Table Of ContentNRCPS-HE-56-07 TUW-07-15 ESI-1992
Non-topological non-commutativity in string theory
Sebastian Guttenberg1,∗, Manfred Herbst2,∗∗, Maximilian Kreuzer3,∗∗∗, and Radoslav
Rashkov4,†
8
0 1 NCSRDemokritos,INP,PatriarchouGregoriou&NeapoleosStr.,15310AgiaParaskeviAttikis,GREECE
0 2 CERN,1211Geneva23,SWITZERLAND
2 3 InstituteforTheoreticalPhysics,TU–Wien,WiednerHauptstr.8–10,1040Vienna,AUSTRIA
n 4 ErwinSchro¨dingerInstituteforMathematicalPhysics,Boltzmanngasse9,1090Vienna,AUSTRIA
a
J
8 Keywords Noncommutativegeometry,D-branes,deformationquantization
1
PACS 02.40.Gh04.62.+v11.25.-w11.25.Uv
] Quantizationofcoordinatesleadstothenon-commutativeproductofdeformationquantization,butisalso
h
at the roots of string theory, for which space-time coordinates become the dynamical fields of a two-
t
- dimensional conformal quantum field theory. Appositely, open string diagrams provided the inspiration
p for Kontsevich’s solution of the long-standing problem of quantization of Poisson geometry by virtueof
e hisformalitytheorem. InthecontextofD-branephysicsnon-commutativityisnotlimited,however,tothe
h topolocial sector. Weshow thatnon-commutative effectiveactionsstillmakesensewhen associativityis
[ lostandestablishageneralized Connes-Flato-Sternheimercondition throughsecondorder inaderivative
expansion.ThemeasureingeneralcurvedbackgroundsisnaturallyprovidedbytheBorn–Infeldactionand
2
reduces tothesymplecticmeasure inthetopological limit, but remainsnon-singular even for degenerate
v
Poissonstructures.AnalogoussuperspacedeformationsbyRR–fieldsarealsodiscussed.
7
6
1
4 Contribution to the proceedings of the BW2007 Workshop ”Challenges Beyond the Standard Model”,
.
2 September2-9,2007,Kladovo,Serbia
1
7
0 1 Introduction
:
v
The non-commutativeproductof deformation quantization [1,2] can be derived from string theory in a
i
X topological limit where the space-time metric is small as compared to the anti-symmetric B-field (the
r ancestor of the Poisson bi-vector) [3–5]. The non-commutative product thus amounts to a summation
a
of the leadingB-field contributionsto the effectiveaction. In thenon-symplecticcase thisinterpretation
is spoiled, however, by the absence of a canonical measure. From the string theory point of view, on
the other hand, associativity is lost for generic backgrounds [6], but the Born-Infeld action provides a
canonical measure [4,7]. We show that the concept of effective actions does not require associativity,
but rather a generalized Connes–Flato–Sternheimer condition called cyclicity [8,9], i.e. commutativity
andassociativityuptosurfaceterms[10]. Cyclicityimplies,however,acompatibilityconditionbetween
the star productandthe measure[9], whichforBorn-Infeldturnsoutto beequivalentto the generalized
MaxwellequationforthegaugefieldontheD-brane[7,10].In[10]wefoundthatcyclicityalsorequiresa
gaugemodificationoftheKontsevichproductatsecondderivativeorderinaderivativeexpansionandwe
discussedtheD-branephysicsrelatedtothesemathematicalstructures.
In section2 of thisnotewe reviewsomeaspects ofdeformationquantizationandformalityin simple
termsbyillustratingtheemergenceofHochschildcocycles,Gerstenhaberbracketsandgaugetransforma-
tionsaccompanyingdiffeomorphismsinderviativeexpansions. Insection3wediscussthestringyorigin
∗ [email protected]
∗∗ [email protected]
∗∗∗ [email protected]
† [email protected]
2 S.Guttenberg,M.Herbst,M.Kreuzer,andR.Rashkov:Non-topologicalnon-commutativity
ofthesestructuresandtheirinterpretationintermsofeffectiveactions,whichrequirestheexistenceofa
measureandageneralizedConnes–Flato–Sternheimerproperty.WhileassociativityisrestrictedtoPoisson
geometry,stringtheorynaturallyintroducestheBorn–Infeldmeasureandkeepscyclicity,atleastthrough
second derivativeorder, independentlyof associativity and without a topologicallimit. We observe that
the results foundin [10] straightforwardlyextend to non-constantdilaton backgrounds. In section 5 we
discuss the Berkovitsstring in generalRR backgroundsand the resulting deformationof superspace. In
section6weconcludewithadiscussionofopenproblemsandworktobedone.
2 Deformationquantization, Kontsevichproduct andformality
The idea of deformation quantization is to emulate the operator product of quantum mechanics by an
associativeproductf⋆gofphasespacefunctionsf,g ∈C∞(M)with
i f⋆g−g⋆f
f⋆g =fg+ ~{f,g} +O(~2) ⇒ lim ={f,g} , (2.1)
2 PB ~→0 i~ PB
where the Poisson bracket can be written for arbitrary phase space coordinates xµ as a bi-derivation
{f,g} =Θµν(x)∂ f∂ gintermsofabi-vectorfieldΘ∈Λ2TM.
