Poincaré Series, 3d Gravity and Averages of Rational CFT
aa r X i v : . [ h e p - t h ] F e b Poincar´e Series, 3d Gravity and Averages of RationalCFT
Viraj Meruliya a Sunil Mukhi a and Palash Singh b a Indian Institute of Science Education and Research,Homi Bhabha Rd, Pashan, Pune 411 008, India b Mathematical Institute, University of Oxford,Woodstock Road, Oxford, OX2 6GG, United Kingdom
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We investigate the Poincar´e series approach to computing 3d gravity partitionfunctions dual to Rational CFT. For a single genus-1 boundary, we show that for certaininfinite sets of levels, the SU(2) k WZW models provide unitary examples for which thePoincar´e series is a positive linear combination of two modular-invariant partition functions.This supports the interpretation that the bulk gravity theory (a topological Chern-Simonstheory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moodyalgebra. We compute the weights of this average for all seed primaries and all relevantvalues of k . We then study other WZW models, notably SU( N ) and SU(3) k , and findthat each class presents rather different features. Finally we consider multiple genus-1boundaries, where we find a class of seed functions for the Poincar´e sum that reproducesboth disconnected and connected contributions – the latter corresponding to analogues of3-manifold “wormholes” – such that the expected average is correctly reproduced. Keywords:
AdS gravity, Modular invariance, AdS/CFT correspondence, Rational con-formal field theory ontents k WZW Models 92.3 SU( N ) k WZW Models 182.3.1 Poincar´e Sums for SU( N ) k k WZW Models 33 k
44D Sage calculations for unitary Virasoro minimal models 48
The problem of computing the partition function of pure Einstein gravity with AdS bound-ary conditions was addressed in [1] using the method of Poincar´e sums (earlier work onthese sums in related contexts can be found in [2], in this context see also [3, 4]). Oneconsiders Euclidean gravity on manifolds whose asymptotic boundary is a torus and as-sumes a semi-classical summation formula. This is physically motivated by the existenceof an SL(2, Z ) family of Euclidean black holes, all of which have a torus boundary, thatare assumed to be the relevant saddle-points of a gravity path integral. The result is aPoincar´e sum whose first term is the contribution from the “vacuum” spacetime, namelyEuclidean AdS , while the remaining terms are modular transforms of this contribution.Subsequently the results of [1] were streamlined and generalised in [5, 6] among otherworks. One of the important generalisations was the fact that the Poincar´e series neednot start with the vacuum contribution. Instead one can start with the partition functionfor any “seed” primary and compute the Poincar´e sum over this. More generally the– 1 –ontributions from different seeds can be added, leading to a large family of candidateAdS gravity partition functions. Unfortunately the results in [1, 6] have certain undesirablefeatures that have proved very difficult, if not impossible, to eliminate even allowing forsuch generalised seeds.Following [1], it was proposed in [7] to examine whether a similar Poincar´e seriesapproach could be used with a different starting point: the existence of unitary minimal-model RCFT’s with central charge c < γ ( a, b, c, d ) are known explicitly. One may define the gravity duals of such theories,though they will not be semi-classical, by treating the identity character of the RCFT asa “seed” primary and assuming its partition function to be the analogue of a “vacuumcontribution” to a gravity partition function. The rest of the partition function is thenobtained by summing a Poincar´e series, which in this case is a finite sum. The result of[7] is essentially a no-go statement about unitary ( m, m + 1) minimal models: beyond thefirst two models, the gravity partition functions are not equal to CFT partition functions.Instead one finds linear combinations of physical CFT partition functions in a few cases,and for many values of m one also gets linear combinations that include unphysical modularinvariants (in the sense of having negative coefficients in their q, ¯ q expansion). Thus, it wasconcluded that most unitary minimal models cannot be interpreted as being dual to AdS gravity .In recent years there has been a realisation that in low dimensions, AdS gravity maybe dual not to a single CFT but to a weighted ensemble of CFT’s [9–12]. As brieflynoted in [11], the result of [7] for unitary minimal models might be an example of thesame phenomenon. This motivates us to examine the Poincar´e sum for various classesof RCFT and try to understand whether it can be interpreted in terms of either a singleRCFT dual or an average over finitely many RCFT’s. In the former case one can claim a(non-semi-classical) AdS/CFT duality, while in the latter one can try to interpret the dualas a weighted average over different physical CFT’s. The latter interpretation requires inparticular that the weights appearing in the average be non-negative, so one has to lookfor classes where this is the case. Next, one may go beyond the case of a single toroidalboundary since additional consistency conditions can arise by considering 3-manifolds withmultiple disconnected torus boundaries, as well as higher-genus boundaries. In particular,with multiple genus-1 boundaries the interpretation as an average of CFT’s should workwith the same weights as for single boundaries. We will find examples that satisfy all ofthese requirements . We expect that this study could teach us valuable lessons, in relativelysimple cases, of how averaging over CFT’s works in the AdS/CFT correspondence.Compared with the general (irrational) case, the Poincar´e sum for RCFT is a finitesum and has to be performed over a coset Γ c \ Γ where Γ = SL(2, Z ) and Γ c is the finite-index subgroup that keeps the seed contribution to the partition function invariant. Thisneeds to be worked out case by case and quickly becomes quite tedious. Also the precise More recently it has been argued in [8] that studying the Poincar´e sum approach with genus-twoboundaries provides additional constraints that exclude even the second ( m = 4) minimal model. We will not consider higher-genus boundaries in this paper and will only comment briefly on this atthe end. – 2 –efinition of Γ c depends on the seed rather than being just a property of the characters.Hence we resort to a more practical approach: we carry out our Poincar´e sum over thecoset Γ( N ) \ Γ where Γ is the principal congruence subgroup of SL(2, Z ), while N is aninteger that we will define (it is similar, but not equal, to the “conductor” N of an RCFTdefined in [13]). This is a single group for each set of characters, independent of whichprimary is used as the seed, and is easily found just be evaluating N . It is a subgroupof the true invariance group Γ c of the given seed partition function, and since it has finiteindex in Γ c (which in turn has finite index in Γ) we will just get the desired result of thePoincar´e sum with an overall factor that can be removed by a normalisation.For the SU( N ) k WZW models the criteria for which one finds different numbers ofRCFT in the sum depend on N and the level k . Depending on these values we find precisecriteria under which the Poincar´e sum yields the partition function of a single RCFT ora linear combination of physical RCFT’s. There are infinitely many values of k for whicheach of these possibilities is realised. Where the Poincar´e sum yields a sum of distinctpartition functions, we will find a method to compute the weights of this average as afunction of N, k , and work it out in the simplest cases where the average is over two CFT’s.We will work things out in considerable detail for the unitary SU(2) k WZW models, andsubsequently discuss aspects of SU( N ) k for N ≥ . We also revisit the minimal (unitaryand non-unitary) series, for which the analysis is quite similar to SU(2) k in many aspects.We then move on to the Poincar´e sum for multiple genus-1 boundaries. We will findthat for carefully chosen seeds, one finds answers such that the partition function is anaverage over products of CFT partition functions with the same weights, for any numberof boundaries. In the irrational case this type of setup implies the existence of “wormhole”contributions from manifolds with multiple