Table Of ContentTWISTED REALITY CONDITION FOR DIRAC OPERATORS
TOMASZBRZEZIN´SKI,NICOLACICCOLI,LUDWIKDA˛BROWSKI,ANDANDRZEJSITARZ
6
ABSTRACT. Motivatedbyexamplesobtainedfromconformaldeformationsofspectraltriples
1
andaspectraltripleconstructiononquantumcones,weproposeanewtwistedrealitycondi-
0
2 tionfortheDiracoperator.
n
u
J
2
1. INTRODUCTION
2
In [3] Connes proposed that real spectral triples should be understood as a noncommuta-
]
A tive counterpart of spin manifolds. This proposal was further supported by establishing in
Q [6] a one-to-one correspondence between classical spin geometries and real spectral triples
. for commutativealgebras of functions over manifolds. Yet, the number of examples of gen-
h
t uine noncommutative real spectral triples is limited. So far, the list includes finite spectral
a
m triples[14, 18], noncommutativetori [3]and theirsubalgebras: noncommutativeBieberbach
[ manifolds [17] and the quantum pillow [2], in some cases with a full classification of in-
equivalent,irreducibleand equivariantreal spectral triples[19, 24, 17]. All furtherexamples
2
v of so-called θ-twisted or isospectral deformations [4, 21, 23] are essentially obtained by us-
4 ing theconstructionof real spectral triples overthenoncommutativetorusor overtheMoyal
0
plane[10]. TheexampleofthePodles´standardquantumsphere[7]wastheonlynon-flatcase
4
7 not obtained as a θ-twist; however, it failed to satisfy some other natural conditions set for
0
spectral geometries. In fact, with the help of Poisson geometry, it was proved in [12] that if
.
1
a noncommutativespectral tripleis a deformation ofthe real spectral tripleof functionson a
0
6 2-dimensionalsmoothmanifold,thentheunderlyingRiemannianmanifoldcanonlybeeither
1 the flat torus or the round sphere. On the other hand, some interesting examples, such as the
:
v quantum group SU (2) [9] allow for an almost real spectral triple, with a slightly modified
q
i
X order-onecondition,which failsto beexact.
r In this note, we propose a modified definition of a real spectral triple, in which only the
a
realitystructureisgeneralised. Thusincontrasttoaninterestingrecentpaper[15]weremain
in the framework of bona fide spectral triples here. We show that this definition allows for
fluctuations ofDirac operators, which do not change thebimoduleof oneforms. Finally, we
illustratethetheory withtwoexamples,whichmotivatetheproposeddefinition.
2. TWISTED REALITY
2.1. Reality twisted by a linear automorphism. Let A be a complex ∗-algebra and let
(H,π) be a (left) representation of A on a complex vector space H. A linear automorphism
ν ofH defines an algebraautomorphism
ν¯ : End(H) → End(H), φ 7→ ν ◦φ◦ν−1.
2010MathematicsSubjectClassification. 58B34,58B32,46L87.
1
2 T.BRZEZIN´SKI,N.CICCOLI,L.DA˛BROWSKI,ANDA.SITARZ
Theinverseofν¯ is φ 7→ ν−1 ◦φ◦ν. Since ν¯ isan algebramap, thecomposite
πν : A π // End(H) ν¯ // End(H)
is an algebra map too, and hence it defines a new representation (H,πν) of A. The map ν is
an isomorphismthat intertwines(H,π) with(H,πν).
Definition 2.1. Let A be a complex ∗-algebra, (H,π) a representation of A, and let D be a
linear operatoron H. Let ν be a linearautomorphismofH, such that there existsan algebra
automorphismνˆ : A → Asatisfying
(2.1) π ◦νˆ = ν¯◦π = πν.
Wesaythatthetriple(A,H,D)admitsaν-twistedrealstructureifthereexistsananti-linear
mapJ : H → H suchthat J2 = ǫid, and,forall a,b ∈ A,
(2.2) [π(a),Jπ(b)J−1] = 0,
(2.3) [D,π(a)]Jν¯2(π(b))J−1 = Jπ(b)J−1[D,π(a)],
(2.4) DJν = ǫ′νJD,
(2.5) νJν = J,
whereǫ,ǫ′ ∈ {+,−}.