PB µ ν
2.1 PolyvectorsandtheSchouten–Nijenhuisbracket
ElementsX ∈ Λ•TM oftheexterioralgebraoverthetangentspaceTM arecalledpolyvectorfieldsand
thereisabilinearoperation,theSchouthen–Nijenhuis(SN)bracket
[X(p),Y(q)]∈Λp+q−1TM for X(p) ∈ΛpTM and Y(q) ∈ΛqTM, (2.2)
thatextendstheLiebracketofvectorfieldstoa gradedbi-derivationofdegree−1onT ∈ Λ•TM. The
JacobiidentityofthePoissonbracketisequivalenttothevanishingoftheSNbracket[Θ,Θ],
2
e{{f,g} ,h} =0 ⇔ [Θ,Θ]=0 with [Θ,Θ]µαβ = e Θµρ∂ Θαβ. (2.3)
PB PB ρ
3
X X
Liederivativesinthedirectionofξ ∈ TM canalsobewrittenintermsoftheSNbracketL X = [ξ,X]
ξ
forallpolyvectorfieldsX ∈Λ•TM.
2.2 MoyalproductandKontsevichgraphs
Incase ofconstantΘ, andhenceinparticularlocallyforDarbouxcoordinates,deformationquantization
canbeachievedbytheMoyalproduct
(f⋆g)(x)=exp i~Θµν∂ ∂ f(y)g(z) (2.4)
2 yµ zν y=z=x
(cid:0) (cid:1)
AfterageneralchangeofcoordinatesinphasespaceΘwillnotstayconstant,whichmotivatestheconsid-
erationofdeformationquantizationforgeneralΘ.ForthesymplecticcasedetΘ6=0theexistenceofastar
producthasbeenshownbyDeWildeandLecompte[11]andthefirstconstructionisduetoFedosov[12].
Some detailsanda historicalassessmentwith referencescanbe foundinthe review[2]. Forthe case of
a generalPoisson structure Θ, which by definition obeys[Θ,Θ] = 0, the constructionof an associative
productisdue to Kontsevich[1]andwill now bediscussed in moredetail. Associativityof this product
is, in fact, a corollaryof the formality theorem, which establishesa quasi-isomorphismof L algebras.
∞
TheformalitymapU mapspolyvectorfieldsT topolydifferentialoperatorsU(T ,...,T )= w D
i 1 n Γ Γ Γ
andisconstructedintermsofgraphsΓandcoefficientswΓ. ThecoefficientswΓaredefinedbyPconvergent
integralsinspiredbyopenstringFeynmandiagrams(cf. section3)withfunctionsinsertedontherealline
andpolyvectorfieldsintheupperhalfplaneasillustratedinfig.2.1,wherethederivativesofthebidiffer-
entialoperatorscorrespondtothearrowspointingatf andg. Thefirsttwographsfig. 2.1agivetheorder
Θ andΘ2 termsof the Moyalproduct,while fig.2.1byieldsfirst derivativecorrectionsfornon-constant
Θ. The latter will be worked out explicitly below. The precise relation of Kontsevich’s construction to
correlationfunctionsoftopologicalsigmamodelsisduetoCattaneoandFelder[5].
3
Θ Θ
Θ Θ Θ
Θ Θ
a) b)
s s s s s s s s
f g f g f g f g
Fig.2.1 Kontsevichgraphsfora)Moyal-typecontributionsandb)derivativecorrections,respectively.
2.3 Hochschildcohomology,Gerstenhaberbracket,andtheformalitytheorem
Ratherthangivingabstractdefinitionsoftheinvolvedmathematicalstructureswenowillustratehowthey
automaticallyshowupinsimplecalculations.Weignoreforamomenttherelation(2.1)toPoissonbrackets
andconsiderageneraldeformationoftheproduct
f⋆g =fg+~B (f,g)+O(~2) with B (f,g)=Bµνf g , f ≡∂ f, (2.5)
1 1 µ ν µ xµ
wherederivativesoffunctionsareabbreviatedbysubscripts. TheO(~)contributiontotheassociator,
f⋆(g⋆h)−(f⋆g)⋆h=~ fB (g,h)−B (fg,h)+B (f,gh)−B (f,g)h +O(~2), (2.6)
1 1 1 1
(cid:16) (cid:17)
hasexactlytheformofaHochschildcoboundary[1]
(δC)(f ,...,f )=f C(f ,...,f )−C(f f ,...,f )+C(f ,f f ,...,f )−... (2.7)
0 p 0 1 p 0 1 p 0 1 2 p
There are, however, equivalences of the resulting deformed associative algebras due to invertible maps
f →Df withdifferentialoperators
D =1+~(Dµ∂ +Dµν∂ ∂ +...)+~2(Dµ∂ +...)+... (2.8)
1 µ 1 µ ν 2 µ
thatrespecttheunitelementD1=1. Theyleadtothefollowingmodificationofthestarproduct,
f →Df ⇒ f⋆′g =D(D−1f ⋆D−1g) (2.9)
andhenceB′(f,g)−B (f,g)=−fD (g)+D (fg)−D (f)g,atorder~,whichisagainaHochschild
1 1 1 1 1
coboundary.ForthespecialcaseD =Dµν∂ ∂ thisimpliesthegaugeequivalenceB′(f,g)−B (f,g)=
1 1 µ ν 1 1
Dµνf g sothatforthefirstorderbidifferentialoperatorB (f,g)=Bµνf g ofeq. (2.5) thesymmetric
1 µ ν 1 1 µ ν
partofBµν canbegaugedawaywithDµν =B(µν). WiththechoiceBµν = iΘµν wethusrecover(2.1).
