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Limits and Degenerations of Unitary Conformal Field Theories 4 Daniel Roggenkamp1,2, Katrin Wendland3 0 0 1 PhysikalischesInstitut, Universita¨tBonn,Nußallee12,D-53115Bonn,Germany. 2 E-mail:[email protected] 2 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United n Kingdom. a 3 Mathematics Institute, UniversityofWarwick,CoventryCV4-7AL,UnitedKingdom. J E-mail:[email protected] 1 1 3 Abstract: In the present paper, degeneration phenomena in conformal field v theoriesarestudied.Forthispurpose,anotionofconvergentsequencesofCFTs 3 is introduced. Properties of the resulting limit structure are used to associate 4 geometric degenerations to degenerating sequences of CFTs, which, as familiar 1 from large volume limits of non-linear sigma models, can be regarded as com- 8 mutative degenerations of the corresponding “quantum geometries”. 0 3 Asanapplication,thelargelevellimitoftheA-seriesofunitaryVirasoromin- 0 imal models is investigated in detail. In particular, its geometric interpretation / is determined. h t - p Introduction e h Limits and degenerations of conformal field theories (CFTs) have occurred in : v various ways in the context of compactifications of moduli spaces of CFTs, in i X particularinconnectionwithstringtheory.Forexample,zerocurvatureorlarge volumelimitsofCFTsthatcorrespondtosigmamodelsareknowntogivebound- r a arypointsoftherespectivemodulispaces[A-G-M,Mo].Theselimitsprovidethe connection between string theory and classical geometry which for instance is used in the study of D-branes. In the Strominger/Yau/Zaslowmirror construction[V-W,S-Y-Z,Gr],boundarypointsplayaprominentrole.Infact,Kontsevich and Soibelman have proposed a mirror construction on the basis of the Stro- minger/Yau/Zaslowconjecture which relies on the structure of the boundary of certain CFT moduli spaces [K-S]. All the examples mentioned above feature interesting degeneration phenomena. Namely, subspaces of the Hilbert space which are confined to be finite di- mensionalforawell-definedCFTachieveinfinitedimensionsinthelimit.Infact, such degenerationsare expected if the limit is formulatedin terms of non-linear sigma models, where at large volume, the algebra of low energy observables is expectedtoyieldanon-commutativedeformationofanalgebra offunctions ∞ A 2 DanielRoggenkamp andKatrinWendland on the target space. The algebra of observables whose energy converges to zero then reduces to at infinite volume. An entire non-commutative geometry ∞ A can be extracted from the underlying CFT, which approaches the target space geometry in the limit [F-G]. By construction, this formulation should encode geometry in terms of Connes’ spectral triples [Co1,Co2,Co3]. By the above, degeneration phenomena are crucial in order to single out an algebrawhichencodesgeometryinCFTs.Anintrinsicunderstandingoflimiting processes in CFT language is therefore desirable. This will also be necessary in ordertotakeadvantageofthegeometrictoolsmentionedbefore,awayfromthose limits.Viceversa,agoodunderstandingofsuchlimitingprocessesinCFTscould allow to take advantage of the rich CFT structure in geometry. The main aim of the present work is to establish an intrinsic notion of such limiting processes in pure CFT language and to apply it to some interesting examples.Tothisend,wegiveadefinitionofconvergenceforsequencesofCFTs, suchthatthecorrespondinglimithasthefollowingstructure:Thereisalimiting pre-Hilbertspace