r
V
S), i.e., the
spaeosympleti spinorvalueddierentialr-forms,r=0;:::;2l. Beausethe
sympletiCliordmultipliation is
~
G-equivariant(seeHabermann, Habermann
[6℄), we an lift it to the assoiated vetor bundle struture, i.e., to let it at
onthe elements from (M;S). Forj
2
=0;1;2,letus denote the vetorbundle
projetions (M;E 2
)! (M;E 2j
2
)byp
2j2 ,i.e.,p
2j2
: (M;E 2
)! (M;E 2j
2
)for
allappropriatej
2
. Thisdenitionmakessensebeauseduetothedeomposition
result(Theorem 2) and Remark below Theorem 2, the
~
G-module ofsympleti
spinorvaluedexterior2-formsismultipliity-free.
LetZ betheprinipalbundleonnetionontheprinipal G-bundle(p:P !
~
prinipal
~
G -bundle(q:Q!M;
~
G). Letusdenotebyr theovariantderivative
assoiatedto
~
Z. Thus,in partiular,r S
atsonthesympletispinorelds.
AnysetionoftheassoiatedvetorbundleS =Q
Sanbeequivalently
onsideredasa
~
G-equivariantS-valuedfuntiononQ. Letusdenotethisfuntion
by
^
, i.e.,
^
: Q ! S. Fora loal adapted sympletiframe s : U ! P, let us
denote by s: U ! Q oneof the lifts of s to Q. Finally, letus set
s :=
^
Æs.
Further for q2 Q and 2 S, letus denote by[q; ℄ theequivalene lass in S
ontaining(q; ). (Asitiswellknown,thetotalspaeS ofthesympletispinor
bundleistheprodutQSmoduloanequivalenerelation.)
Lemma4. Let(M;!;r)beaFedosovmanifoldadmittingametapletistru-
ture. Then for eah X 2 X(M), 2 (M;S) and a loal adapted sympleti
frames:U !P,wehave
r S
X
=[s;X(
s )℄
{
2 l
X
i=1 [e
i+l :(r
X e
i ): e
i :(r
X e
i+l
):℄and
r S
X
(Y:)=(r S
X
Y):+X:r S
Y :
Proof: SeeHabermann,Habermann[6℄.
Theurvaturetensoronsympletispinoreldsisdenedbytheformula
R S
(X;Y)=r S
X r
S
Y
r
S
Y r
S
X
r
S
[X ;Y℄
;
where2 (M;S)andX;Y 2X(M).
Inthenextlemma,apartoftheationofR S
onthespaeofsympletispinors
isdesribedusingjust thesympletiRiiurvature tensoreld .
Lemma5. Let(M;!;r)beaFedosovmanifoldadmittingametapletistru-
ture. Thenforasympletispinoreld2 (M;S),wehave
p 20
R S
= {
2l
ij
!
k l
k
^ l
e
i :e
j ::
Proof: SeeKr ysl[11℄.
4. SympletiKilling spinor elds
In this setion, we shall fous our attention to the sympleti Killing spinor
elds. Morespeially,weomputethesympletiKillingspinoreldsonsome
Fedosovmanifoldsadmittingametapletistrutureandderiveaneessaryon-
ditionsatisedbyasympletiKillingspinoreld.
Let (M;!;r) bea Fedosov manifoldadmitting a metapletistruture. We
allanon-zerosetion2 (M;S)sympleti Killingspinoreld if
r S
=X:
for a omplex number 2 C and eah vetor eld X 2 X(M). The omplex
numberwillsometimes bealled sympletiKillingspinornumber. (Allowing
the zero setion to be a sympleti Killing spinor would make the notion of a
sympletiKillingspinornumbermeaningless.)