1 1 1 1 2
ReturningtotheKontsevichgraphsfig. 2.1wenowwanttoworkoutthederivativecorrectionsthatare
neededforassociativityatorder~2. ForthispurposewedefinetheMoyalpart
[f⋆g]≡fg+i~Θµνf g − ~2ΘµαΘνβf g −... (2.10)
2 µ ν 8 µα νβ
ofaproductf ⋆gastheresultofdroppingalltermswithderivativesactingonΘ. Then
f⋆g =[f⋆g]−~2Θµρ∂ Θαβ(a[f ⋆g ]+b[f ⋆g ])+O ∂2 (2.11)
ρ αµ β α βµ
(cid:0) (cid:1)
where O ∂2 only countsderivativesacting on Θ and the coefficientsω of the two graphsin fig. 2.1b
Γ
areaand(cid:0)b,r(cid:1)espectively. InsteadofdeterminingthesecoefficientsfromintegralsoverΘintheupperhalf
planewedeterminethembyimposingassociativity.Thefirstderivativeorderpartoff⋆(g⋆h)is
1
−~2Xµαβ af ⋆(g⋆h) +bf ⋆(g⋆h) + f ⋆(g ⋆h )+af⋆g ⋆h +bf⋆g ⋆h ) (2.12)
µα β α µβ µ α β µα β α µβ
4
h i
withXµαβ =Θµρ∂ Θαβ,andtheO(∂)contributionsto(f⋆g)⋆hare
ρ
1
−~2Xµαβ a(f⋆g) ⋆h +b(f⋆g) ⋆h − (f ⋆g )⋆h +af ⋆g ⋆h+bf ⋆g ⋆h) (2.13)
µα β α µβ α β µ µα β α µβ
4
(cid:2) (cid:3)
sothatf⋆(g⋆h)−(f ⋆g)⋆h=~2[f ∗g ∗h ] (a−1)Xµαβ+(a−b)Xαµβ−(b+1)Xβµα .Associativity
µ α β 4 4
(cid:16) (cid:17)
impliesthatthecoefficientof[f ∗g ∗h ]vanishes.UsingtheantisymmetryofΘwefirstobservethatX
µ α β
4 S.Guttenberg,M.Herbst,M.Kreuzer,andR.Rashkov:Non-topologicalnon-commutativity
ξ Θ [Θ,Θ]
a) s b) s s c) s s s
Fig.2.2 Dressingsofa)Liederivative,b)Poissonbracketandc)associator,respectively.
cannotbetotallyantisymmetric.Thussymmetrizationinµα,αβandβµimpliesb=2a− 1, a=2b+ 1
4 4
anda+b=0,respectively.Theuniquesolutionisa=−b= 1 . Hence
12
1
f ⋆(g ⋆h)−(f ⋆g)⋆h=− ~2[f ∗g ∗h ] e Θµρ∂ Θαβ +O(∂2) (2.14)
µ α β ρ
6
µXαβ
sothat[Θ,Θ]=0isanecessaryconditionfortheexistenceofanassociativedeformation.TheKontsevich
productthroughsecondderivativeorder(setting~=1)
1 1
f⋆g = [f⋆g]− Θµγ∂ Θνρ f ⋆g +f ⋆g + ∂ Θµρ∂ Θνσ[f ⋆g ]
γ µν ρ ρ µν σ ρ µ ν
12 24
(cid:2) (cid:3)
i i
+ Θµγ∂ Θνδ∂ Θρλ [f ⋆g −f ⋆g ]− ΘµγΘνδ∂ ∂ Θρλ f ⋆g −f ⋆g
γ δ µρ νλ νλ µρ γ δ µνρ λ λ µνρ
48 48
(cid:2) (cid:3)
1 1
+ (Θµγ∂ Θρλ)(Θνδ∂ Θστ) f ⋆g +2f ⋆g +f ⋆g +O(∂3) (2.15)
2122 γ δ µρνσ λτ µρτ λνσ λτ µρνσ
(cid:2) (cid:3)
hasbeendeterminedin[10]usingtheknowncoefficient 1 ofthegaugetermandsymmetryundercom-
24
plex conjugation combined with the exchange of f and g. Note that each term in (2.15) comprises the
contributionsofaninfinitenumberofgraphswithMoyal-typeadditionstotheclassicalpartof⋆,sincethis
formulaholdstoallordersintheundifferentiatedΘ’s.
