aa r X i v : . [ h e p - t h ] J un Simplicity condition and boundary-bulk duality
Fen Zuo ∗ Abstract
In the first-order formulation, general relativity could be formally viewed as the topological BF theorywith a specific constraint, the Plebanski constraint. BF theory is expected to be the classical limit of theCrane-Yetter (CY) topological state sum. In the Euclidean case, the Plebanski constraint could be lifted inan elegant way to a quantum version in the CY state sum, called the simplicity condition. The constrainedstate sum is known as the Barrett-Crane (BC) model. In this note we investigate this condition from thetopological field theory side. Since the condition is in fact imposed on the faces, we want to understandit from the viewpoint of the surface theory. Essentially this condition could be thought of as resultingfrom the boundary-bulk dualtiy, or more precisely from the recent “bulk=center” proposal. In the languageof topological phases, it corresponds to the diagonal anyon condensation, with the BC model being thecondensed phase.The CY state sum, and correspondingly the BC model, is usually constructed with a modular tensor cat-egory (MTC) from representations of a quantum group at roots of unity. The category of representationsof quantum group and that of modules of the corresponding affine Lie algebra are known to be equivalentas MTCs. This equivalence, together with the simplicity condition in the BC model, guarantees the con-struction of a full 2d rational conformal field theory through the Fuchs-Runkel-Schweigert formalism. Wethus obtain a full-fledged 4d quantum geometry, which we name as “stringy quantum geometry”. Someattracting features of such a new geometry are briefly discussed. ∗ Email: . INTRODUCTION The essence of general relativity (GR) is believed to be that it is established without any a-priori background. One would thus expect that the quantum theory must inherit this key property.Topological quantum field theory (TQFT) [1, 2] provides a natural framework for a background-independent formulation of quantum gravity [3]. Indeed, GR has close relation to a specific topo-logical theory, which is manifest in the first-order formalism. With the frame field e and the spinconnection A , the Einstein-Hilbert action could be compactly written as S GR ( e, A ) ∼ Z M ( e ∧ e ∧ F ) , (1)where the wedge product is supposed to take over both the spacetime part and the internal part.Apparently, this could be viewed as the topological BF theory [4, 5]: S BF ( B, A ) = Z M ( B ∧ F ) , (2)with the constraint B = e ∧ e. (3)This is essentially the Plebanski formulation for GR, so we may call (3) the Plebanski constraint.The simplification for the canonical quantization with the Ashtekar variable [6] indicates thatwe should also express the constraint through chiral variables. Then it becomes clear that theconstraint simply says that the B field must be a simple bivector, which is an equal-weight com-bination of the self-dual/anti self-dual parts [7]. Moreover, the simplicity of the bivector deducesthat the two parts have the same norm [7]. This could then be lifted to a constraint in the quantumversion of the BF-theory, namely the Crane-Yetter (CY) construction [8], at least in the Euclideancase. The resulting constrained state-sum defines the Barrett-Crane (BC) model [7].Let us make this procedure more explicit for 4d Euclidean gravity as in [7]. Taking the spincover Spin (4) of the internal gauge group SO (4) , the self-dual and anti self-dual parts factorize asSpin (4) ∼ = SU (2) × SU (2) . (4)As in 3d [9], in order to obtain finite state sums [10], one has to deform the corresponding universalenveloping algebra to get the “quantum” group. The deformation parameter q is supposed to berelated to the cosmological constant [4]. When q takes special values at roots of unity, the categoryof irreducible representations of the quantum group possesses the nice structure of a “modular2ensor category” (MTC) [11], which could then be used to construct a CY state sum. In the BCmodel it turns out that one has to choose opposite braiding of the two quantum groups [7]. Denote C ≡ Rep ( U q sl ) and C − a copy of C but with reversed braiding. Then it means, here one hasactually chosen the Deligne tensor product C ⊠ C − . As a MTC, C contains only finitely manysimple objects. Number the set of simple objects of C by I . The first part of the simplicityconstraints, called the diagonal one, is imposed by selecting specific objects of the form X i × X i ,for any i ∈ I . Since in CY state sum one colors the triangles in the simplicial decomposition bythe objects, the diagonal condition is essentially imposed on the surfaces. The non-diagonal partis related