boundaries [9, 14–16] and we propose analogous“wormhole” seeds in the RCFT context.Let us now briefly review the approach of [1, 6] (henceforth referred to as MWK) tothe calculation of the semi-classical partition function of AdS Einstein gravity, as well asits generalisation to the RCFT case in [7]. The MWK approach sums a seed over a setof manifolds denoted M γ where γ ∈ Γ = SL(2, Z ). Of these, M is thermal AdS and theothers are Euclidean BTZ black holes. The most obvious starting point is to take thermalAdS as the seed. Because the BTZ black holes are SL(2, Z ) transforms of thermal AdS ,we have: Z γ ( τ, ¯ τ ) = Z ( τ γ , ¯ τ γ ) (1.1)where: γ = a bc d ! ∈ Γ , τ γ = aτ + bcτ + d (1.2)Thus one would naively write the gravity partition function as a Poincar´e sum: Z ( τ, ¯ τ ) = X γ ∈ Γ Z γ ( τ, ¯ τ ) = X γ ∈ Γ Z ( τ γ , ¯ τ γ ) (1.3) While discussing the “gravity duals” of WZW models, we will make some tentative comments on whatthis should mean. – 3 –owever, SL(2, Z ) elements of the form: γ = n ! (1.4)leave Z invariant, and these generate a group isomorphic to Z . So the sum above isreplaced by a Poincar´e sum over coset representatives: Z ( τ, ¯ τ ) = X γ ∈ Z \ Γ Z ( τ γ , ¯ τ γ ) (1.5)The rest of the calculation proceeds by evaluating Z using the action and symmetriesof thermal AdS , and then evaluating the sum. After regularisation the result is finitebut unfortunately has several undesirable features including negativity of coefficients thatshould correspond to degeneracies of states. A relatively recent work [17] noted a large-scale occurence of negative coefficients in the q -series. One class of remedies that has beenproposed [6, 18, 19] is to cure the negativity by generalising from the identity seed to a sumover suitably selected primary seeds. A different remedy [20] is based on a relation (viaKaluza-Klein reduction) to the analogous problem for AdS , which suggests one shouldsum over additional 3-manifolds – however the compatibility of this proposal with modularinvariance is not yet clear.To find gravity duals for rational CFT, the approach pioneered in [7], starts from aslightly different perspective. This work considers the unitary Virasoro minimal modelswith central charge 0 < c <
1. At such small values of c the gravity dual (if such a concepthas a meaning) would have an AdS radius ℓ AdS = G N c ∼ G N which is far from the semi-classical regime. So one gives up the idea of summing over manifolds that are classicalsolutions. In its place, one considers the partition function Z Id ( τ, ¯ τ ) = | χ Id | coming fromsecondaries over the identity primary (after null vectors are removed). Defining Γ c ⊂ Γ asthe subgroup of Γ that preserves Z Id , one evaluates the Poincar´e sum: Z ( τ, ¯ τ ) = X γ ∈ Γ c \ Γ Z Id ( τ γ , ¯ τ γ ) (1.6)Unlike the semi-classical case this sum is finite, because for RCFT Γ c is a finite-indexsubgroup of Γ and therefore the coset has a finite number of representative elements.When applied to the ( m, m + 1) Virasoro minimal models, the results of [7] are asfollows. For m = 3 , m ≥ m one starts to encounter unphysicalpartition functions (those whose coefficients in the q, ¯ q series are either non-integral whenthe identity state is normalised to unity, or non-positive, or both) and these appear in the– 4 –oincar´e sum .In what follows, we examine the Poincar´e series approach for rational CFT, focusingprimarily on SU( N ) k WZW models but also revisiting the case of Virasoro minimal models.In Section 2 we explicitly evaluate the Poincar´e sum for a number of models, workingsuccessively with SU(2) k , SU( N ) and SU(3) k . For the first two cases we develop anapproach to compute the weights of the resulting average over CFT’s as a function of k, N respectively and identify infinite families of models (and seed primaries) for whichthese weights are non-negative. Thereafter we revisit the minimal models and computethe weights of the average for cases where there are two physical invariants. In Section3 we study the Poincar´e series for multiple genus-1 boundaries, focusing on SU(2) k . Theconclusions are in Section 4, while the Appendices provide some basic data on the relevantCFT’s as well as some additional tables