TheGerstenhaberbracketofpolydifferentialoperatorsP isthecommutator[P ,P ]withrespecttoan
i 1 2
appropriatedefinitionof the compositionP ◦P of degree−1. For bidifferentialoperatorsthe bracket
1 2
thusyieldsatridifferentialoperator,andinthespecialcaseP =P =⋆thebracketbecomesproportional
1 2
totheassociator 1[⋆,⋆](f,g,h)=f⋆(g⋆h)−(f⋆g)⋆h.Theformalitymapcanberegardedasadressing,or
2
quantization,ofpolyderivationsT ∈ Λ•TM tohigherorderpolydifferentialoperatorsP . Theformality
i i
theoremensuresthatthismapisanL quasi-isomorphismwherethe(homotopy)Liealgebrastructures
∞
arerelatedtotheSNbracketandtheGerstenhaberbracket,respectively.
Thecasesofvectorfieldsξ,PoissontensorsΘandrankthreetensorsJ ∈Λ3TM showninfig.2.2are
ofparticularinterest.ThequantizationofΘyieldsthestarproduct(2.15).SincetheSNbracketismapped
totheGerstenhaberbracket,J =[Θ,Θ]aswellasitsquantizationvanishinthecaseofaPoissonstructure
[Θ,Θ] = 0. Since[⋆,⋆]istheassociatorthisestablishesassociativityoftheKontsevichproduct(compare
fig.2.2ctoourresult(2.14)atleadingorder~2).
ξ Θ ξ ξ Θ Θ Θ ξ Θ Θ Θ
Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ
r r r r r r r r r
f f f f f f f f f
(a) (bc) (AB) (CD) (EFG) () (HIJ) (KL) (MNP)
Fig.2.3 Samplegraphsfordressedcoordinatetransformationsthroughsecondderivativeorder.
For vectorfieldsξ the classical term is the Lie derivative,which amountsto a changeof coordinates.
Its quantization yields an equivalence transformation D of the form (2.9) so that the Kontsevich prod-
ξ
ucttransformscovariantlyunderchangesofcoordinatesonlyupto gaugeequivalence. For infinitesimal
transformations
∂ (f⋆ g)=D f⋆g+f⋆D g−D (f⋆g) with x˙µ =ξµ. (2.16)
t t ξ ξ ξ
Infig.2.3weenumeratethegraphsthatcontributetoinfinitesimaltransformations(i.e.linearinξ)through
secondderivativeorderinΘ. Notethattheadditionallinescorrespondingtoderivativesactingonξandf
5
canleadtodifferenttensorstructure,asindicatedbycoefficientsA,...,P forthetermswithtwoderiva-
tivesonΘ’s,plusaninfinitenumberofadditionalMoyal-typecontributions.Throughfirstderivativeorder
1
D =ξα∂ + ξα Θµν∂ Θρβ∂ ∂ +O(∂2). (2.17)
ξ α 24 µρ ν α β
AtsecondderivativeorderthegraphsdefinedifferentialoperatorsD containing(non-Moyal)termswith
ξ
up to fivederivatives,D f = ξαf +...+Pξα ∂ Θβµ∂ ΘργΘδν∂ Θσεf butmanycoefficients
ξ α ρσ ν µ ν αβγδε
maybezero.
3 Openstrings, Born–Infeldelectrodynamics and non-commutativity
InordertorelatetheKontsevichproduct(2.15)tostringtheorywestartwiththePolyakovactionforclosed
stringsmovinginacurvedbackgroundwith2-formfieldB. Inconformalgauge
1
S = d2z∂Xµ∂¯Xν g (X)+B (X) , (3.1)
P 2πα′ ZΣ (cid:16) µν µν (cid:17)
whereXµ : Σ −→ M mapstheclosedworldsheetΣto thetargetmanifoldM. Notethatthisactionis
invariantunderthegaugetransformationδ B =dΛ.
Λ
Whenweconsideropenstrings, wehavetointroduceworldsheetswith boundariesandspecifya hy-
persurface in M, i.e. a D-brane, to which the end points of open strings are mapped. In the following
we will only consider space-filling branes. By Stokes’ theorem, (3.1) is not gauge invariant anymore,
X∗δ B = X∗Λ,andwehavetointroduceacompensatorfieldAattheboundary,whichturnsout
Σ Λ ∂Σ
RtobeaU(1)gaRugefieldwithfieldstrengthF =dA. Theassociatedaction,
S = X∗A= dt∂ XµA (X)= X∗F, (3.2)
A t µ
Z Z Z
∂Σ ∂Σ Σ
thenrestoresgaugeinvarianceof(3.1)bysettingδ A=− 1 Λ+dλ.
Λ,λ 2πα′
AsaconsequenceofgaugesymmetrytheeffectiveactiondependsonthefieldsAandB onlythrough
thegaugeinvariantquantitiesF =B+2πα′F andH =dB =dF. ForslowlyvaryingfieldsF andgthe
effectivetheoryontheD-braneisBorn–Infeldelectrodynamics[13]governedby
S = dDx det(g +F ). (3.3)
BI µν µν
ZM q
Letushaveacloserlookatthequantizationof(3.1)and(3.2)ontheupperhalfplane,conformallyequiv-
alent to the disk. We split the embedding map into fluctuations around a constant mode, Xµ(z,z¯) =
xµ +ζµ(z,z¯), andorganizetheperturbativequantizationintermsofaderivativeexpansionintheback-
groundfields. Moreover,we regardthemetricg(x) andthecurvatureF(x) asaclassicalbackgroundin
ordertoensureconformalinvariance.