whichcarriestheactionofaVirasoroalgebra,andsimilar ∞ H to ordinary CFTs to each state in we assign a tower of modes. Under an ∞ H additionalconditionthelimitevenhasthestructureofafullCFTonthesphere. This is the case in all known examples, and in particular, our notion of limiting processes is compatible with deformation theory of CFTs. If the limit of a converging sequence of CFTs has the structure of a CFT on thesphere,butisnotafullCFT,thenthisisduetoadegenerationasmentioned above.In particular,the degenerationof the vacuum sector can be used to read offageometryfromsuchadegeneratelimit.Namely,inourlimitsthealgebraof zeromodesassignedtothosestatesin withvanishingenergyiscommutative ∞ H andcanthereforebeinterpretedasalgebraofsmoothfunctionsonsomemanifold M.Theasymptoticbehaviouroftheassociatedenergyeigenvaluesallowstoread offadegeneratingmetriconM andanadditionalsmoothfunctioncorresponding to the dilaton as well. Moreover, being a module of this commutative algebra, can be interpreted as a space of sections of a sheaf over M as is explained ∞ H in [K-S]. Simple examples which we can apply our techniques to are the torus models, where our limit structure yields geometric degenerations of the corresponding target space tori à la Cheeger-Gromov [C-G1,C-G2]. In this case, is the ∞ H spaceofsectionsofatrivialvectorbundleovertherespectivetargetspacetorus. Similarstatementsaretruefororbifoldsoftorusmodels,onlythatinthiscasethe fiber structure of over the respective torus orbifold is non-trivial. Namely, ∞ H the twisted sectors contribute sections of skyscraper sheaves localized on the orbifold fixed points. Our favorite example, which in fact was the starting point of our investiga- tions, is the family of unitary Virasorominimal models. Some of their structure constants have a very regular behaviour under the variation of the level of the individualmodels.WeusethistoshowthattheA-seriesofunitaryVirasoromin- imalmodelsconstitutesaconvergentsequenceofCFTs.Allfieldsinitslimitthe- oryatinfinitelevelcanbeconstructedintermsofoperatorsinthe su(2) WZW 1 model. The sequence degenerates, and the limit has a geometric interpretation in the above sense on the interval [0,π] equipped with the (dilaton-corrected) metric g(x)= 4 sin4x (in fact, the x-dependent contribution is entirely due to π2 the dilaton). This also allows us to read off the geometry of D-branes in these LimitsandDegenerations ofUnitaryConformalFieldTheories 3 models. Though this means that the vacuum sector of our limit is well under- stood,it remains aninteresting openproblemto investigatethe full fusion rules in detail, in particular whether an appropriate limiting S-matrix can be found. A different limit for the A-series of unitary Virasoro minimal models at infi- nitelevelwasproposedin[G-R-W,R-W1,R-W2].Itisdescribedbyawell-defined non-rationalCFT of centralchargeone, which bears some resemblance to Liou- ville theory. In particular, its spectrum is continuous, but degenerations do not occur.Ourtechniquescanalsobeusedtodescribethis latterlimit.Therelation between the two different limit structures is best compared to the case of a free boson,compactifiedona circle of largeradius,where apartfromthe degenerate limit described above one can also obtain the decompactified free boson. While the limit investigated in this article has the advantage that it leads to a consis- tent geometric interpretation, the one which corresponds to the decompactified free boson gives a new well-defined non-rational CFT. Thisworkisorganizedasfollows:InSect.1weexplainhownon-commutative geometries can be extracted from CFTs, after giving a brief overview of some of the basic concepts.Sect. 2 contains our definitions ofsequences,convergence, and limits, and is the technical heart of this paper. Moreover, the geometric interpretationsofdegeneratelimitsarediscussed.Sect.3isdevotedtothestudy oftorusmodelsandorbifoldsthereof,whereweexemplifyourtechniques.InSect. 