with the results of our calculations. Let us start by defining the Poincar´e sum for an arbitrary RCFT. We start with a setof holomorphic characters χ ( τ ) , χ ( τ ) , · · · , χ P − ( τ ). These form a vector-valued modularform (VVMF) under Γ = SL(2, Z ). Thus, under a modular transformation: γ = a bc d ! ∈ Γ (2.1)we have: τ → τ γ = aτ + bcτ + d (2.2)and: χ i ( τ γ ) = P − X j =0 ( M γ ) ij χ j ( τ ) (2.3)for some matrix M γ . The above relation can be written more briefly as χ ( τ γ ) = M γ χ ( τ ).Each character has the behaviour: χ i ( τ ) = q α i (cid:0) a ( i )0 + a ( i )1 q + a ( i )2 q + · · · (cid:1) (2.4)for some rational exponents α i and non-negative integers a ( i ) n . In a unitary theory, α is thesmallest of the α i and is identified with − c where c is the central charge. The remaining α i are then identified with h i − c where h i are the holomorphic conformal dimensions.Now we pick one of the characters, not necessarily the identity, and consider: Z ii ( τ, ¯ τ ) ≡ | χ i ( τ ) | (2.5) We note in passing that the existence of modular-invariant partition functions with negative integralcoefficients seems to be a non-holomorphic version of the “quasi-characters” extensively studied in [23–25]. It seems that negative integrality is a rather widespread feature in the context of both holomorphicvector-valued modular forms and non-holomorphic modular invariants. – 5 –e think of this as the partition function corresponding to the “spinless seed primary” i of dimension ( h i , h i ). It is not modular-invariant by itself. We can also allow non-diagonal primaries with dimensions ( h i , h j ) for j = i as long as h j − h i is an integer. Thecorresponding seed partition function would be: Z ij ( τ, ¯ τ ) ≡ χ i (¯ τ ) χ j ( τ ) + cc (2.6)Next we take Z seed to be any of the various seed primaries as above, or a linear combinationof them. The relevant Poincar´e series is then: Z ( τ, ¯ τ ) ≡ X γ ∈ Γ c \ Γ Z seed ( τ γ , ¯ τ γ ) (2.7)where Γ c is the subgroup of Γ that leaves Z seed invariant.The result of the sum in Eq. (2.7) is manifestly modular invariant. Thus it must bea linear combination of all allowed modular invariants for the given set of characters. Abasis for these is: Z J ( τ, ¯ τ ) = χ (¯ τ ) I J χ ( τ ) ≡ P − X i,j =0 χ i ( I J ) ij χ j (2.8)were I J is a basis of matrices that lead to modular-invariants Z J and J runs over thenumber of independent matrices of this type. It follows that: Z ( τ, ¯ τ ) = X J c J Z J ( τ, ¯ τ ) (2.9)where c J are some basis-dependent coefficients. Such invariants sometimes have all positivecoefficients in their q, ¯ q expansion. Then they can correspond to the partition function ofa CFT, but only if they satisfy some additional requirements such as having a uniquevacuum state. In this case we refer to them as “physical”. Alternatively they may havesome negative Fourier coefficients, in which case they cannot correspond to a CFT.Thus the possible results of performing a Poincar´e sum are of three types: (I) whenthe index J only takes one value and the invariant corresponds to a physical RCFT, we getthe partition function of that RCFT, (II) when the index J ranges over multiple modularinvariants that all correspond to physical RCFT’s, we get a linear combination of partitionfunctions of these different RCFT’s, (III) when the index J runs over both physical andunphysical invariants, we get a result that cannot be interpreted as an average over RCFT.Within type (II) there are two sub-classes: if all the c J ≥
0, the linear combination can beinterpreted as a weighted average. This case can be called type (IIa). On the other hand ifat least one of the c J is negative then the averaging interpretation is no longer tenable, thisis called (IIb). This classification applies independently for any Z seed , which, as indicatedabove can be associated to a single seed primary or a sum over seed primaries.In view of our discussion above, we will consider types (I) and (IIa) to be physicallyacceptable, with the latter corresponding to an average over RCFT’s with definite weights,while types (IIb) and (III) will be rejected as unphysical . Our primary goal is to explore– 6 –he type (IIa) behaviour in considerable detail