ThevariationoftheactionrequiresthemixedDirichlet–Neumannboundarycondition
g ∂ Xν −F ∂ Xν =0, (3.4)
µν t µν n ∂Σ
(cid:12)
whichleadstothefollowingprop(cid:12)agatorforfluctuationsattheboundary(τ,τ′ ∈∂Σ):
1
hζµ(τ)ζν(τ′)i=− Gµν(x)ln|τ −τ′|2+iπΘµν(x)ǫ(τ −τ′) , (3.5)
2π
n o
whereweintroducedG(µν)+Θ[µν] :=(g +F )−1andthesignfunctionǫ(τ)=τ/|τ|.
µν µν
Inthelimitwheng vanishes(withF keptfinite)[4],theactionS +S istopological. Onlythe
µν µν P A
second part in (3.5) survives, and the non-commutativeproducton the D-brane world volume becomes
apparent.ForconstantbackgroundsitistheMoyalproduct.Forvaryingbackgroundswenotice,however,
thattheEinsteinequationsforthebackgroundfieldsrequireH =dF =0inthetopologicallimit[14],i.e.
F isasymplecticformwithPoissonstructureΘ = F−1. Theresultingnon-commutativeproductisthen
theassociativeproduct(2.15)duetoKontsevich[1].
6 S.Guttenberg,M.Herbst,M.Kreuzer,andR.Rashkov:Non-topologicalnon-commutativity
4 Associativity,cyclicinvarianceand effective actions
From the string theory point of view, the assumption of Θ(x) being a Poisson structure is not natural.
Theonlyconditiononthebackgroundfieldsshouldcomefromconformalinvariance,orequivalentlythe
classical equationsof motion. Therefore,it is preferableto definethe non-commutativeproductwithout
takingthetopologicallimit.Itisclearthatthefirstterminthepropagator(3.5)shouldplayasecondaryroˆle
inthisdefinition,whichsuggeststoconsidertwo(off-shell)vertexoperatorsatadistanceτ′−τ = 1[7],
i.e. 1
f(x)◦g(x):= Dζ e−S[X=x+ζ]f(X(0))g(X(1)). (4.1)
|g+F| Z
TheBorn–Infeldmeasurepintheprefactoriscancelledby(worldsheet)1-loopdiagrams.Athigherderiva-
tiveordersofthebackgroundfieldsthemeasuregetscorrected[15].
Letuscommentonsomepropertiesofthisproduct.
• AnimmediateconsequenceofgivinguponΘ(x)beingPoissonisthelossofassociativity,sothata
sumoverdifferentconfigurationsofbracketswillappearinopenstringscatteringamplitudesandin
theeffectiveaction.Inthetopologicallimit,thenon-commutativeproduct(4.1)becomestheKontse-
vichproduct,uptogaugeequivalence,D(f ⋆g)=Df ◦Dg,sothatassociativityisrestored.
• Aswasarguedin[10],thevariationalprincipleforthelow-energyeffectivetheoryrequiresthatthe
non-commutativeproductiscyclic,i.e.
Ωf ◦g = Ωf ·g and Ω(f ◦g)◦h= Ωf ◦(g◦h), (4.2)
Z Z Z Z
M M M M
whereΩisameasure,whichrequires∂ (ΩΘµν)=0.Fromastringtheorypointofviewthemeasure
µ
ΩistheBorn–Infeldmeasurethatappearedin(3.3),i.e. Ω = |g+F|,andcyclicityfollowsfrom
thegeneralizedMaxwellequationassociatedwiththeBorn–Infpeldaction(3.3):
1
∂ ( |g+F|Θµν)=0 ⇐⇒ GρσD F − ΘρσH λF =0. (4.3)
µ ρ σµ ρσ λµ
2
p
This is in line with the assumption of a classical background, which ensures conformal invariance
and,inparticular,cyclicinvarianceofdiskamplitudes. Noticethatifweincludethedilatonφinthe
background,themeasureismodifiedto e−φ |g+F|.
For Poisson structures the second conditionpin (4.2) follows from associativity, and the first is due
toConnes–Flato–Sternheimer[16]. Infact, foranyvolumeformΩsubjectto∂ (ΩΘµν) = 0there
µ
exists a star-product that satisfies cyclic invariance (4.2) [9]. However, in contrast to the physical
contextabove,thereisnocanonicalmeasureforPoissonstructures.
• In [7] an explicit computation of the product (4.1) was given to first derivative order, ∂Θ, in the
backgroundfield,buttoallordersinΘ. In[10]itwasshownthatthecyclicinvariance(4.2)uniquely
fixesthenon-commutativeproducttosecondderivativeorder,withtheresult
1
f ◦g =f⋆g− ΘµρΘνσ∂ ∂ (logΩ)f g . (4.4)
ρ σ µ ν
24
Thefirstcontributionisthesameexpression(2.15)astheKontsevichproductbutwithoutthePoisson
constraintonΘandthesecondisagaugetermthatisneededtoensurecyclicinvariance.