If (A,H,D) admits a grading operator γ : H → H, γ2 = id, [γ,π(a)] = 0, for all a ∈ A,
γD = −Dγ, and ν2γ = γν2, thenthetwistedreal structureJ is alsorequired tosatisfy
(2.6) γJ = ǫ′′Jγ,
whereǫ′′ isanothersign.
This purely algebraic definition of twisted reality is motivated by and aimed at being ap-
plicable to spectral triples. In this case H is a Hilbert space, and π is a star representation
of A by bounded operators. The operator D is assumed to be a (essentially) selfadjoint op-
erator on the dense domain H with a compact resolvent, and such that for every a ∈ A the
0
commutators[D,π(a)] arebounded. TheoperatorJ isantiunitary. Thegradingγ (in caseof
an evenspectraltriple)is self-adjoint. Besides theusualanalyticproperties ofthedataofthe
spectral triple, the map ν is then a (essentially) selfadjoint invertibleoperator defined on the
domain containing H and with the requirement that ν¯ maps π(A) into bounded operators.
0
More precisely, ∀a∈A, νπ(a)ν−1 preserves H and is bounded on H , and so extends to an
0 0
element of B(H). Furthermore (2.4)and (2.5)should be understoodas operators H → H .
0 0
In such a case we shall say that a spectral triple admits a ν-twisted real structure, or simply
thatit isaν-twisted real spectraltriple.
Thesignsǫ,ǫ′,ǫ′′ determinetheKO-dimensionmodulo8intheusualway[3]. Often(2.2)
is called the order-zero condition, (2.3) is called the twisted order-one condition, while we
shallreferto(2.4)astothetwistedǫ′-condition,andto(2.5)astothetwistedregularity(with
the adjective twisted omitted if ν = id). Note that the twisted order-one condition (2.3) can
beequivalentlystated as
(2.7) [D,π(a)]Jπ(νˆ(b))J−1 = Jπ(νˆ−1(b))J−1[D,π(a)].
TWISTEDREALITYCONDITIONFORDIRACOPERATORS 3
Furthemore,thetwistedorder-oneconditioncanbeunderstoodasrightA-linearityasfollows.
Given a representation π of a ∗-algebra A in H, a linear automorphism ν of H and an anti-
linearautomorphismJ ofH, onecan definearightA-modulestructureonH by
(2.8) h·a := Jν¯(π(a∗))J−1(h) = πJ◦ν(a∗)(h), for alla ∈ A, h ∈ H.
We denote H with this A-module structure by HJ,ν. Then the first-order condition (2.3)
simply states that, for all a ∈ A, the commutators [D,π(a)] are right A-linear maps from
HJ,ν2 toHJ,id.
Therelation(2.1)betweenνˆandν canbealsotermedas “ν implementsanautomorphism
νˆin therepresentation π".
Remark 2.2. One can contemplate a more general notion of the ν-twisted reality defined as
in Definition 2.1 except the existence of νˆ satisfying (2.1). Note, however, that if (H,π)
is a faithful representation, then π is a monomorphism and hence A is isomorphic to π(A).
Therefore, ifin addition,
(2.9) Im(πν) ⊆ Im(π),
then the restriction of ν¯ to π(A) defines a unique automorphism νˆ of the algebra A that
satisfies (2.1). All the examples discussed in Section 3 come with faithful representations
π and automorphisms ν satisfying condition (2.9). Finally, we note in passing that one can
always workwithafaithful representationofthealgebra A/kerπ.
2.2. Twistedfluctuations. InthesetupofDefinition2.1,letΩ1 beabimoduleofone-forms:
D
1
Ω := π(a )[D,π(b )], a ,b ∈ A ,
D i i i i
( )
i
X
wheretheindicesirun overafiniteset.