to the tetrahedra itself, or the dual four-valent vertices. A four-valent vertex could besplit as two three-valent vertices, with one intermediate link. The three-valent vertices are coloredwith the intertwiners for U q sl , which are completely fixed up to normalization. One is left witha single intermediate link. Then the non-diagonal condition is specified in the same way as thediagonal one. In this way one obtains a unified description for all the links, either the original onesdual to the triangles, or those coming from splitting. Such a unified description could be easilyextended to general decomposition with arbitrary polyhedra [12, 13]. It is also vividly figured as“left-handed area = right-handed area” [14].One could further simplify the above quantum condition a little more. The idea is similar tothe Skein-theory description based on handle-decomposition [15]. Since in the CY state-sum onlyfinite sums and products are involved, one could change the order of them arbitrarily. Also it isimplicitly assumed that each allowed simple object appears in the final sum once and just once.Therefore one could color each link with a single special object: A cl ≡ ⊕ i ∈ I X i × X ∨ i . (5)Here we have intensively replace the second factor by its dual, which does not make any differencesince the representations are self-dual. Then the intertwiner for the individual simple objects mustalso be lifted to a specific intertwiner for three A cl ’s. This should in principle be possible, since allthe simple objects are treated on the same footing. At the end the only thing that matters would bethe normalization of the intertwiners, which we neglected here. In principle one should be able togive a rigorous proof of this following the procedure in [15], which will not be pursued here. Themain purpose of this paper is to explain the special nature of this object A cl from a holographicviewpoint. This will be the focus of the next section.3 I. “BULK=CENTER”A. “bulk=center”
The CY construction is of the kind of “absolute” theory called in [2]. In general we will need the“relative” theory [2] or extended ones as advocated in [3, 16] using higher categories. Even theseextended types still have two shortages: first, a TQFT constructed with some non-trivial highercategory could turn out to be classical, which can not be justified directly from the category data;Secondly, although a formal description of holographic principle could be established [17], it is notclear how it is concretely realized in extended TQFTs. To overcome these difficulties, one needs toperform some kind of the state-operator correspondence, and to change the focus from the Hilbertspaces to the observables. From the physical point of view, the Hamiltonian formulation [18–20]provides a nice solution to these problems. In particular, the Hamiltonian formulation allows fora concrete categorification of the holographic principle, which is encapsulated in the so-called“bulk=center” relation [21–23]. While the original proposal is restricted to the gapped phases [21,22] , a model-independent exposition is given recently [23] , which shows that the relation isuniversal (which is also briefly mentioned in [22]). In particular, a rigorous construction for thegapless edges of (2 + 1) d bulk topological phases is made recently in [27], by generalizing thenotion of enriched monoidal category [28]. It is proved that the “bulk=center” relation is indeedvalid in such a gapless case [28]. In the following we shortly summarize their main idea here.In the Hamiltonian formulation [18–20] of TQFTs, the topological excitations of various codi-mensions could be identified as the physical observables. In ( n + 1) -dimension, this is compactlydescribed by a unitary fusion n -category, which could also be viewed as a unitary ( n + 1) -categorywith one simple object [21, 22]. One could then study the relation between the theories in differentspacetime dimensions. It is discovered that this could be elegantly captured by the mathematicalnotion of “ Z n -center” of a ( n − fusion category. Namely if a ( n + 1) -dimensional bulk theoryis holographically/physically related to a n -dimensional boundary theory, the bulk excitations aregiven by the Z n -center of the boundary excitations, which themselves are captured by a ( n − fu-sion category. In this rough statement we have somehow relaxed the required conditions specified Another issue is the so-called perturbative gravitational anomaly in k + 3 spacetime dimensions [21]. In (2 + 1) d this would correspond to the framing anomaly [24, 25]. This is expected to be resolved/restored [24][20, 26] whenone lifts it to a (3 + 1) d bulk theory, and thus will be neglected in this paper.