Ifwewantto usethenon-commutativeproduct(4.1)tocomputestringS-matrixelementswehaveto
imposeon-shellconditionsnotonlyonthebackgroundfieldsbutalsoonthevertexoperatorinsertions.In
thepresentcontextthevertexoperatorsarefunctions,f(X),andthustheon-shellconditionistheonefor
anopenstringtachyonT(x)
1 1
(cid:3)T = ∂ ΩGµν∂ T =− T. (4.5)
Ω µ ν α′
(cid:0) (cid:1)
Thisfixesthekinetictermforthelow-energyeffectiveaction.Theresultis
1 1 8
S =− Ω Gµν∂ T ∂ T − T2− T (T ◦T) , (4.6)
2go2 ZM n µ ν α′ r9α′ o
wherethecubictachyoninteractionwasfoundin[17]bycomputing3-pointamplitudes.
7
5 Superstrings and non-anticommutative superspace
The superstring in Green-Schwarz(GS) related formulationsis an embeddingof a string in superspace.
It thusappearsnaturalthat, in additionto non-commutativityof space-time coordinates, there shouldbe
a mechanism that deformsthe anticommutationof the fermionic superspace coordinates. Indeedsuch a
mechanism exists. Independentlyof string theory,special cases of non-anticommutingsupercoordinates
were already considered by van Nieuwenhuizen and others in [18] (N = 1 SUSY, see [19]). A more
2
generalansatzwaspresentedin[20].Afterindicationsin[21,22]thatsimilarstructuresoriginatefromthe
superstring, this could eventuallybe shown in [23] for a string in four dimensions(with six dimensions
compactified on a Calabi-Yau) and was generalized in [24] to ten dimensions. In both cases a constant
RR-field-strength was considered and turned out to be responsible for the nonanticommutativity of the
supercoordinates. The calculations where performed in different versions of the covariant superstring
[25–27]. Thisnon-(anti)commutativitycan againbeimplementedvia a star product,now onsuperspace
(see [28] and referencestherein). For non-constantbackgroundfields (but in the topologicallimit), this
correspondsto a graded generalization of Kontsevich’s associative star product. A derivation from a σ-
modelwithsuper-targetspacealongthelinesofCattaneoandFelder[5]waspresentedin[29]. Theeffect
ofaconstantRR-potential(notfieldstrength)onthedeformationofthebosonicspacewasalreadystudied
in [31]. In the followingwe sketch how non-anticommutativityof superspace arises from the Berkovits
purespinorsuperstring[24].
AlthoughwewillconsideranopenstringwithtypeIsupersymmetry,wewanttocoupleittothetype
IIbulkfields(seee.g.[31]).InparticulartheRR-fieldsbelongtothebulkandwilltakeovertheroleofthe
B-field inthefermioniccase. Itisthereforenecessarytoembedthestringintoa typeIIsuperspacewith
coordinatesxM = (xm,θµ,θˆµˆ). InthissectionGreekletterswillbereservedforfermionicindiceswhile
bosonicindicesaredenotedbyLatinletters. Inconformalgauge,theGSactioninflatbackgroundreads
S ≡ d2z 1Πaη Πb+L (conformalgauge)
GS 2 z ab ¯z WZ
R
L ≡ −1Πa θγ ∂¯θ−θˆγ ∂¯θˆ + 1(θγa∂θ)(θˆγ ∂¯θˆ)−(z ↔z¯), (5.1)
WZ 2 z a a 2 a
(cid:16) (cid:17)
whereΠa arethesupersymmetricmomenta. Theycanbedescribedasthepullbackofthebosonicpart
z/z¯
ofthesupervielbein Πa
EA ≡dxME A fl=at dxa+dθγaθ+dθˆγaθˆ, dθα , dθˆαˆ (5.2)
M
z }| {
(cid:0) (cid:1)
to the worldsheet. Lettersfromthe beginningof the alphabetshalldenote“flat indices”(with respectto
thelocalframe),whilelettersfromtheendofthealphabetwilldenote“curvedindices”.Thisdistinctionis
morerelevantforthecurvedbackgroundtobediscussedlater.Asusual,γa denotestheoff-diagonalchiral
αβ
block of the 10-dimensionalDirac gammamatrix Γa, in a representationwhere it is real and symmetric
(i.e. gradedantisymmetric)intheindicesαandβ.
The Wess-Zumino term L is responsible for the existence of a local fermionic symmetry, the κ-
WZ
symmetry. Indeed,thetheorycontainsanumberoffermionicconstraintsd ,dˆ . Onlyhalfofeachset,
zα z¯α
however,isfirstclassandtheconstraintalgebraisthereforenotclosed:
{d (σ),d (σ′)} ∝ 2γa Π δ(σ−σ′). (5.3)
zα zβ αβ za
Beingaspinorinanirreduciblerepresentation,d cannotcovariantlybeseparatedintofirstandsecond
zα
classandthusdoesnotallowcovariantquantization.Alongstruggletoovercomethisproblemresultedin
theinventionofthepurespinorstring[27,30]asanalternativeformalism.