4wepresentourresultsonthe A-seriesofunitaryVirasorominimalmodels.We endwithadiscussioninSect.5.Severalappendicescontainbackgroundmaterial and lengthy calculations. Acknowledgments. It is a pleasure to thank Gavin Brown, Jarah Evslin, José Figueroa-O’Farrill,Matthias Gaberdiel, Maxim Kontsevich,Werner Nahm, An- dreasRecknagel,MichaelRo¨sgen,VolkerSchomerus,GérardWatts and the ref- eree for helpful comments or discussions. We also wish to thank the “abdus salam international center for theoretical physics” for hospitality, since part of this work was performed there. D. R. was supported by DFG Schwerpunktprogramm1096and by the Marie Curie Training Site “Strings, Branes and Boundary Conformal Field Theory” at King’s College London, under EU grant HPMT-CT-2001-00296. K. W. was partly supportedunder U.S. DOE grantDE-FG05-85ER40219,TASK A,at the University of North Carolina at Chapel Hill. 1. From geometry to conformal field theory, and back to geometry StringtheoryestablishesanaturalmapwhichassociatesCFTstocertain,some- times degenerate geometries. Conversely, one can associate a geometric interpretation to certain CFTs, and the latter construction is made precise by using Connes’ definition of spectral triples and non-commutative geometry. In Sect. 1.1 we very briefly remind the reader of spectral triples, ex- plaining how they encode geometric data. Somewhat relaxing the conditions on spectraltripleswedefinespectral pre-tripleswhichwillbeusedinSect.1.2. There, we recall the basic structure of CFTs and show how to extract spectral pre-triples from them. If the spectral pre-triple defines a spectral triple, then this will generate a non-commutative geometry from a given CFT. In Sect. 1.3 weexplainhowinfavorablecaseswecangeneratecommutativegeometriesfrom 4 DanielRoggenkamp andKatrinWendland CFTs. In the context of string theory, this prescription gives back the original geometric data of the compactification space. Much of this Sect. 1 consists of a summary of known results [Co2,F-G,Co3, Re,K-S], but it also serves to introduce our notations. 1.1. From Riemannian geometry to spectral triples. For a compact Riemannian manifold(M,g),whichforsimplicityweassumetobesmoothandconnected,the spectrum of the associated Laplace-Beltramioperator ∆ :C (M) C (M) g ∞ ∞ −→ encodes certain geometric data of (M,g). However, in general one cannot hear the shape of a drum, and more information than the set of eigenvalues of ∆ is g needed in order to recover (M,g). Bythe Gel’fand-Naimarktheorem,thepointsettopologyofM iscompletely encoded in C0(M)=C (M): We can recover each point p M from the ideal ∞ ∈ of functions which vanish at p. In other words, given the structure of C (M) ∞ as a C∗-algebra and its completion C0(M), M is homeomorphic to the set of closed points of Spec( ), where is the sheaf of regular functions on M. M M O O Connes’ dual prescription uses C∗-algebra homomorphisms χ:C∞(M) C, −→ instead, such that p M corresponds to χp:C∞(M) C with χp(f):=f(p); ∈ −→ the Gel’fand-Naimark