in many examples, as well as to find generalformulae for the coefficients c J in families of theories where such behaviour holds.It remains to find the appropriate coset to sum over when performing a Poincar´e sumfor RCFT. As indicated above, it is sufficient to search for a finite-index subgroup of Γ thatleaves the seed partition function Z seed invariant. This may be smaller than the largestpossible such group, that we have earlier called Γ c , but this does not matter since the resultfor the Poincar´e sum will be the same up to an overall factor.We need a few definitions. The conductor N of an RCFT is defined as the LCM of thedenominators of the exponents α i defined in Eq. (2.4) above. Then, it is well-known [13]that the principal congruence subgroup Γ( N ) leaves each character separately invariant : χ i ( τ γ ) = χ i ( τ ) , γ ∈ Γ( N ) (2.10)In general, Γ( N ) is a proper subgroup of the kernel of the modular representation (thelargest subgroup of Γ that preserves the characters).However, this result is too strong for us because we are interested in the subgroup thatpreserves Z ij = χ i χ j (where | h i − h j | is integer) rather than χ i itself. For this, we myconsider the larger subgroup Γ( N ) where N is defined as follows. Picking α to be themost negative exponent (it will eventually be identified with − c ) we define the conformaldimensions associated to our characters by h i = α i − α . Now we define N to be the LCMof the denominators of the h i . We will refer to N as the semi-conductor .All elements of Γ( N ) preserve each character up to an overall phase, which impliesthat Γ( N ) is contained in Γ c . The argument is as follows. The proof of [13] that Γ( N )preserves all characters makes use of the standard action of the generators T, S of SL(2, Z )on the characters. Of these, T acts as a phase: T : χ i → e πiα i χ i (2.11)where α i = − c + h i . Now we can modify the T generator by defining: T ′ = ωT (2.12)where ω = e πi c . Then T ′ and S also form a representation of SL(2, Z ), and we can findthe subgroup that preserves the characters under this action. By the same arguments asbefore, this group will be Γ( N ). It follows that the subgroup that preserves charactersupto a common phase, in the usual action, is Γ( N ). Since the seed partition function isblind to this phase, Γ( N ) is therefore a subset of Γ c . It may be noted that N divides theconductor N , and therefore Γ( N ) is a superset of Γ( N ).For diagonal seeds of the form Z ii = | χ i | one can find a larger invariance group thanΓ( N ). For this one uses the fact that T is not an element of Γ( N ). However it multiplieseach character by a (distinct) phase, so it preserves every seed partition function. T generates a hyperbolic subgroup of Γ, of which the independent elements that are not in Various congruence subgroups, among which we will make use of Γ( N ) and Γ ( N ) in particular, aredefined in Appendix A. – 7 –( N ) are T, T , · · · , T N − . We can now combine T with Γ( N ) and find the resultinggroup, which turns out to be Γ ( N ) (this is defined in Appendix A). This follows fromthe inclusions Γ( N ) ⊂ Γ ( N ) and T n ∈ Γ ( N ), from which we see that the combinationhas at least N times the number of elements of Γ( N ). Because [Γ ( N ) : Γ( N )] = N it also has at most this number of elements, therefore it must be precisely the congruencesubgroup Γ ( N ).In what follows, we will work with both Γ( N ) and Γ ( N ) at different points. Theformer group has the advantage of being normal, so left- and right-cosets are the same.Some of the calculations are also simpler. However the latter is a larger group and thereforehas a smaller index in Γ. As a result the calculation of the Poincar´e sum has fewer termsand this economy is helpful when working with larger values of N .Thus, we finally define the Poincar´e sum for an RCFT as: Z ( τ, ¯ τ ) ≡ N X γ ∈ Γ sub \ Γ Z seed ( τ γ , ¯ τ γ ) (2.13)where Γ sub will be either Γ( N ) or Γ ( N ). In each case this will be a finite sum. Here N is a normalisation factor such that the result has a leading term normalised to unity.Though our notation does not show this explicitly, the normalisation factor will dependon the seed. Note that