Berkovits’purespinorformalismhastwobasicingredients.Thefirstisafreeactionoftheform
S = d2z 1∂xmη ∂¯xn+∂¯θαp +∂θˆαˆpˆ
free 2 mn zα z¯αˆ
= R d2z 1Πaη Πb +L +∂¯θαd +∂θˆαˆdˆ , (5.4)
2 z ab z¯ WZ zα z¯αˆ
R
where p , pˆ are independent variables and d ≡ p − (γ θ) ∂xa − 1θγa∂θ − 1θˆγa∂θˆ and
zα z¯α zα zα a α 2 2
its hatted counterparthave the same algebra as the constraints of the G(cid:0)S-string. In addition, this a(cid:1)ction
8 S.Guttenberg,M.Herbst,M.Kreuzer,andR.Rashkov:Non-topologicalnon-commutativity
coincidesclassicallywith theGS-actionford = dˆ = 0. Thesecondbasic ingredientarethe BRST
zα z¯αˆ
operators
Q= dzλαd , Qˆ = dz¯λˆαˆdˆ , (5.5)
zα z¯αˆ
which implemeHnt in some sense thHe constraints d = dˆ = 0 cohomologically. λα and λˆαˆ are
zα z¯αˆ
ghost fields of even parity. Containing also second class constraints, the above BRST operators fail
to be nilpotent in general. This can be repaired by constraining the ghost fields to be so-called pure
spinors, obeying λγaλ = λˆγaλˆ = 0. Like the fermionic coordinates, the ghost fields should be left
and right-moving respectively and one thus adds the corresponding ghost term to the free action (5.4):
S =S + d2z ∂¯λαω +∂λˆαˆω +L (λγaλ)+Lˆ (λˆγaλˆ).Theimplementationofthepure
ps free zα z¯αˆ zz¯a zz¯a
spinor constraiRntswith the help of Lagrangemultipliersimmediatelyreveals(byvaryingwith respectto
the ghost) a gaugesymmetry of the antighostsof the formδ ω = µ (γaλ) which correspondsto
(µ) zα za α
the (first-class)pure-spinorconstraints. Becausethe fieldequationsarebasicallyfree,onegetsfreefield
operatorproductsafterquantization. Fortheantighostfield,thisstatementisrestrictedtogaugeinvariant
operatorsliketheghostcurrentortheLorentzcurrent. Apartfromthecentralcharges,theirOPEslookas
if there was no pure spinor constraint. To determine the centralcharges, one has to solve the constraint
once(seee.g.[27]).
Inordertocompletethedescriptionfortheopenstring,westillneedboundaryconditions.Forvanishing
backgroundanaturalchoiceistosetθ = θˆandλ = λˆ attheboundary. Thiscanbeimplementedbythe
variationofaboundarytermthatoneshouldaddtotheaction. Thepreciseformofthisboundarytermis
fixedbyN =1supersymmetry,BRSTinvarianceandtheantighostgaugesymmetry.Asthefinalformof
theboundaryactionisquitelengthyandnotveryilluminating,wereferto[32]forfurtherdetails.
Theopenstringinageneralbackgroundofbulkandboundaryfieldsconsistsofabulkpartofthesame
form as a closed string in generalbackgroundand an additional boundarypart. The closed pure spinor
superstringin generalbackgroundwasstudied first byBerkovitsandHowe in [33]. Alreadyat classical
level, conservation and nilpotency of the BRST charges implement the type II supergravity constraints.
Those,inturn,guarantee1-loopquantumconformalinvarianceofthetheory[34].Thepresentationofthe
bulkpartinthefollowingisbasedon[35]. Thestartingpointisthemostgeneralclassicallyconformally
invariantaction:
S = d2z 1∂xM(G (։x)+B (։x))∂¯xN +∂¯xME α(։x)d +∂xME αˆ(։x)dˆ +
bulk MN MN M zα M z¯αˆ
Z 2
+d Pαβˆ(։x)dˆ +λαC βγˆ(։x)ω dˆ +λˆαˆCˆ βˆγ(։x)ωˆ d +
zα z¯βˆ α zβ z¯γˆ αˆ z¯βˆ zγ
+ ∂¯λβ+λα∂¯xMΩ β(։x) + ∂λˆβˆ +λˆαˆ∂xMΩˆ βˆ(։x) ωˆ +
Mα Mαˆ z¯βˆ
(cid:16) (cid:17) (cid:16) (cid:17)
+1L (λγaλ)+ 1Lˆ (λˆγaˆλˆ)+λαλˆαˆS ββˆ(։x)ω ωˆ (5.6)
2 zz¯a 2 z¯zaˆ ααˆ zβ z¯βˆ
The variable ։x containsxm,θµ andθˆµˆ. In additionto theaction, we needthetwo BRST operators. In
principletheycouldcontainbackgroundfieldsaswell,butitisalwayspossibletoreparametrized and
zα
dˆ suchthattheyhavethesameformasintheflatcase. Consistencyoftheequationsofmotionwiththe
z¯αˆ
purespinorconstraintsrequiresthatthebackgroundfieldsΩMαβ andΩˆMαˆβˆareeachasumofaspinorial
Lorentz-transformationanddilatationin the lasttwo indices. Theycan thusbe regardedasLorentzplus
scaleconnections.Thispropertyalsoestablishestheantighostgaugesymmetryinthegeneralcase. BRST
invarianceoftheactionrequiresthatthesymmetrictwo-tensorisoftheformG = E aη E b. The
MN M ab N
backgroundfields E a, E α andE αˆ can then be combinedto a single objectE A andregardedas
M M M M
supervielbein. BRST invariance and nilpotencyof the BRST transformationsput several restrictions on
thebackgroundfieldswhichturnouttobeequivalenttothetypeIIsupergravityconstraints[33–35].