theorem ensures that for every commutative C∗-algebra there exists a Hausdorff space M with = C0(M). M is compact if is A A A unital. Example 1.1.1 Let R R+, then M = S1 = R1/ with coordinate x x+2πR has the ∈ R ∼ ∼ Laplacian ∆=−ddx22. Its eigenfunctions |miR,m∈Z, obey ∀m∈Z: |miR: x7→eimx/R ; 12∆|miR = 2mR22|miR; (1.1.1) m,m′ Z: m R m′ R = m+m′ R, ∀ ∈ | i ·| i | i and they form a basis of C0(M) and C (M) with respect to the appropriate ∞ norms.Anysmoothmanifoldis homeomorphictoS1,equippedwiththeZariski R topology, if its algebra of continuous functions has a basis fm, m Z, which ∈ obeys the multiplication law fm fm′ =fm+m′. · To recover the Riemannian metric g on M as well, we consider the spectral triple H:=L2(M,dvolg),H := 12∆g,A:=C∞(M) , where H is viewed as self-adjoint operator which is densely defined on the Hilbert space H, and (cid:0) (cid:1) A is interpreted as algebra of bounded operators which act on elements of H by pointwise multiplication. Following [Co2,F-G,Co3], we can define a distance functional d on the topological space M by considering g := f G :=[f,[H,f]]= f2 H +H f2 +2f H f f F ∈A − ◦ ◦ ◦ ◦ (cid:8) (cid:12) obeys h C(cid:0)∞(M): Gfh (cid:1)h . (cid:12) ∀ ∈ | |≤| | One now sets (cid:9) x,y M: d (x,y):= sup f(x) f(y) . (1.1.2) g ∀ ∈ | − | f ∈F In Ex. 1.1.1 with M =S1R one checks that for all f,h∈C∞(M):Gfh=(f′)2h, and in general G h = g( f, f)h. In fact, by definition [B-G-V, Prop. 2.3], f ∇ ∇ LimitsandDegenerations ofUnitaryConformalFieldTheories 5 any second-order differential operator O satisfying [f,[1O,f]]=g( f, f) is a 2 ∇ ∇ generalized Laplacian. Using the time coordinate of a geodesic from x to y and truncating and smoothing it appropriately one checks that (1.1.2) indeed givesbackthegeodesicdistancebetweenxandywhichcorrespondstothemetric g. In other words, (M,g) can be recoveredfrom the spectral triple (H,H, ). A More generally, consider a spectral triple (H,H, ) with H a Hilbert space, A H a self-adjoint positive semi-definite operator, which on H is densely defined with H0,0 := ker(H) ∼= C, and A a C∗-algebra of bounded operators acting on H. In fact, in the above let us assume that M is spin and replace H = 21∆g by the corresponding Dirac operator and H = L2(M,dvolg) by the Hilbert D space H′ of square-integrable sections of the spinor bundle on M. Note that H can be calculated from , see (1.1.3) and (1.1.5). Moreover, we assume that D (H′, , ) obeys the seven axioms of non-commutative geometry [Co3, D A p.159]. Roughly speaking, these axioms ensure that the eigenvalues of H have the correct growth behaviour (1.1.4), that defines a map on with D ∇ A f : f :=[ ,f]: H′ H′; h : [ f,h]=0, (1.1.3) ∀ ∈A ∇ D → ∀ ∈A ∇ where in the aboveexamples f acts onH′ by Clifford multiplication,and that ∇ givessmoothcoordinatesonan“orientablegeometry”;furthermore,thereare A finiteness and reality conditions as well as a type of Poincaré duality on the K- groups of . If all these assumptions hold, then by (1.1.2) the triple (H′, , ) A D A defines a non-commutative geometry à la Connes [Co1,Co2,Co3]. If the algebra is commutative, then the triple (H′, , ) in fact defines a unique ordinary A D A Riemanniangeometry(M,g)[Co3,p.162].Theclaimthatthe differentiableand the spin structure of (M,g) can be fully recovered is detailed in1 [Re]. Following [F-G], instead of studying spectral triples (H′, , ), we will D A be less ambitious and mainly focus on triples (H,H, ), somewhat relaxing the A defining conditions: Definition 1.1.2 Wecall(H,H, )aspectral pre-tripleifHisapre-HilbertspaceoverC,H A is a self-adjoint positive semi-definite operator on H with H0,0 := ker(H) ∼= C, and A is an algebra of operators acting on H. Since H0,0 ∼=C∋1, we can view ֒ H by A A 1. A → 7→ · If additionally the eigenvalues of H have the appropriate growth behaviour, i.e. for some γ R and V R: ∈ ∈ N(E):=dimC ϕ H Hϕ=λϕ , N(E)E→∞V Eγ/2, (1.1.4)  ∈  ∼ · λ E M≤ (cid:8) (cid:12) (cid:9)  (cid:12)  then (H,H, ) is called a spectral pre-triple of dimension γ. A If there exists an operator which is densely defined on a Hilbert space H′ D that carries an action of with (1.1.3) such that A f,h : f, h =2 f,Hh (1.1.5) H H ∀ ∈A h∇ ∇ i ′ h i and such that (H′, , ) obeys the sevenaxioms of non-commutative geometry, D A thenwecall(H,H, )aspectral tripleoranon-commutative geometry A 1 WethankDiarmuidCrowleyforbringingthispapertoourattention. 6 DanielRoggenkamp andKatrinWendland of dimension γ. Remark 1.1.3 Note that our condition (1.1.5) for the operator H does not imply H = 1∆ on 2 g L2(M,dvol ). In fact, H will in generalbe a generalized Laplacianwith respect g to a metric g = (g ) in the conformal class of g. More precisely, we will have ij dvol =e2Φdvol with Φ C (M), and with g 1 =(gij), g g ∞ − ∈ e e e2H =−e−2Φ detg−1 ∂iee2Φ deteg gij∂j (1.1.6) i,j p X p e ee with respect to local coordinates,in accordwith (1.1.5). We call g the dilaton corrected metric with dilaton Φ. Note that g is easily read off from the symbol of H, allowing to determine Φ from dvol =e2Φdvol . g g e AgeneralizationofConnes’approach,whichisnaturalfromourpointofview,is e givenin[Lo].There,theDiracoperatoronthespinorbundleofM isreplacedby the Dirac type operator =d+d∗ on H′ =Λ∗(T∗M). Since ∆g = 2, (1.1.3)– D D (1.1.5) remaintrue,but the list ofaxioms reducesconsiderablyto the definition ofaRiemanniannon-commutativegeometry[Lo,III.2].However,ourmain emphasis lies on the recovery of the metric structure (M,g) rather than the differentiable structure. Similarly, in [K-S] the main emphasis lies on triples (M,R+g,ϕ), where ϕ: M is a map into an appropriate moduli space −→M M of CFTs. Itwillbeeasytoassociateaspectralpre-tripletoeveryCFT.Usingdegener- ations of CFTs in the spirit of[K-S],one canoften associatespectralpre-triples ofdimensionγ =ctoaCFTwithcentralchargec.Ageneraltheorem,however, whichallowsto associatenon-commutativegeometriesto arbitraryCFTs seems out of reach. In all cases we are aware of where a non-commutative geometry is obtained from CFTs, this is in fact proven by deforming an appropriate commutative geometry. In Sect. 4, we present a non-standard example of this type which should lead to interesting non-commutative geometries by deformation. 1.2. SpectraltriplesfromCFTs. Wedonotattempttogiveacompletedefinition of CFTs in this section; the interested reader may consult, e.g., [B-P-Z,M-S2, Gi,M-S1,F-M-S,G-G]. Some further properties of CFTs that are needed in the main text are collected in App. A. A unitary two-dimensional conformal field theory (CFT) is speci- fied by the following data: – a C–vector space of states with scalar product . This scalar product H h·|·i is positive definite, since we restrict our discussion to unitary CFTs; – an anti-C-linear involution on , often called charge conjugation; ∗ H – anactionoftwocommutingcopies ir , ir ofaVirasoroalgebra(A.1)with c c V V centralcharge2c Ron ,withgeneratorsLn, Ln, n Z,which3 commutes ∈ H ∈ 2 As a matter of convenience, we always assume left and right handed central charges to agree. 