depending on the choice of Γ sub , the values of the coefficients c I inEq. (2.8) will be scaled by an overall constant.Let us now describe the method to derive the coefficients c J (this was used in [7] forspecific minimal models). Consider a general seed for the Poincar´e series. This can bewritten as: Z seed ( X, τ, ¯ τ ) = χ (¯ τ ) X seed χ ( τ ) (2.14)where X seed is a P × P matrix. Now under a modular transformation we have: Z seed ( X, τ γ , ¯ τ γ ) = ¯ χ (¯ τ γ ) X seed χ ( τ γ ) = ¯ χ (¯ τ ) ( M † γ X seed M γ ) χ ( τ ) (2.15)The result of performing a Poincar´e sum over Z seed will be denoted Z ( X, τ, ¯ τ ) to highlightits dependence on the seed X (this is the modular invariant partition function that in othercontexts we just called Z ( τ, ¯ τ )). Then: Z ( X, τ, ¯ τ ) = X γ Z seed ( X, τ γ , ¯ τ γ ) = ¯ χ (¯ τ ) (cid:16) X γ M † γ X seed M γ (cid:17) χ ( τ ) (2.16)Thus, the modular-invariant matrix P γ M † γ X seed M γ determines what linear combinationsof invariants will appear in the final answer. Referring to Eq. (2.8), we have: X γ M † γ X seed M γ = D X J =1 c J I J (2.17)– 8 –ow, we can define an inner product on the space of matrices as: d JK = Tr( I J I K ) (2.18)Using this in the above equation, we have: X J d KJ c J = X γ Tr( I K M † γ X seed M γ )= X γ Tr( I K X seed )= | Γ sub \ Γ | Tr( I K X seed ) (2.19)where in the intermediate step we used modular invariance of the I J . This matrix equationcan be inverted to give: c J = | Γ sub \ Γ | X K d − JK Tr( I K X seed ) (2.20)These are the desired coefficients which, after normalisation, should be interpreted as theweights with which we average different CFT’s. k WZW Models
We now show that in the unitary SU(2) k WZW models there are infinitely many realisationsof both type (I) and type (IIa) behaviour as described in the previous section. In the lattercase we will be able to compute the coefficients c J for generic models within such familiesand classify the cases where they are non-negative.Before embarking on the calculation let us briefly remark on the generalisation fromVirasoro to current-algebra minimal models. By generic rules of the AdS/CFT correspon-dence, this should mean that the corresponding gravity theory will have gauge fields forthe corresponding Lie algebra, in this case SU(2). So at first one might think the gravitytheory should be Einstein gravity coupled to a Yang-Mills field. However, it has beensuggested in [11] in a related case, that where a current algebra is present in the CFT, thedual theory could be pure Chern-Simons theory. The logic is that in the boundary theorythe energy-momentum tensor is not an independent field but a composite of the currentsvia the Sugawara construction, hence the bulk should have no independent metric field,but only gauge fields that are the counterparts of boundary currents. Since the bulk theoryis topological we do have a general coordinate invariant theory, something like a gravitytheory without gravitons. This in turn is not so strange, given that AdS Einstein gravityanyway has no local graviton degrees of freedom in the bulk.Applying that logic here, the bulk dual for the SU( N ) k WZW model should be twocopies of pure SU( N ) Chern-Simons theory, one representing left-movers and the otherrepresenting right-movers of the CFT. Denoting the gauge fields by matrix valued 1-forms– 9 –s A = A µ dx µ , A = A µ dx µ , the action is then: S bulk = k π (cid:18)Z tr (cid:0) A ∧ d A + A ∧ A ∧ A (cid:1) − Z tr (cid:0) A ∧ d A + A ∧ A ∧ A (cid:1)(cid:19) (2.21)This difference action is well-known to lead to a parity-conserving theory with an SU( N ) k × SU( N ) k current algebra on its boundary, and has found applicability in the context ofmultiple membranes in M-theory as well as more generally in the context of a novel Higgsmechanism in 3d (for a review and references to the original works, see [26]). The noveltyhere is that we are not simply considering the path integral on a fixed 3-manifold, butperforming a Poincar´e sum that is to be thought of as the analogue (for small ℓ AdS ) of asum over 3-manifolds.Now we turn to a discussion of SU (2) k WZW models in the present context. The basicfeatures of the models are summarised in