4n the original formulation [21, 22]. But this does not cause any real problems since later we onlyapply it in the low-dimensional cases, which have been analyzed explicitly [21]. We will not recallthe rigorous definition of the Z n -center, but only outline some properties. In the special case when n = 2 , it just coincides with the ordinary monoidal/Drinfel’d center Z . The Z n -center was shownto possess an important property, namely the center of center is trivial. This can then be employedto define a generalized cohomology theory for the theories in different spacetime dimensions. Atheory is called closed if the center of its excitations is trivial, otherwise it is anomalous; in otherwords, a closed boundary theory defines a trivial bulk theory, while an anomalous boundary neces-sarily gives rise to nontrivial bulk excitations and a bulk description is really necessary. A theory iscalled exact if its excitations are the center of the excitations of a boundary theory; in other words,it has a dual boundary description. B. 4-3-2 Now we could revise the various topological state sums in physical space-time dimensionswith the above holographic principle. This has essentially been done in the Hamiltonian approach,so we only need to translate them back to the “path-integral” formulation. Let us start with thepreviously mentioned CY state sum, which is expected to be the quantum version of the BF -theory. Originally it is constructed from a MTC , e.g., Rep ( U q sl ) for q at roots of unity, thusa special kind of -category. One would then wonder if the CY construction satisfies the aboveboundary-bulk duality, so let’s go to the observables. It turns out that the CY-theory is classical [15,32][20], thus contains no non-trivial topological excitation at all. In other words, it is characterizedby the trivial category . And if we want it to be dual to a boundary theory, the boundary theorymust be closed, or anomaly free.Since the state space of the BF -theory is given by the Chern-Simons (CS) func-tional [24][33][4], it is natural to expect that the two are holographically related. The mathematicalformulation of the CS theory [24] is believed to be the Reshetikhin-Turaev (RT) TQFT [25], whichcould be further extended to give a (3 − − extended theory. From the handle-decompositiondescription [15, 26], it is clear that CY and RT theory are indeed directly related. In particular, The title is borrowed from [29]. We are not going to talk about the more general construction with a premodular category [30, 31], since they willnot be directly related to the BF -theory anymore. C . However, no Hamiltonian formulation of RT theoryhas been realized, and thus it is hard to characterize it with the excitations. Since the RT theoryassigns the objects of C to the circle S [34–37], which is the lowest level in the extended theory,one may argue that the basic excitations are just specified by the category itself. Later we willsee that this is indeed the case for a special class of MTCs. If we accept such an identificationof excitations, the boundary-bulk duality asserts that the MTC must have a trivial center. In otherwords, its Z -center is trivial in the notion of [21, 22], Z ( C ) = . (6)In order to get a boundary description, we should further impose the exactness condition: C = Z ( C ) , (7)for C a unitary fusion category . As stated before, the Z -center coincides with the monoidalcenter of a fusion category. To simplify the notation, we will just write C instead of C , and themonoidal center instead of the Z -center. Thus C = Z ( C ) . Notice that the modular structure of C is automatically guaranteed when C is unitary fusion. Furthermore, if C is modular we have thefollowing braided equivalence: C = Z ( C ) ∼ = C ⊠ C − , (8)where C − is again obtained from C by reversing all the braiding. At this moment we only needits fusion structure. It is well known that from a unitary fusion category C one could constructthe Turaev-Viro (TV) state sum [10], which is a convergent form of the Ponzano-Regge state sumfor 3d quantum gravity [9]. The relation with RT invariants has been extensively studied [15, 26].In [34–37] it is shown that it could be extended as a (3 − − TQFT, and the extended theory isequivalent to RT-theory based on Z ( C ) : Z TV , C ∼ = Z RT ,Z ( C ) . (9)More importantly, we have an elegant Hamiltonian realization of TV-theory, the Levin-Wenmodel [19], which is constructed with the same data C . A nice review of this model is given According to [38], this can be relaxed to a braided equivalence. Moreover, based on the notion of enriched monoidalcategories, it has been proved that such an equivalence can be established for general MTCs [27, 28].