3 Theindexingofallmodesbelowcorrespondstoenergy, nottoitsnegative. LimitsandDegenerations ofUnitaryConformalFieldTheories 7 with .TheVirasorogeneratorsL andL arediagonalizableon ,suchthat 0 0 dec∗omposes into eigenspaces4 H H = , (1.2.1) H Hh,h h,Mh R, ∈ h h Z − ∈ and we set := 0 if h h Z. The decomposition(1.2.1) is orthogonal Hh,h { } − 6∈ with respect to ; h·|·i – a growth condition for the eigenvalues h, h in (1.2.1): For some ν R+ ∈ and V R: ∈ E R+: > dim E→∞exp V√E . (1.2.2) ∀ ∈ ∞  Hh,h ∼ (h+Mh)ν E (cid:16) (cid:17) ≤   In particular, for all h, h∈R we have Hh∗,h ∼=Hh,h, and we define ˇ∗ := ∗ ; H Hh,h hM,h R ∈ – a unique ∗-invariant vacuum Ω ∈ H0,0 ∼= C, as well as a dual Ω∗ ∈ Hˇ∗ characterized by (A.2); – a map C : ˇ∗ ⊗2 C that encodes the coefficients of the opera- H ⊗H −→ tor product expansion (OPE), such that C(,Ω, ): ˇ∗ C, (Ψ,χ) Ψ(χ), (1.2.3) · · H ⊗H−→ 7−→ i.e. the induced map is the canonical pairing. The OPE-coefficients C obey (A.10) and (A.12) and can be used to define an isomorphism ˇ Hψ −→Hψ∗∗, s. th. ∀χ∈H: ψ∗(χ)=C(ψ∗,Ω,χ)=hψ|χi. (1.2.4) 7−→ There are many properties of the map C, like the sewing relations, that have to be fulfilled for reasons of consistency, and which we will not indulge to list explicitly.SomepropertiesofCFTsthatfollowfromtheseconsistencyconditions should be kept in mind, however: – ϕ isalowestweightvector(lwv)withrespecttothe actionof ir , ir , c c ∈H V V i.e. a primary state, iff for all5 n N 0 : L nϕ=0, L nϕ=0. For any Z–graded algebra = ∈n w−e d{efi}ne − − L L n Z ∈ := L, (1.2.5) ± n L L n>0 ±M L := ker − = ϕ n N 0 , w n: wϕ=0 . H L ∈H ∀ ∈ −{ } ∀ ∈L− In other words, setting (cid:8) = ir(cid:12)= ir ir by abuse of notatio(cid:9)n, ir (cid:12) c c V L V V ⊕V H denotes the subspace of primary states in . H 4 Inthiswork,werestrictourinvestigations tobosonicCFTs. 5 Weagreeon0∈N,asarguedin[Bo,IV.4.1,R.6.2]. 8 DanielRoggenkamp andKatrinWendland – TheOPE,whichweencodeinthemapC asintroducedabove,allowstoasso- ciatetoeachϕ∈Hatowerϕµ,µ,µ,µ∈R,oflinearoperatorsϕµ,µ: Hh,h −→ , called (Fourier) modes, see (A.13). In particular, the elements Hh+µ,h+µ Ln, Ln, n Z, of irc, irc can be interpreted as the Fourier modes of ∈ V V theholomorphicandantiholomorphicpartsT ,T oftheenergy-momentum tensor. Moreover,Ω acts as identity on , and all other modes of Ω act 0,0 H by multiplication with zero. By abuse of notation we write T =L Ω , 2 2,0 ∈H T =L Ω for the Virasoro states in . 2 0,2 ∈H H A sextuple = ( , , Ω, T, T, C) with , , Ω, T, T, C as above specifies a C H ∗ H ∗ CFT. Two CFTs = ( , , Ω, T, T, C) and ′ = ( ′, ′, Ω′, T′, T′, C′) are C H ∗ C H ∗ equivalent, if there exists a vector space homomorphism I : , such ′ H −→ H that I:(Ω, T, T) (Ω′, T′, T′) and ′ =I , C′ =C (I∗ I I). Instead of prim7→ary states in ir∗, below◦,∗we will be◦ inte⊗rest⊗ed in primary V H stateswithrespecttoalargeralgebrathan ir,namelythe(generic)holomor- V phic and antiholomorphicW-algebra ∗ ∗,see(A.15).By(1.2.5)the W ⊕W primary states with respect to a subalgebra of ∗ ∗ are W W ⊕W W = ker − = ϕ n N 0 , w n: wϕ=0 . H W ∈H ∀ ∈ −{ } ∀ ∈W− To truncate the OPE(cid:8)to pri(cid:12)maries note that by (1.2.2) for give(cid:9)n ϕ and χ ∈ Hh,h, we have ϕµ,µχ 6=(cid:12) 0 for a discrete set of (µ,µ) ∈ R2. He∈ncHe, whenever the set I (ϕ,χ):= (µ,µ) R2 ψ W: ψ∗(ϕµ,µχ)=0 W ∈ ∃ ∈H 6 is finite, we candefine the(cid:8)truncated O(cid:12)PEϕ(cid:24)ψ asthe orthogona(cid:9)lprojectionof (cid:12) ϕ χ onto : µ,µ W H (µ,µ) I (ϕ,χ) ∈PW o W := ϕ W χ W: I (ϕ,χ) < ; H ∈H ∀ ∈H | W | ∞ o ϕ(cid:8) W, χ(cid:12) W : (cid:9) (1.2.6) ∀ ∈H ∀ (cid:12)∈H ϕ(cid:24)χ s.th. ψ : ψ (ϕ(cid:24)χ)=C(ψ ,ϕ,χ). W W ∗ ∗ ∈H ∀ ∈H Let us remark that the above definition of (cid:24) may well be too restrictive: By introducing appropriate (partial) completions of the relevant vector spaces one can attempt to replace our finiteness condition in (1.2.6) by a condition on o normalizability and thereby get rid of the restriction to W. Although in most o H ofourexampleswefind W = W,fortheorbifoldsdiscussedinSect.3.2, W o H H H − W containsalltwistedgroundstates.Thelatterdonotenterintothediscussion H of the zeromode algebra,which is relevantfor finding geometricinterpretations (see Sect. 2.2). Summarizing, our definition of (cid:24), above, is well adapted to our purposes, though it may be too restrictive in general. By construction, o ϕ W: ϕ(cid:24)Ω =Ω(cid:24)ϕ=ϕ. ∀ ∈H Let us extract a spectral pre-triple from a CFT = ( , , Ω, T, T, C). By C H ∗ definition of the adjoint (see (A.5), (A.11)), L acts as self-adjoint operator on 0 H, and L†1 = L−1. Moreover, 2L0 = [L†1,L1] shows that L0 is positive semi- definite, and similarly for L . Therefore, to associate a spectral pre-triple to a 0 LimitsandDegenerations ofUnitaryConformalFieldTheories 9 CFT , we will always use H := L +L , which by the uniqueness of Ω obeys 0 0 C ker(H)=H0,0 ∼=C. Following [F-G], we let H:= W =ker − H W denotethespaceofprimariesin withrespecttoanappropriatesubalgebra e H oW oftheholomorphicandantiholomorphicW-algebras.Moreover,toeveryϕ W ∈H we associate an operator Aϕ on H which acts by the truncated OPE ϕ(cid:24) as in o (1.2.6). The operators Aϕ, ϕ W, generate our algebra with the obvious ∈ H A vector space structure and with ceomposition of operators as multiplication: e o o ϕ W: Aϕ: H H, Aϕ(χ):=ϕ(cid:24)χ; := Aϕ ϕ W . (1.2.7) ∀ ∈H −→ A ∈H D (cid:12) E It is not hard to seee thate(H,H, ) obeys Def. e1.1.2 thu(cid:12)(cid:12)s defining a spectral A o pre-triple. As a word of caution, we remark that in general for ϕ,χ W, ∈ H Aϕ Aχ =Aϕ(cid:24)χ. e e ◦ 6 Severalother attempts to associatean algebra to a CFT can be found in the literature. Truncation of the OPE to leading terms, as suggested in [K-S, 2.2], givesastraight-forwardalgebrastructurebutseemsnottocaptureenoughofthe algebraicinformationencodedinthe OPE:Onthe onehand,ifallstatesin W H are given by simple currents, e.g. for the toroidal theories focused on in [K-S], then truncation of the OPE to leading terms is equivalent to our truncation (1.2.6). On the other hand, for the example that we presentin Sect. 4, it is not, andweshowhowourtruncation(1.2.6)givesaconvincinggeometricinterpreta- tion.Forholomorphicvertexoperatoralgebras,Zhu’s commutative algebra isacommutativeassociativealgebrawhichcanbe constructedusingthe normal ordered product by modding out by its associator (see [Zh,B-N,G-N]), and it is isomorphic to the zero-mode algebra[B-N]. Although to our knowledgeZhu’s commutativealgebrahasnotbeengeneralizedtothenon-holomorphiccase,itis very likely that such a generalization would yield the same geometric interpre- tations for degenerate CFTs that we propose below; this is related to the fact thatAϕ Aχ =Aϕ(cid:24)χ holdsfortherelevantstatesinthesedegenerateCFTs,see ◦ Lemma 2.2.3 and Prop. 2.2.4. Summarizing, our truncation (1.2.6), which goes back to [F-G], seems to unite the useful aspects of both the truncation