Appendix B and we can perform the Poincar´esums using this data. The main information required is the form taken by the S and T matrices when acting on the characters χ λ (note that we are now labelling the charactersby λ = 2 j + 1, the multiplicity of the isospin j representation, which goes from 1 to k + 1).We will find that the prime factors of the shifted level n = k + 2, also known as the“height”, determine the behaviour of the Poincar´e sum in the corresponding model. Itis easily verified that the semi-conductor, defined in Section 2.1, takes the simple form N = 4 n , except for k = 1 in which case N = 4. We started by calculating the Poincar´esums for the seed corresponding to each individual character for the values k = 1 , , . . . , Z A whichexists for all n , Z D which exists for all even n , and Z E which exists for n = 12 , ,
30, inaddition to unphysical invariants that exist for various values of n . Where our answerinvolves only physical invariants we write it as c A Z A + c D Z D + c E Z E where any one or twoof the coefficients c A , c D , c E may vanish. These coefficients are only defined up to an overallnormalisation in view of the fact that we summed over cosets of Γ by a group Γ( N ) ⊂ Γ c .We make the following observations about the results in this Table: • Within a given model, several distinct seeds give the same linear combination ofinvariants. • The vacuum primary seed always leads to a linear combination involving non-negativeintegers unless there are unphysical invariants at the given level. • For n = 4 ρ, ( ρ ≥ c D c A = 1 for the vacuum seed. • For n = 4 ρ + 2 , ( ρ ≥ c D c A = − for the seed χ . • For certain values of n and for specific seeds, one finds only Z A or only Z D as theanswer. – 10 –e will now explain these observations and generalise them to arbitrary values of n = k + 2. For this we briefly review some details of the results of [21, 22]. Modularinvariants made of the SU(2) k characters at any fixed k can be written as follows: Z δ = 12 n X λ,λ ′ =1 ¯ χ λ (Ω δ ) λλ ′ χ λ ′ (2.22)where δ is a positive integer that divides n . The range of indices on the characters and thematrices Ω δ has been doubled, with a constraint: they are defined modulo 2 n and obeythe relations χ λ = χ λ +2 n = − χ − λ . These relations imply in particular that χ n = 0.The matrices Ω δ are as follows. For a given δ we define α = [ δ, n/δ ] (the GCD of thetwo integers) and ω = 1 mod nα . Then:(Ω δ ) λλ ′ = ( , α ∤ λ or α ∤ λ ′ P α − ξ =0 δ λ ′ , ωλ +2 ξn/α , otherwise (2.23)Not all the Z δ are linearly independent. For example, for δ = n we have α = 1 , ω = 1, so(Ω n ) λλ ′ = δ λλ ′ , while for δ = 1 we have α = 1 , ω = −
1, so (Ω ) λλ ′ = δ λ, − λ ′ . This meansthat Z n = − Z . In general, Ω δ and Ω n/δ will have the same value of α but ω changes signto − ω . Hence the non-zero matrix elements of Ω n/δ can be written as: (cid:0) Ω n/δ (cid:1) λλ ′ = α − X ξ =0 δ λ ′ , − ωλ +2 ξn/α ( α | λ and α | λ ′ ) (2.24)Changing variables by ξ = α − ξ ′ we have that (cid:0) Ω n/δ (cid:1) λλ ′ = α X ξ ′ =1 δ λ ′ , − ωλ − ξ ′ n/α +2 n = α − X ξ ′ =0 δ λ ′ , − ωλ − ξ ′ n/α ( α | λ and α | λ ′ ) (2.25)So, when we compute the partition function, we find: Z n/δ = 12 n X λ =1 α − X ξ ′ =0 ¯ χ λ χ − ωλ − ξ ′ n/α = − n X λ =1 α − X ξ ′ =0 ¯ χ λ χ ωλ +2 ξ ′ n/α = − Z δ (2.26)What we saw above for the case of δ = 1 , n was a special case of this linear relation.To write a formula for the number of invariants, define the divisor function: σ ( n ) = X δ | n n . Now if σ ( n ) is even then eachdivisor δ has a distinct counterpart nδ and in this case there are σ ( n )2 invariants. Howeverif σ ( n ) is odd then δ and nδ must be the same for one value of δ , say δ = m . This meansthat n = m . Using the relation (2.26), we have that Z m = − Z m , so Z m = 0. Therefore,– 11 –or odd values of σ ( n ), we have only ( σ ( n ) − / Z δ = ( σ ( n )2 , σ ( n ) even σ ( n ) − , σ ( n ) odd (2.28)We now consider those heights n that give rise to 1 , One Physical Invariant