The standard fluctuation of a spectral triple (A,H,D) consists of adding to the Dirac op-
erator D a selfadjoint one-form α ∈ Ω1 . In the case of a real spectral triple, the fluctuated
D
Dirac operator, D, becomes D + α + ǫ′JαJ−1, where α + ǫ′JαJ−1 is selfadjoint. For a
ν-twistedreal spectraltriplewesetthefluctuated DiracoperatorD to be:
α
D := D +α+ǫ′νJαJ−1ν,
α
with the requirement that α + ǫ′νJαJ−1ν is selfadjoint. We shall often use the shorthand
notationα′ = νJαJ−1ν.
Proposition 2.3. If (A,H,D) with J ∈ End(H) is a ν-twisted real spectral triple, then
(A,H,D ) with (the same) J is also a ν-twisted real spectral triple. If (A,H,D) is even
α
with grading γ, then (A,H,D ) is even with (the same) grading γ. The composition of
α
twistedfluctuationsisa twistedfluctuation.
Proof. AsaperturbationofDbyaboundedselfadjointoperator,thefluctuatedDiracoper-
atorD is selfadjoint,has boundedcommutatorswithπ(a) ∈ Aandhas compact resolvent.
α
First, we shall demonstrate that a fluctuation of the fluctuated Dirac operator is also a
fluctuation. Inotherwords,thebimoduleofoneformsisindependentofthechoiceofα. Let
4 T.BRZEZIN´SKI,N.CICCOLI,L.DA˛BROWSKI,ANDA.SITARZ
ustakeany a ∈ Aandα ∈ Ω1 , and compute:
D
[α′,π(a)] = νJαJ−1νπ(a)−π(a)νJαJ−1ν
= νJαJ−1νπ(a)−ν(ν−1π(a)ν)JαJ−1ν
= νJαJ−1νπ(a)−νπ(νˆ−1(a))JαJ−1ν
= νJαJ−1νπ(a)−νJαJ−1π(νˆ(a))ν
= νJαJ−1νπ(a)−νJαJ−1 νπ(a)ν−1 ν = 0,
by (2.1), (2.2)and (2.7). Therefore, foranyα ∈ Ω1 and(cid:0)a ∈ A: (cid:1)
D
[D ,π(a)] = [D,π(a)]+[α,π(a)],
α
butthenthelastcommutatorisagainaone-form. Forthisreason,thefluctuatedDiracopera-
torD does satisfy theν-twisted order-one condition(2.3). Thestatement about the grading
α
γ iseasily verified.
To finish the proof, it remains only to check that D satisfies the compatibility relation
α
withJ (2.4), thatis:
D Jν = ǫ′νJD .
α α
SinceD itselfsatisfies(2.4), suffices itto check itforα+ǫ′α′:
(α+ǫ′α′)Jν = αJν +ǫ′νJαJ−1νJν
= αJν +ǫ′νJα
= ǫ′νJ α+ǫ′J−1ν−1αJν
= ǫ′νJ (α+ǫ′α′).
(cid:0) (cid:1)
wherewehaveused(2.5). ⊔⊓
3. EXAMPLES
The definitions in Section 2.1 are motivated by two classes of examples. The first class
arises from conformally rescaled spectral geometries, which (albeit in a different setting)
werefirstproposedbyConnesandTretkoff[5]andledtotwistedspectraltriples. Thesetting
using spectral triples instead was proposed by Da˛browski and Sitarz [8] for more general
modifications. It appears, however, that for conformal rescaling the setup is quite robust,
not only keeping most of the properties of spectral geometries (like the Hochschild cycle
condition[22]),butalsomakingtheconformallyrescalednoncommutativetoriintoquantum
metricspaces in thesenseofRieffel [16].
Thesecondclassofexamplescomesfromq-deformedgeometries,wheremodulartwisting
automorphismsnaturallyappear and are significantfor cycliccohomology[11] and modular
Fredholmmoduleconstructions,asexplainedforexamplein[20]. Itisworthpointingoutthat
theoperatorimplementingtherealityisintrinsicallyrelated totheTomita-Takesakioperator,
whichis usuallynontrivialin q-deformedgeometry.