6n [20]. Roughly speaking, the model is built by labeling the edges in a 2d space lattice with ob-jects of C . The morphisms and associators are treated as operators acting on the lattice. Combiningthese operators one could construct a Hamiltonian, which essentially imposes “gauge-invariant”constraints on the vertices and “flatness” constraints on the plaquettes. The meaning of theseconstraints become more clear in an equivalent model, Kitaev’s quantum double model [18], con-structed with some weak Hopf algebra H as the “gauge group”. The two are related according tothe Tannaka-Krein reconstruction theorem [39] C ∼ = Rep ( H ) . (10)The equivalence of the extended versions of the two models, and with the extended TV/RT-theory,have been firmly established [40–43]. Now with these Hamiltonian realizations, one could explic-itly identify the various excitations, which appear whenever the constraints in the Hamiltonian areviolated. It turns out they are characterized by the monoidal center Z ( C ) . Intuitively, this corre-sponds in the extended TV/RT theory to the data assigned to the circle S [34, 36, 37]. This alsoprovides some evidence of our previous proposal that the excitations in general RT-theory couldbe identified with the constructing data C .The power of the Hamiltonian formulation is much beyond this. They could be further gener-alized to describe also the boundary theory, which are elaborated in [44, 45]. Here a very nice no-tion, as a straightforward categorization of modules over rings, module categories over a monoidalcategory [46], perfectly fit in. As we mentioned, the Levin-Wen model is built with a 2d spacelattice, with the edges colored with objects in C . On the 1d boundary of the lattice, it is naturalto color the boundary edge with objects of the module category C M , and the vertices with theaction morphisms. Moreover, the boundary excitations are specified by the monoidal category ofendofunctors of C M , C ∨ M := F un C ( M , M ) . It is shown that C and C ∨ M are Morita equivalent, which immediately leads to the boundary-bulkduality [44][38, 45], C = Z ( C ) ∼ = Z ( C ∨ M ) . . Frobenius algebra, full center and anyon condensation However, the description of the boundary theory with the module category is a little too formal.Also the relation with the ordinary 2d open-closed TQFT [47, 48] is not clear. It would be moresatisfactory if an intrinsic formulation using some structure within C could be established. Toachieve this we need some kind of classification of the module category C M . This is providedin [46] through the introduction of an association algebra A in C . The category of right A -modulesMod C ( A ) have a natural structure of module category over C . In [46] the inverse statement isproved: for any semisimple module category C M , there exists a semisimple algebra A in C suchthat C M ∼ = Mod C ( A ) as module categories. The algebra A is determined up to Morita equivalence, and can be chosento be connected. Connectivity means A has a unique unit. Thus we could characterize the modulecategories with different algebras in C .We want to focus on the special situation when C is modular, to see what additional propertythe algebra A will inherit. In this case we have (8), which enables us to use the folding trick [49].With this the topological boundary condition of RT-theory with Z ( C ) is mapped to the topologi-cal surface operator in a RT-theory constructed directly with C [38, 49]. The surface operator isshown manifestly, using open-string scattering diagram, to be a special symmetric Frobenius alge-bra (SSFA) in the MTC C [49]. Actually [49] gives a very nice illustration of such a complicatednotion. It is then rigorously proved in [38] that to any such surface defect there is associated aMorita equivalence class of SSFAs.A similar analysis [49] shows that closed-string scattering diagrams define a special kind ofboundary-bulk map for the given boundary