For the moment, we restrict ourselves to a glance at the propagator. I.e., we are interested in the
quadratic part of the action and do not yet need all the constraints. Expanding the coordinates around
a constant zero mode, restricting to vanishing zero mode for the fermionic coordinates and the ghosts,
choosingaparametrizationwhichcorrespondstotheWZ-gaugeandrestrictingtothequadraticpart,one
9
arrivesat
S | = d2z 1∂ζm e a(→x)η e b(→x)+B (→x) ∂¯ζn+d Pαβˆ(→x)dˆ (5.7)
qu θ=λ=0 2 m ab n mn zα z¯βˆ
R (cid:16) (cid:17)
+∂¯ζmψ α(→x)d +∂¯ζµδ αd +∂ζmψˆ αˆ(→x)dˆ +∂ζµˆδ αˆdˆ +∂¯λβω +∂λˆβˆωˆ
m zα µ zα m z¯αˆ µˆ z¯αˆ zβ z¯βˆ
withxM(z,z¯) = xM +ζM(z,z¯). AtthisstageitbecomesvisiblethattheRamond-Ramond(RR)fields
Pαβˆ will enter the propagatorbetween the fermionic coordinates. This observation was made for con-
stant RR-fields in [23] for four dimensions (with six compactified on a Calabi-Yau) and in [24] for ten
dimensions.Theassociatedanticommutationrelationswerefoundtobe
{θα,θˆβˆ}∝Pαβˆ. (5.8)
TurningonthefieldstrengthF modifiestheboundaryconditionsforallworldsheetfieldsandalsoleads
toaRRbackgrounddependentshiftinthenoncommutativityparameterΘmn[31].
Forgeneralbackgrounds,oneneedstochecktheconsistencyoftheboundaryactionwiththebulkBRST
transformationsandthepurespinorconstraints. Alreadyfortheopenpurespinorstringinanopenstring
backgroundthisisalongstory,whichwasdiscussedbyBerkovitsandPershinin[32]. Inadditiontothe
boundarytermthatwasmentionedbeforetheyaddtheintegratedopenstringvertexoperatoroftheform
1
V ∝ dτ θ˙αA (x,θ )+ΠmB (x,θ )+d+Wα(x,θ )+ (N )β(γF)α(x,θ ) (5.9)
Z + α + + m + α + 2 + α β +
to the action. The worldline fields with index ’+’ are just suitable linear combinations of the left and
rightmoversand(N )β ∝λβω+.TheobjectsA ,B ,WαandFαβareN =1backgroundsuperfields.
+ α + α α m
TheconsistencyrequirementsoftheboundaryactionwithBRSTinvarianceandthepurespinorconstraint
leadstothefieldequationsofsupersymmetricBorn–Infeldforthesebackgroundsuperfields.
Inordertogeneralizetheresult(5.8)tonon-constantbulkfieldsonehastobecomeyetmoregeneral,
combiningtheboundarypartV withthebulkaction(5.6)andstudyingtheconsistentboundaryconditions
andfieldequations.Thisisworkinprogress.
6 Conclusion
Inthisnotewe gaveanintroductiontotheKontsevichproductanddiscussedourproposalforageneral-
izationtothenon-associativecase. Weestablishedcyclicitythroughsecondderivativeorder,whichallows
forthenon-commutativeproducttobeusedintheconstructionofeffectiveactions. Wecheckedthatour
previousresults[10]generalizetonon-constantdilatonbackgrounds,withtheonlymodificationbeingthe
prefactorexp(−φ)inthemeasure. We alsoreviewedtheexistingresultsandideasaboutgeneralizations
tosuperstrings,whichhavebeeninvestigatedsofarforconstantbackgroundfields.
There is a number of obviousdirections for further work. For the bosonic string a non-commutative
generalization of the gauge field effective action should be constructed, which presumably is related to
derivativecorrectionstothemeasure. Thenon-abeliancaseshouldalsohaveinterestingimplicationsfor
commutativenon-abelienBorn-Infeldactions. A quite demandingtask will be the generalizationof our
resultstosuperstringsincurvedRRandB-fieldbackgrounds.Onthemoremathematicalside,itwouldbe
interestingtoestablishcyclicitytoallordersinthederivativeexpansionandifpossibleexplicitlyconstruct
thenon-associativeproduct.
Acknowledgements We would like to thank Giovanni Felder, Karl-Georg Schlesinger and Ricardo Schiappa for
helpfuldiscussions.S.G.wassupportedbytheEuropeanNetworkonRandomGeometry,EECGrantNo.MRTN-CT-
2004-005616. M.K.acknowledgessupportbytheAustrianResearchFundsFWFgrantP19051-N16.
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