TWISTEDREALITYCONDITIONFORDIRACOPERATORS 5
3.1. ConformallytransformedDiracoperators. Letusassumethatwehavearealspectral
triple(A,H,D,J)withreality operatorJ andfixed signsǫ,ǫ′. Let k ∈ π(A) bepositiveand
invertiblesuch thatk−1 is alsobounded,and letk′ := JkJ−1.
Proposition3.1. If(A,H,D,J)isarealspectraltriple,whichsatisfiesorder-onecondition,
thenfor:
D = k′Dk′, ν(h) = (k−1k′)(h),
k
the triple (A,H,J,D ,ν) is a ν-twisted real spectral triple. If furthermore (A,H,D,J) is
k
even withgradingγ, then (A,H,D ,J,ν) iseven with (thesame)gradingγ.
k
Proof. Since k and k′ are bounded operators, it is clear that ν¯ maps bounded operators to
boundedoperators, anddueto(2.2), forall a ∈ A:
ν¯(π(a)) = k−1π(a)k.
Clearly νˆ(a) = (k˜)−1ak˜,where k˜ ∈ A issuchthat π(k˜) = k, satisfies(2.1).
By the properties of the real spectral triple (A,H,D,J), k′(DomD) ⊂ DomD and the
operator D is obviously selfadjoint on the domain DomD, and due to (2.2) has bounded
k
commutators with arbitrary a ∈ A. Furthermore, D has compact resolvent: this is obvious
k
if D is invertible since then D−1 = k−1D−1k−1 and D−1 is compact and k−1 bounded by
k
assumption. Otherwise, we can employ some invertible bounded perturbation of D on the
kernel ofD.
WeshownowthatD satisfies thetwistedorder-onecondition(2.3):
k
Jπ(b)J−1[D ,π(a)] = Jπ(b)J−1JkJ−1[D,π(a)]JkJ−1
k
= k′[D,π(a)]k′J(k−2π(b)k2)J−1 = [D ,π(a)]Jν¯2(π(b))J−1.
k
Nextwecheck (2.5):
νJν = k−1JkJ−1Jk−1JkJ−1 = J.
Finally,ifJD = ǫ′DJ thenforD wehave:
k
JD = Jk′J−1JDk′ = ǫ′kDJk′ = ǫ′k(k′)−1D (k′)−1kJ,
k k
sothat (2.4)issatisfied
νJD = ǫ′D Jν.
k k
Thestatementaboutthegradingγ iseasilyverified. ⊔⊓
Remark3.2. ClearlytheKO-dimensionof(A,H,D ,J,ν)isthesameasof(A,H,D,J). In
k
the‘classical’caseofamanifoldM and(commutative)A = C∞(M)representedintheusual
way on L2-sections of some vector bundle on M with Ad being the complex conjugation,
J
theconformaltwistsν arealways trivialas JkJ−1 = k forapositivek and henceν = id.
Remark 3.3.1 Usingargumentssimilarto thoseintheproofofProposition3.1, onecan show
that if (A,H,D,J,ν) is a ν-twisted real spectral triple which satisfies the twisted order-one
condition, then, for all k as in Proposition 3.1 such that ν¯(kk′) = kk′, (A,H,D ,J,µ) is a
k
µ-twistedreal spectral triple,where
D = k′Dk′, µ(h) = k′νk−1(h).
k
Thegradingγ, ifitexists,is againunchanged.
1WethankTomaszMaszczykforraisingthispoint.
6 T.BRZEZIN´SKI,N.CICCOLI,L.DA˛BROWSKI,ANDA.SITARZ
3.2. Twisted reality ofDiracoperators onquantum cones. Thecoordinatealgebraofthe
quantumdiscO(D ) isacomplex∗-algebragenerated by z, subjectto therelation
q
(3.1) z∗z −q2zz∗ = 1−q2,
whereq ∈ (0,1);see[13]. O(D )can beunderstoodas aZ-graded algebra
q
O(D ) = O(D ) ,
q q n
n∈Z
M
with the degrees given on the generators by |z| = −|z∗| = 1. For any N > 1, the quantum
coneO(CN)isdefinedasthe∗-algebrageneratedbyy andself-adjointxsubjecttorelations:
q
N−1 N
(3.2) xy = q2Nyx, yy∗ = 1−q−2lx , y∗y = 1−q2lx ;
l=0 l=1
Y (cid:0) (cid:1) Y(cid:0) (cid:1)
see [1]. The map O(CN) → O(D ) defined on the generators as x 7→ 1 − zz∗, y 7→ zN
q q
providesanidentificationoftheconewiththesubalgebraofO(D )consistingofallelements
q
ofdegreeamultipleofN, i.e.,
O(CN) ∼= O(D ) .