condition. More precisely, it relates the boundarycondition, represented by the SSFA A in C , to its derived center, which is a commutative SSFA Z ( A ) in Z ( C ) . The precise definition of such a notion is provided in [50], called full center ofan algebra in a monoidal category. It is further proved that the full center is the property of themodule category Mod C ( A ) , and thus depends only on the Morita equivalence class. Now therelation with the well-known classification of the 2d open/closed TQFT [47, 48] becomes clear.One may say that a Morita-equivalent class of SSFAs in a MTC, together with its full center,defines an anomalous open/closed TQFT [51]. This is the essential part of the TQFT constructionof rational conformal field theory (RCFT) in the series of papers [51–53], which we denote as8uchs-Runkel-Schweigert (FRS)-formalism.It will be more clear to consider the boundary-bulk map for a fixed boundary condition fromthe viewpoint of anyon condensation [54–58]. In particular, a systematical bootstrap analyzes ofanyon condensation is performed in [56] in a full categorical language. According to the analyzes,the SSFA A could be considered as the vacuum of the boundary phase. The previously mentionedconnectivity of A guarantees that the boundary phase is stable. Boundary excitations are describedby the unitary fusion category of A -bimodules in C , which is again Morita equivalent to C . The fullcenter Z ( A ) is the condensed set of anyons, or vacuum of the condensed phase, which is definedas a condensable algebra in Z ( C ) . This is an equivalent notion as a commutative SSFA in Z ( C ) ,with the additional physical condition that Z ( A ) be connected. The boundary theory correspondsto the special anyon condensation from the Z ( C ) -phase to the trivial -phase. In this case, thefull center/condensable algebra Z ( A ) is a Lagrangian algebra in Z ( C ) , i.e., ( dim Z ( A )) = dim Z ( C ) . (11)Here the quantum dimensions of an individual object and of the category itself are defined as usualin a MTC. D. Simplicity condition as diagonal anyon condensation
Now we know the 4d theory could be holographically described by the boundary data of aconnected SSFA A in a MTC C . But how to explicitly obtain these algebras? While the generalsituation could be quite complicated, there always exists a simple and special construction: theso-called Cardy case. It is also referred to as the “charge conjugation construction”. In this case A is simply taken to be the tensor unit of C , or its bimodules in C : A = X ⊗ X ∨ , X ∈ C . (12)According to [49], these objects represent the transparent/invisible surface operators. The corre-sponding full center is given by Z ( A ) = ⊕ i ∈ I X i × X ∨ i , (13)where I labels the equivalent classes of simple objects in C . It is not difficult to check that theLagrangian condition (11) is indeed satisfied. In 2d RCFTs, this corresponds to the so-called9diagonal modular invariant” [59]. Accordingly, this is called a diagonal anyon condensation fortopological phases [58].Now let us fix C = Rep ( U q sl ) for q at roots of unity, as in the BC model. According to(8), the bulk excitations are then given by the center Z ( C ) ∼ = C ⊠ C − , which provides a directinterpretation of the opposite braiding of the two factors in the BC model (see also the discussionin [12]). Moreover, the special object A cl (5) for the simplicity condition is just Z ( A ) in Z ( C ) forthe choice (12). This explains one of the doubts I mentioned in a recent note [60]. Similar ideahas appeared recently [61].In summary, the BC model is a sub-sector of CY theory satisfying Cardy-case boundary con-dition on any possible d boundaries/domain walls. Alternatively speaking, the BC model is adiagonal anyon-condensed phase of the CY theory. The condensed phase is topologically triv-ial in the bulk, but on the d domain walls of these trivial phases, a full closed RCFT naturallyemerges [27]. A general construction of boundary/bulk RCFTs from anyon condensation is estab-lished recently [27]. This full RCFT structure is the main topic of the next section. III. STRINGY QUANTUM GEOMETRY