Letusnote that oneanrewriteequivalentlythepreedingdening equation
forasympletiKillingspinorinto
r S
=F +
:
Indeed, if this equation is satised, we get by inserting the loal vetor eld
X =X i
e
i
theequation r S
X =
X (
i
e
i
:)= i
(X)e
i
:=X i
e
i
:=X:,
i.e.,thedeningequation. Conversely,oneanprovethatr S
X
=X:isequiv-
alent to
X r
S
=
X (F
+
). Beause this equation holds for eah vetoreld
X,wegetr S
=F +
. Weshallallboththedeningequationandtheequiv-
alentequation forasympletiKilling spinoreld the sympletiKillingspinor
equation.
Inthenextexample,weomputethesympletiKillingspinorsonthestandard
sympleti2-plane.
Example 1. Let us solve the sympleti Killing spinor equation for the stan-
dard sympleti vetorspae (R 2
[s;t℄;!
0
) equipped with the standard at Eu-
lideanonnetion r. Inthis ase, (R 2
;!
0
;r)is also aFedosov manifold. The
bundleof sympletiframein TR 2
denes aprinipal Sp(2;R)-bundle. Beause
H 1
(R 2
;R) = 0, we know that there exists, up to a bundle isomorphism, only
one metapleti bundle over R 2
, namely the trivial prinipal Mp(2;R)-bundle
R 2
Mp(2;R) !R 2
andthusalsoauniquemetapletistruture:Mp(2;R)
R 2
! Sp(2;R)R 2
given by (g;(s;t)) := ((g);(s;t)) for g 2 Mp(2;R) and
(s;t) 2R 2
. LetS !R 2
bethesympleti spinorbundle. In this aseS ! R 2
isisomorphitothetrivialvetorbundleSR 2
=L 2
(R)R 2
!R 2
. Thus,we
may think of asympleti spinoreld asof amapping :R 2
! S=L 2
(R).
Letus dene :R 3
!C by (s;t;x) :=(s;t)(x) foreah (s;t;x) 2R 3
. One
easily showsthat is a sympleti Killing spinor if and only if the funtion
satisesthesystem
s
= {x and
t
=
x :
If=0, thesolutionof this systemofpartial dierentialequationsis nees-
sarily (s;t;x)= (x),(s;t;x)2R 3
, forany 2L 2
(R) .
If 6=0, let us onsider the independent variable and orrespondingdepen-
dentvariable transformations=s;y =t+ 1
x,z =t 1
x and (s;t;x) =
e
(s;t+ 1
x;t 1
x) = e
(s;y;z). The Jaobian of this transformation is
2= 6= 0 and the transformation is obviously a dieomorphism. Substitut-
transformedsystem
e
s
= {
2
2
(y z) e
e
y +
e
z
= (
e
y
1
+
e
z (
1
)):
(Let us notie that the substitution we have used is similar to that one whih
isusually usedto obtainthed'Alemebert's solutionofthewaveequationin two
dimensions.) Therstequationimplies
e
z
=0,andthus e
(s;y;z)= (s;y)fora
funtion . Substitutingthisrelationintotheseondequationofthetransformed
system,weget
s
= {
2
(y z) 2
:
Thesolutionofthisequationis (s;y)=e {
2
2
(y z)s e
(y)forasuitablefuntion e
.
Beauseofthedependeneoftherighthandsideofthelastwrittenequationonz,
weseethat doesnotexistunless=0or e
=0(Moreformally,onegetsthese
restritionsbysubstitutingthelast writtenformula for into therstequation
ofthetransformedsystem.) Thus, neessarily =0or=0. Thease=0is
exludedbytheassumptionat thebeginningofthisalulation.
Summing up, we have proved that any sympleti Killing spinor eld on
(R 2
;!
0
;r)isonstant,i.e.,foreah(s;t)2R 2
,wehave(s;t)= forafuntion
2L 2
(R ). TheonlysympletiKilling spinornumberiszerointhisase.
Remark. More generally, one an treatthe aseof a standardsympletive-
torspae(R 2l
[s 1
;:::;s l
;t 1
;:::;t l
℄;!