q q nN
n∈Z
M
ThroughoutwewillviewO(CN)as asubalgebraofO(D ) inthisway.
q q
UsingtheZ-grading ofO(D ) wedefineadegreecountingalgebraautomorphism,
q
ν : O(D ) → O(D ), a 7→ q|a|a,
q q
forallhomogeneouselementsofO(D ). Obviously,ν preservesthedegreesofhomogeneous
q
elements, and hence restricts to an automorphism of O(CN). It is also compatible with the
q
∗-structurein thesensethat,
(3.3) ν ◦∗◦ν = ∗.
Themaps ∂ ,∂ : O(D ) → O(D ), defined ongenerators ofthediscalgebraby
− + q q
(3.4) ∂ (z) = z∗, ∂ (z∗) = 0, ∂ (z) = 0, ∂ (z∗) = q2z,
− − + +
extendtothewholeofO(CN)as ν2-skew derivations,i.e., bythetwistedLeibnizrule,
q
(3.5) ∂ (ab) = ∂ (a)ν2(b)+a∂ (b),
± ± ±
for all a,b ∈ O(D ). As a consequence of (3.4) and (3.5), the maps ∂ have degrees ±2,
q ±
respectively,and sinceν isthedegree-countingautomorphism,
(3.6) ν ◦∂ ◦ν−1 = q±2∂ .
± ±
Thecombinationof(3.3),(3.4)and (3.5)alsoyields
(3.7) ν(∂ (a)∗) = ν−1(∂ (a∗)).
± ∓
Set
H = O(D ) , H = O(D ) , H = H ⊕H ,
+ q nN+1 − q nN−1 + −
n∈Z n∈Z
M M
TWISTEDREALITYCONDITIONFORDIRACOPERATORS 7
i.e., H is the external direct sum of two vectorspaces H and H . Since O(CN) is spanned
+ − q
by homogeneous elements of degrees that are multiples of N, and ν preserves the degrees,
bothH and hencealsoH areleft O(CN)-modules(representations)by theaction
± q
(3.8) π(a)(h ) = ν2(a)h , forall a ∈ O(CN), h ∈ H .
± ± q ± ±
Furthermore,ν restrictstoanautomorphismofthevectorspaceH. Note,however,thatsince
ν is an automorphism of the graded algebra O(D ) and the action of O(CN) is defined by
q q
themultiplicationin O(D ), theinducedautomorphismν¯ of the(linear) endomorphismring
q
takes aparticularlysimpleformon π(a),
(3.9) ν¯(π(a)) = π(ν(a)), forall a ∈ O(CN),
q
so that νˆ satisfying (2.1) is simply the restriction of ν to O(CN). Since ∂ have degree ±2,
q ±
respectively,
(3.10) ∂ (H ) ⊆ H ,
± ∓ ±
and the∗-operation reversesdegrees,we can defineoperators
(3.11) D : H → H, (h ,h ) 7→ −q−1∂ (h ), q∂ (h ) ,
+ − + − − +
(3.12) J : H → H, (h ,h(cid:0) ) 7→ −h∗,h∗ , (cid:1)
+ − − +
and alsothegradingoperator,
(cid:0) (cid:1)
(3.13) γ : H → H, (h ,h ) 7→ (h ,−h ).
+ − + −
Proposition3.4. Withthesedefinitions,(O(CN),H,D)isan(algebraic)evenspectraltriple
q
ofKO-dimensiontwo with ν-twisted realstructureJ andgradingγ.