At the end of [7], the authors made an interesting observation/suggestion:"It is interesting that the sort of tensor categories that go into the state sum we are proposing areso similar to the ones invented in constructing string theories [62]. Terms in our state sum can beinterpreted as diagrams in string perturbation theory, by connecting together the diagrams we areassociating to the -simplexes, and interpreting elements in the representations as string states."In this section we want to make a first attempt towards realizing explicitly the above suggestion.Actually we have already used a lot of familiar elements from string theory or 2d conformal fieldtheories (CFTs): open-string/closed-string scattering diagrams, diagonal modular invariants, etal. And many mathematical notions we have employed, such as module category, associativealgebra in a monoidal category, and full center, are motivated (or partially motivated) by the TQFTconstruction of RCFTs [51]. Thus one would expect that the correspondence between generaltypes of anyon condensation and modular invariants should somehow be a consequence, ratherthan a coincidence [54, 58]. Indeed, the exact relation has been established recently in the generalcase [27]. Here we only focus on the special case relevant for the BC model.Up to now we have been focusing on the MTC C = Rep ( U q sl ) for q at roots of unity. For10any years people believe that such a MTC, for the representations of the quantum deformationof a general simple Lie algebra g , is equivalent to the MTC C ˆ g of modules of the affine Lie algebra ˆ g at level k . However, it takes quite a long time for the mathematicians to rigorously prove this.Only until recently it becomes clear that the complete proof for the general situation could indeedbe achieved [63]. Let us recall the precise statement as follows [64–69] [63, 70, 71]:For a general simple Lie algebra g , the following equivalenceRep ( U q g ) ∼ = C ˆ g , (14)as MTCs exists, for q = e i π/m ( k + h ∨ ) . (15)Here m is squared ratio of a long root and a short root, and h ∨ is dual Coxeter number. See [72, 73]for some interesting discussions on this theorem. As emphasized in [63], the construction of thetensor category structure for the modules of the affine Lie algebra is indeed a vertex-algebraicproblem, which requires the general theory of Vertex Operator Algebra (VOA). While on thequantum group side, it is still a pure algebraic construction based on Lie algebra representationtheory. That may explain why it takes such a long time to finally establish the MTC structure ofthe modules of affine Lie algebras for general simple Lie algebras.Now with the above theorem, we could consider all the data in constructing the BC modelas from the category of the modules of the affine Lie algebra b su (2) k with positive integer level k . As the loop extension of the Lie algebra, affine Lie algebra allows us to include a richlocal structure into the state sum, in contrast with the quantum group. In fact, the affine Liealgebra b su (2) k together with a Morita-equivalence class of the SSFA (12) allows us to constructa full CFT, namely Wess-Zumino-Witten model, on any surfaces [51, 53]. The advantage of sucha construction is that it is expected to be completely background independent/coordinate free.In [74], it is proposed that a full 2d CFT could be viewed as a “stringy algebraic geometry”. Herewe could imitate this notion and call the 4d spacetime with full CFT structure on the 2d surfacesas a “stringy quantum geometry”. In the following I will try to sketch some attractive features ofthis new geometry. When we consider open (1 + 1) d boundary supporting chiral gapless excitations, then this enrichment would berequirable rather than desirable [27].