0
)equippedwiththestandardatEulidean
onnetionr. OnegetsbysimilarlinesofreasoningthatanysympletiKilling
spinorforthisFedosovmanifoldisalsoonstant,i.e.,
(s 1
;:::;s l
;t 1
;:::;t l
)= ;
for(s 1
;:::;s l
);(t 1
;:::;t l
)2R l
and 2L 2
(R l
). Butweshallseethisresultmore
easilybelowwhenwewillbestudyingtheprolongatedequationmentionedinthe
Introdution.
Now,in order tomakeaonnetionof thesympletiKillingspinorequation
tosomeslightlymoreknownequations,letus introduethefollowingoperators.
Theoperator
D: (M;S)! (M;S); D:= F r S
is alled sympletiDira operator and its eigenfuntions are alled sympleti
Diraspinors. Letus notie that thesympleti Diraoperator wasintrodued
Theoperator
T: (M;S)! (M;E 11
); T:=r S
p 10
r S
isalled(the rst)sympletitwistoroperator.
In the next theorem, the sympleti Killing spinor elds are related to the
sympletiDiraspinorsandtothekernelofthesympletitwistoroperator.
Theorem6. Let(M;!;r)beaFedosovmanifoldadmittingametapletistru-
ture. A sympletispinoreld 2 (M;S)isasympletiKillingspinoreldif
and only if is a sympletiDiraspinor lying in the kernelof thesympleti
twistoroperator.
Proof: Weprovethisequivaleneintwosteps.
(1) Suppose2 (M;S)isasympletiKillingspinortoasympletiKilling
number2C. Thusitsatisestheequationr S
=F +
. Applyingthe
operator F tothebothsidesofthepreeding equationandusingthe
denition of thesympleti Diraoperator, we getD = F F +
=
( H+F +
F )= H= ( {l)={ldue to thedenition of
H and Lemma3(2)(a)and(3). ThusisasympletiDiraspinor.
Now, weomputeT. Using thedenition of T,wegetT=(r S
p 10
r S
)=(F +
p
10
F +
)=p 11
F +
=0,beauseF +
2 (M;E 10
)
duetoLemma 3(2)(a).
(2) Conversely,let2 (M;E 00
)beinthekernelofthesympletitwistorop-
eratorandalsoasympletiDiraspinor. Thus,wehaver S
p
10
r S
=
0 and D = F r S
= for a omplex number 2 C. From
the rst equation, we dedue that := r S
2 (M;E 10
). Beause
F +
j (M;E 00
)
is surjetive onto (M;E 10
) (see Lemma 3(2)(b)), there ex-
istsa 0
2 (M;E 00
)suh that =F + 0
. Letus omputeF +
F =
F +
F F + 0
=F +
(H F +
F ) 0
=F +
( {l 0
)= {l ,where we have
used thedening equation forH and Lemma 3(2)(a)and (3). Thus we
get
F +
F ={l : (1)
FromthesympletiDiraequation,weget= F . Thus F +
F
=F +
. Using theequation (1), weobtain {l = F +
, i.e., r S
=
{
l F
+
. Thus,isasympletiKillingspinortothesympletiKilling
spinornumber {=l.
Inthenexttheorem, wederive thementionedprolongationofthesympleti
Killing spinor equation. It is a zeroth order equation. More preisely, it is an
equation for the setions of the kernel of an endomorphism of the sympleti
spinorbundleS!M. AsimilaromputationiswellknownfromtheRiemannian
Theorem 7. Let (M ;!;r) be a Fedosov manifold admitting a metapleti
struture and a sympleti Killing spinor eld 2 (M;S) to the sympleti
Killingspinornumber. Then
ij
e
i :e
j
:=2l 2
:
Proof: Let2 (M 2l
;S)beasympletispinorKillingeld,i.e.,r S
X
=X:
foraomplexnumberandanyvetoreldX 2X(M). ForvetoreldsX;Y 2
X(M),wemaywrite
R S
(X;Y) = (r
X r
Y r
Y r
X r
[X ;Y℄ )
= r
X
(Y:) r
Y
(X:) [X;Y℄:
= (r
X
Y):+Y:(r
X
) (r
Y
X): X:r
Y
: [X;Y℄:
= T(X;Y):+ 2
(Y:X: Y:X:)
= T(X;Y):+{ 2
!(X;Y)={ 2
!(X;Y);
wherewehaveusedthesympletiKillingspinorequationandtheompatibility
ofthesympletispinorovariantderivativeandthesympletiCliordmultipli-
ation(Lemma4).