Proof. The order zero condition (2.2) follows immediately by the definition of J and the
fact that ∗ is an anti-algebra involution, the condition (2.5) is a straighforward consequence
of(3.3). Checkingthetwistedorder-onecondition(2.3)andtwistedǫ′-condition(2.4)isabit
moreinvolvedand usesall theproperties(3.3)–(3.9). Specifically, forallh ∈ H ,
± ±
νJD(h ) = q±1ν(∂ (h )∗) = q±1ν−1 ∂ (h∗) ,
± ∓ ± ± ±
by (3.7). On theotherhand,
(cid:0) (cid:1)
DJν(h ) = ±D ν−1(h∗) = q∓1∂ ν−1(h∗ ) = q±1ν−1 ∂ (h∗) ,
± ± ± ± ± ±
where the first equality follows by (3.3) and the third one by (3.6). This proves the twisted
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
ǫ′-condition (2.4). To prove the twisted order-one condition (2.3) observe that, for all a,b ∈
O(CN) and h ∈ H ,
q ± ±
[D,π(a)](h ) = ±q±1 ∂ ν2(a)h −ν2(a)∂ (h )
± ∓ ± ∓ ±
= ±q±1∂ ν2(a) ν2(h ) = ±q±5ν2(∂ (a)h ),
(cid:0)∓ (cid:0) (cid:1) ± (cid:1)∓ ±
by theq-skew-derivationproperties(3.5)(cid:0)and(3.(cid:1)6), and
Jπ(b)J−1(h ) = h ν2(b)∗ = h ν−2(b∗),
± ± ±
by (3.3). In viewofthelatterand (3.9),
Jν¯2(π(b))J−1(h ) = h ν−4(b∗).
± ±
8 T.BRZEZIN´SKI,N.CICCOLI,L.DA˛BROWSKI,ANDA.SITARZ
PuttingallthistogetheroneeasilychecksthatJ andDsatisfythetwistedorder-onecondition
(2.3). The statement about the grading γ is easy to see and we have J2 = ǫ = −1, ǫ′ = 1,
ǫ” = −1, which correspondsto theKO-dimension2. ⊔⊓
The above construction of the twisted reality structure for a Dirac operator is not particu-
larly specific to the case of quantum cones, but is a special case of a more general situation.
Thekeyroleisplayedbythefactthatthemodulestructureisinducedfromthemultiplication
ofagraded ∗-algebrawhich admitsq-skewderivationscompatiblewiththe∗-structure.
Let G be a group and A = ⊕ A be a G-graded ∗-algebra, and set B := A , where e is
g∈G g e
theneutralelementofG. Thegradingisassumedtobecompatiblewiththe∗-structureinthe
sensethat,forall g ∈ G,
A∗g ⊆ Ag−1,
so,in particular,B is a∗-subalgebraofA. Let G ⊂ G andset
+
G := {g−1 | g ∈ G }, H = A , H = H ⊕H ,
− + ± g + −
gM∈G±
i.e., H is the external direct sum of two vector spaces H and H . The definition of G
+ − −
ensures that H∗ ⊆ H . Let ν be a graded (i.e., degree preserving) algebra automorphismof
± ∓
Athatsatisfies(3.3). Let∂ : A → Abelinearmapsthatsatisfy(3.5),(3.6),(3.7)and(3.10).
±
IfweviewH asaleftB-modulebythemultiplicationinAorasin(3.8),thentheoperatorD
defined by (3.11) has the twisted reality structure J given by (3.12) and the grading γ given
by (3.13).
ACKNOWLEDGMENTS
T. Brzezin´ski acknowledges support of INdAM–GNFM and would like to thank SISSA
for hospitality. N. Ciccoli acknowledges support of INdAM–GNSAGA. A. Sitarz thanks
Università degli Studi di Perugia for hospitality. N. Ciccoli, L. Da˛browski and A. Sitarz
acknowledgesupportoftheNCNgrant2012/06/M/ST1/00169. Theauthorsaremostgrateful
tothereferees forvery helpfulcomments.
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