11e already see that the BC model is completely fixed once we choose for all the faces, originalor intermediate, the special object (5). This is because the U q sl intertwiner for three simpleobjects is fixed once the normalization for all the simple objects are chosen [13]. While thislooks very elegant, it also becomes a fatal weakness of the model: it lacks degrees of freedomto accommodate some allowed quantum data like the face angles [75, 76]. Also it is suspectedthat the correlations between neighboring simplices are too limited [75, 77]. Various relaxationsto the original model have been made to improve these situation [76–78]. Here the introductionof the CFT structure may provide a new solution, and without modifying the elegant topologicalstructure.First at the topological level, we have just reinterpreted the original model with some newmathematical structure, but with no essential changes. In particular, the whole structure is still fullyconstrained by the three-valent intertwiners, or structure constants of the three-point correlationfunctions in RCFT. This also has a nice interpretation in the new mathematical setting: they aregiven by the fusing matrices of the category of the A -bimodules [52]. For the Cardy case, theexplicit expressions for both the fusing matrices and the structure constants are derived [52].But the affine Lie algebra, viewed as a VOA, has in addition to the topological structure alsocomplex analytic structure. These may enable us to capture the required local correlations andquantum data. First consider the faces, or the dual links. At the tensor category level, they are col-ored by (5) as the direct sum of simple objects. But each simple object corresponds to a conformalfamily, which possesses an infinite tower of descendants. For a large excitation level, the numberof descendants increases exponentially with the level. Based on this, it is suggested in [60] thatthis huge correlation across the faces could explain the Bekenstein-Hawking formula for the vac-uum entanglement entropy. These descendants are completely hidden in the two and three-pointconformal blocks. In other words, the corresponding two- and three-point functions have universaldependence on the coordinates of insertion points, which are completely fixed by conformal in-variance. The situation changes for the four-point conformal blocks, in which the contributions ofdifferent descendants could be distinguished by the different coordinate dependence. This signifiesthat the tetrahedra, or more properly four-punctured sphere, has its own degrees of freedom notfixed by the structure constants. Further gluing five such punctured spheres, one obtains a fattened -simplex. This is a genus- Riemannian surface, as shown in [79–81]. Correspondingly, on theCFT side one describes this with the vacuum partition function on this surface. One would expectthe dependence of the partition function on the moduli space captures the underlying geometry.12ne could continue to glue different fattened -cells across punctured spheres. It is a little difficultto imagine the process, but a very nice picture is provided in the FRS-formalism [51]. One makesthe double cover of the punctured sphere, with the punctures identified. Then one could easilyconnect the two pieces to the two neighboring fattened -cells. Since the punctured spheres havetheir own degrees of freedom, one expects that these data would be transmitted during this gluingprocedure. In summary, in the FRS-formalism we have very nice factorization/sewing propertiesfor the CFT correalators, which perfectly match this gluing procedure for the spacetime geometry.It would be interesting to notice that the above situation is quite similar to the recent Witten-Costello construction of integrable lattice model [82, 83]. Starting from the CS theory, they intro-duce the loop extension (without central extension) of the gauge group. By properly recombiningthe loop parameter and one space coordinate, they obtain a mixed holomorphic/topological the-ory, which results the long-expected full-fledged Yang-Baxter equation with the spectral parame-ter [84, 85]. Since the loop parameter mixes with one of the space coordinate, the original threedimensional symmetry in CS theory is reduced to the 2-dimensional symmetry in the integrablelattice models [82, 83].The comparison with the Witten-Costello construction would be quite inspiring for the deter-mination of the symmetry algebra here. Eventually this provides an alternative interpretation ofthe decomposition (4). So I would like to discuss a little more of the choice of the Lie algebra atthis point, which is unfortunately ignored in the first version of [60]. More generally, this would berelated to the notion of so-called quantum symmetry. While the original internal symmetry groupis SO (4) in 4d Euclidean case, after loop and central extension we are left with ˆ g = b su (2) k , with su (2) the symmetry algebra and