Thus R S
= { 2
!. Beause of Lemma 3(2)(), we know that the right
hand side is in (M;E 20
). Thus also R S
=p 20
R S
. Using Lemma 5, we get
{
2l
!
ij
e
i :e
j :={
2
!. Thus ij
e
i :e
j
:=2l 2
andthetheoremfollows.
Remark. Letus reallthat in theRiemannian spingeometry(positivedenite
ase),theexisteneofanon-zeroKillingspinorimpliesthatthemanifoldisEin-
stein. Further, let us notie that if the sympleti Rii urvature tensor is
(globally)diagonalizablebyasympletomorphism, theprolongatedequationhas
theshapeof the equation for eigenvaluesof the Hamiltonianof an ellipti l di-
mensional harmoni osillator with possibly varying axes lengths. An example
ofadiagonalizablesympletiRiiurvaturewill betreatedin Example 3. Al-
though,inthis asetheaxiswillbeonstantandtheharmoniosillatorwill be
spherial.
Now, wederiveasimpleonsequeneofthepreeding theorem in theaseof
FedosovmanifoldsofWeyltype,i.e.,=0.
Corollary 8. Let (M;!;r) be a Fedosov manifold of Weyl type. Let (M;!)
admitametapletistrutureandasympletiKillingspinoreldtothesym-
pletiKillingspinornumber. ThenthesympletiKillingspinornumber=0
andisloallyovariantlyonstant.
Proof: Follows immediately from the preeding theorem and the sympleti
Killingspinorequation.
Example2. Let usgobaktotheaseof (R 2l
;!
0
;r) from RemarkbelowEx-
strutureisovariantlyonstant,i.e.,in fat onstantinthis ase,and anysym-
pletiKillingnumberiszero. Inthisase,weseethattheprolongatedequation
fromTheorem7makesitpossibletoomputethesympletiKillingspinorelds
without any big eort, ompared to the alulations in Example 1 where the
2-planewastreated.
In the next example, we ompute the sympleti Killing spinor elds on S 2
equippedwiththestandardsympletistrutureandtheRiemannianonnetion
of theround metri. This is anexample of aFedosov manifold (speied more
arefullybelow)forwhih oneannotuse Corollary8,beauseitis notofWeyl
type. Butstill,oneanuseTheorem7.
Example 3. Considerthe roundsphere (S 2
;r 2
(d 2
+sin 2
d 2
)) ofradius r >
0, being the longitude a the latitude. Then ! := r 2
sind^d is the
volume form of theround sphere. Beause ! is also asympleti form,(S 2
;!)
is a sympletimanifold. Let us onsider the Riemannian onnetion r of the
round sphere. Then r preservesthe sympleti volume form ! being ametri
onnetionoftheroundsphere. Beauseristorsion-free,weseethat(S 2
;!;r)is
aFedosovmanifold. Now,wewillworkinaoordinatepathwithoutmentioning
itexpliitly. Letusset e
1 :=
1
r
ande
2 :=
1
rsin
. Clearly,fe
1
;e
2
gisaloal
adaptedsympletiframeanditisaloalorthogonalframeaswell. Withrespet
tothisbasis,theRiiform ofrtakestheform
[ ij
℄
i;j=1;2
=
1=r 0
0 1=r
:
LetusonsiderS 2
astheomplexprojetivespaeCP 1
. Itiseasyto seethat
the(unique)omplexstrutureonCP 1
isompatiblewiththevolumeform. The
rst Chern lass of thetangent bundle to CP 1
is known to beeven. Thus, the
sympletimanifold(S 2
;!)admitsametapletistrutureandwemayonsidera
sympletiKillingspinoreld2 (S 2
;S)orrespondingtoasympletiKilling
spinornumber. Beause thersthomologygroupofthesphereS 2
is zero,the
metapletistrutureisuniqueandthusthetrivialone.Beauseofthetrivialityof
theassoiatedsympletispinorbundleS!S 2
,wemaywrite(m)=(m;f(m))
wheref(m)2L 2
(R ) foreahm2S 2
. UsingTheorem7andthepresriptionfor
theRiiform,weget that ij
e
i :e
j
:[f(m)℄= 1
r
H[f(m)℄=2 2
f(m),where H =
2
x 2
x 2
isthequantumHamiltonianoftheonedimensionalharmoniosillator.