of the OPE to leading terms and Zhu’s algebra. AnapplicationofTauber’stheoremknownasKawamata’stheorem[Wi,Thm. 4.2]showsthatthegrowthcondition(1.2.2)ensurestheeigenvaluesofH toobey (1.1.4) for γ =ν. In general,γ will not coincide with the central chargec of the CFT, but in many examples with integral c we find γ = 2c, see e.g. Ex. 1.2.2 below. So far, we have shown: Proposition 1.2.1 To any CFT = ( , , Ω, T, T, C) of central charge c which obeys (1.2.2) C H ∗ with γ = ν R, after the choice of a subalgebra of the holomorphic and ∈ W antiholomorphic W-algebras, we associate a triple o H:= W =ker −, H :=L0+L0, := Aϕ ϕ W . H W A h ∈H i (cid:16) (cid:12) (cid:17) Then (H,He, ) is a spectral pre-triple of dimenseion γ =ν(cid:12) as in Def. 1.1.2. A e e 10 DanielRoggenkamp andKatrinWendland The operator := L1 +L1: H obeys (1.1.5) as well as a Leibniz rule. ∇ → H However,forgeneralCFTsweareunabletoshowthat(H,H, )givesaspectral A tripleofaspecificdimension,i.ee.anon-commutativegeometryaccordingtoDef. 1.1.2. need not, in general, act by bounded operatores, andewe are unable to A check all seven axioms of Connes’ or their reduction in [Lo], including the fact that eis a C∗-algebra. Neither are we aware of any attempt to do so in the A literature, see also [F-G-R] for a discussion of some unsolved problems that this approaech poses. For toroidal CFTs, the above construction indeed gives a C∗-algebra of A bounded operators [F-G]. We illustrate this by e Example 1.2.2 Let R,R R+, denote the circle theory at radius R, i.e. the CFT with centrCal cha∈rge c = 1 that describes a boson compactified on a circle6 of radius R. All possess a subalgebra = u(1) u(1) of the holomorphic and an- R tiholomCorphic W-algebra7 (see AWpp. B), an⊕d the pre-Hilbert space of R R H C decomposes into irreducible representations of . The latter can be labeled by W left- and right handed dimension and chargeh , Q and h , Q of their lwvs, R R R R where h = 1Q2, h = 1Q2. The space of primaries of with respect to R 2 R R 2 R CR W is H := HW = spanC |QR;QRi:= Q22R,QR ⊗ Q22R,QR n (cid:12) E (cid:12) E(cid:12) e ∃m,n∈Z: (cid:12)(cid:12)QR = √12 mR(cid:12)(cid:12)+nR , Q(cid:12)(cid:12) R = √12 mR −nR , (cid:0) (cid:1) (cid:0) (cid:1)o see (B.4). To obtain the spectralpre-triple associatedto by Prop.1.2.1from R C H= W we need to consider the truncated OPE (1.2.6). By (1.2.7) and (B.6), H orthonormalizing the Q ;Q as in (B.3), we have | R Ri e A|QR;QRi◦A|Q′R;Q′Ri = (−1)(QR+QR)(Q′R−Q′R)/2A|QR+Q′R;QR+Q′Ri = A . (1.2.8) |QR;QRi(cid:24)|Q′R;Q′Ri o We see that W = W and := Aϕ ϕ W is generated by the Aϕ H H A h ∈ H i with ϕ as a vector space, i.e. (cid:24) defines an (associative!)product on , ∈HW (cid:12) HW whichsimplifiesthesituationconesiderablyi(cid:12)ncomparisontothegeneralcase.The algebra is clearly non-commutative.It is a straight-forwardnon-commutative A extensionofthe product(1.1.1)ofthealgebraofsmoothfunctions onthecircle, taking weinding and momentum modes into account. In fact, is the twisted A group algebra Cε[Γ] of the u(1) u(1)- charge lattice ⊕ e Γ = (QR;QR)=(Qm,n;Qm,n)= √12 mR +nR ; mR −nR m,n∈Z n (cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1)(cid:12)(cid:12) (1o.2.9) (cid:12) 6 Our normalizations are such that the boson compactified on a circle of radius R = 1 is describedbythesu(2)1 WZWmodel. 7 To clear notations, our symbol galways denotes the loop algebra associated to the Lie groupGwithLiealgebrag,andgk denotes itscentral extensionoflevelk.