b su (2) k the spectrum-generating algebra. To get a complete theorywith the symmetry, we need to consider the full field algebra ˆ g ⊗ ˆ g [86]. Then the correspondingmodules C ˆ g ⊗ ˆ g ∼ = C ˆ g ⊠ ( C ˆ g ) − ∼ = Z ( C ˆ g ) (16)provide the required bulk excitations. Similar consideration for the group structure of the BCmodel has been noticed in the early literature [87].If one accepts this logic, then it will be natural that in the Lorentzian case the Lie algebra su (1 , is to be used. See [88, 89] for recent discuss on such a choice. Extending to the affineLie algebra and making the doubling to the full field algebra then gives b su (1 , k ⊗ b su (1 , k . Thisis the starting point of [60], which is chosen there simply by analogy with the investigation of (2 + 1) d black holes [90]. However, compared to the Euclidean case there is a crucial difference13ere. Now b su (1 , k is not rational any more, and correspondingly the category of modules donot have the structure of a MTC. Then the relations (8) and (16) do not hold. The folding trickand the surface operator/boundary condition correspondence also become ambiguous. But fromthe viewpoint of the 2d CFT, this may still be the natural choice. Also the transparent objects(12) and the corresponding diagonal modular invariant may still exist. If this is accepted, onecould then follow the derivation there to get the Bekenstein-Hawking formula for the vacuumentanglement entropy. It should be pointed out that no simplicity condition has been imposed insuch a derivation (which is of course not a proper choice) [60]. But it is easy to show that furtherimposing this changes at most the sub-leading logarithmic term.An immediate question will be, when will the 4d rotational/Lorentzian symmetry be restored?And a similar question could be asked in the Witten-Costello construction, in which part of answeris already known. We leave this question to the future study. IV. DISCUSSION
In this paper I have illustrated that the simplicity condition in the BC model corresponds to thespecial symmetric Frobenius algebra in C = Rep ( U q sl ) for the Cardy case. In other words, itcorresponds to a diagonal anyon condensation from the original CY topological phase character-ized by Z ( C ) ∼ = C ⊠ C − . The equivalence for the MTC structure between quantum groups andthe affine Lie algebras allows us to reinterpret the whole theory with the modules of the affine Liealgebra b su (2) k . Employing the general VOA framework and the FRS-formalism, this enables us toconstruct a full CFT on any surfaces. Essentially this makes it possible to introduce local geomet-rical data into the spin-foam framework, which has been one of the long-awaited goal of quantumgravity [5]. I have roughly sketched the geometrical structure with the usual simplicial decom-position, which is also the strategy adopted in the FRS formalism. As I mentioned, perhaps thehandle-decomposition could provide a better and more general description of the geometry, just asof the topological structure [15]. The topological theory based on handle-decomposition has beenrecently developed in [91]. It would be interesting to further enrich the chain-mail diagram in thisdescription with VOA data.We have given some arguments suggesting that in the Lorentz case we should follow the sameprocedure with b su (2) k replaced by b su (1 , k . Unfortunately, the affine Lie algebra is no longerrational any more. So many nice properties of those mathematical structure in the rational case14re lost. One even does not know if the whole scheme is still feasible. Some preliminary analysishas been made in my recent note [60], as I mentioned. This is possible due to the observationthat, in the large- k limit, the resulting theory could be formally viewed as a gauge-fixed version of (2 + 1) d closed-string theory. Some familiar techniques from string theory are then employed tomake the state-counting for the surface degrees of freedom. It is hoped that by fusing together thecombinatoric/topological construction with the geometric aspect of string theory, further progresscould be made. As we show, this is also what Barrett and Crane have suggested/expected almost20 years ago. Acknowledgments
Much of this work has been done during my visit at Yau Mathematical Sciences Center, Ts-inghua University. The hospitality of YMSC is acknowledged. I want to thank Si Li in particularfor the invitation, for explaining many relevant mathematical notions, and for encouraging me tocomplete the subject. I am grateful to Liang Kong and Hao Zheng for sending me the updatedversion of their recent work and illustrating some new results therein. I have also benefitted fromthe discussions with Yi-Hong Gao, Ling-Yan Hung, Wei Song, Jun-Bao Wu, Jian Zhou and manyothers. The work was partially supported by the National Natural Science Foundation of Chinaunder Grant No. 11405065. [1] G. B. Segal.
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