ThesolutionsoftheSturm-LiouvilletypeequationH[f(m)℄=2r 2
f(m),m2S 2
,
arewellknown. The eigenfuntionsof H are theHermitefuntions f
l
(m)(x) =
h
l
(x):=e x
2
=2 d l
dx l
(e x
2
)form2S 2
andx2R andtheorrespondingeigenvalues
are (2l+1),l2N
0
. Thus2r 2
= (2l+1)and onsequently
={ r
2l+1
:
Usingthefat thatfe
1
;e
2
gis aloal orthonormalframeandr ismetriand
torsion-free,weeasilyget
r
e1 e
1
=0 r
e1 e
2
=0
r
e
2 e
1
= ot
r e
2 r
e
2 e
2
= ot
r e
1 :
From the denition of dierentiability of funtions with values in a Hilbert
spae, we see easily as a onsequene of the preeding omputations that any
sympletiKillingspinor eld is neessarilyofthe form(m) =(m;(m)f
l (m))
forasmoothfuntion2C 1
(S 2
;C) . SubstitutingthisAnsatzintothesympleti
Killingspinorequation,wegetforeahvetoreld X2X(S 2
)theequation
r
X (f
l
)=(X)f
l +r
X f
l
=(X:f
l ):
Dueto Lemma4,wehaveforaloal adaptedsympletiframes:U S 2
!
P =Sp (2;R)S 2
,
r
X f
l
=[s;X(f
l )
s
℄ {
2 [e
2 :(r
X e
1 ): e
1 :(r
X e
2 ):℄f
l :
(See the paragraph above Lemma 4 for an explanation of the notation used in
thisformula.)
Beausem7!(m;f
l
(m)) isonstantasasetionofthetrivialbundleS!S 2
,
therstsummandofthepreedingexpressionvanishes. ThusforX =e
1
,weget
(e
1 )f
l +
{
2 [e
2 :(r
e
1 e
1 ): e
1 :(r
e
1 e
2 ):℄f
l
=(e
1 :f
l ):
Using the knowledge of the valuesof r
e
1 e
j
, for j = 1;2, omputed above, the
seond summandat theleft hand sideofthe last writtenequation vanishesand
thus,weget
1
r
f
l
={xf
l :
Thisequation implies(;)= (x;)e {rx
for x suh that h
l
(x)6=0and a
suitablefuntion . (Thesetofsuhx2R,suhthath
l
(x)6=0istheomplement
in R ofanite set.) Beause r>0is givenand isertainly non-zero(seethe
presription for above), the only possibility for to be independent of x is
=0. Therefore=0and onsequently=0. On theotherhand,=0(the
zerosetion)islearlyasolution,butaordingtothedenitionnotasympleti
Killing spinor. Thus, there is no sympleti Killing spinor eld on the round
sphere.
Remark. Inthefuture,oneanstudy holonomyrestritionsimpliedbytheex-
istene ofasympletiKilling spinor. Oneanalsotry toextend theresultsto
generalsympletionnetions,i.e.,todroptheonditiononthetorsion-freeness
orstudyalsothesympletiKillingeldsonRiitypeFedosovmanifoldsadmit-
tingametapletistrutureinmoredetail.
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Sokolovsk
a83,Praha8,Czeh
Republi
E-mail: kryslkarlin.m.uni.z
(Reeived September9,2011 , revised January23,2012 )