First let us ask when there is a unique modular invariant. From (2.28), we know that thishappens when σ ( n ) = 2, which means n = p for some prime p , or σ ( n ) = 3 in which case n = p for some prime p . In these cases the unique modular invariant function is the onecorresponding to Ω n (defined in Eq. (2.23)) and it is the diagonal invariant Z A . When thisis the case, the Poincar´e sums that we compute must necessarily be proportional to Z A ,regardless of the seed. The central charges for these cases are of the form: c = 3 − p or c = 3 − p (2.29)for some prime p . In Table 1 we list the first few examples of models which fall under thiscategory. The models are labelled by the height n = k + 2. Level ( n, c , N ) Poincar´e sum (3 , ,
12) 24 Z A (4 , ,
16) 32 Z A (5 , ,
20) 36 Z A (7 , ,
28) 24 Z A (9 , ,
36) 54 Z A (11 , ,
44) 54 Z A (13 , ,
52) 84 Z A (17 , ,
68) 108 Z A (19 , ,
76) 120 Z A (23 , ,
92) 144 Z A (25 , , Z A (29 , , Z A (31 , , Z A (37 , , Z A – 12 –41 , , Z A (43 , , Z A Table 1 : Poincar´e sums for SU (2) k WZW models with unique modular invariantThus, these Poincar´e sums are candidate partition functions for a gravitational theorydual to the diagonal SU(2) k theory. It is intriguing that, unlike the Virasoro minimalmodels, here we find an infinite family of unitary models that pass the first test to be dualto a gravity theory. Two Physical Invariants
Next we consider the case when there are exactly two modular invariants. From (2.28) wesee that this is the case when σ ( n ) = 4 or 5. For the former case, we have n = pq or n = p where p, q are primes, while for the latter we have n = p . Since the diagonal invariant Z A always exists, this tells us there is exactly one more invariant for all these values of n . However, the second invariant is not necessarily physical. In fact, [21, 22] tells us thatwhenever there are two physical invariants, they must be as follows: Z A = χ Ω n χ, Z D = χ (Ω + Ω n ) χ (2.30)Therefore we must consider only those sub-cases of n = pq, p , p for which 2 divides n (sothat Ω can exist). These consist of one infinite family with n = 2 p and two sporadic caseswith n = 8 ,
16. Clearly these cases have no unphysical invariants. In Table 2 we list severalexamples of models which fall into this category, displaying the result of the Poincar´e sumfor all possible diagonal seeds of the form:( X seed ) λ,λ ′ = δ λλ δ λ ′ λ (2.31)(there can also be non-diagonal seeds corresponding to primaries with spin, these will comeup later). The identity seed corresponds to λ = 1. In the Table, the seed will be labelledby its value of λ . Data ( n, c , N ) Value of λ Poincar´e sum (6 , ,
20) 1 , Z A + Z D )2 , Z A − Z D )3 48 Z D (10 , ,
40) 1 , , , Z A + Z D )2 , , , Z A − Z D )5 96 Z D (14 , ,
56) 1 , , , , ,
13 8 (8 Z A + 5 Z D )– 13 – , , , , ,
12 64 (2 Z A − Z D )7 144 Z D (22 , ,
88) odd , = 11 24 (4 Z A + 3 Z D )even 96 (2 Z A − Z D )11 240 Z D (26 , , , = 13 8 (14 Z A + 11 Z D )even 112 (2 Z A − Z D )13 288 Z D (34 , , , = 17 24 (6 Z A + 5 Z D )even 144 (2 Z A − Z D )17 384 Z D (38 , , , = 19 8 (20 Z A + 17 Z D )even 160 (2 Z A − Z D )19 432 Z D Table 2 : Poincar´e Sums for SU (2) k WZW model with two modular invariantsWe see in Table 2 that the linear combinations appearing for the case when the vacuumis the seed primary have positive coefficients for Z A and Z D . From the perspective ofAdS/CFT, this means that the gravity partition function could plausibly be interpreted asan average of the partition functions of an ensemble of physical CFTs. Here the ensemblehas just 2 elements, a consequence of our choice of symmetry algebra. If we write thePoincare sum as c A Z A + c D Z D , then the relative size of the coefficients c A and c D , providesthe measure for how to average over the two theories.Following the discussion in the previous section, we now derive the coefficients c A , c D ,namely the linear combinations of Z A and Z D that appear in the last column of Table 2for any seed. This gives a successful prediction of all the results in Table 2 and extendsthem to n = 2 p for arbitrary prime p .Referring to Eq. (2.20), we see that the matrix d JK is a 2 × I J . Recall fromthe discussion around Eq. (2.30) that Ω n and Ω exist. Let us now look at the modularinvariant functions that we get from these matrices. We start with Ω n , so α = 1 and ω = 1and the invariant is: Z n = 12 n X λ,λ ′ =1 ¯ χ λ (Ω n ) λλ ′ χ λ ′ = 12 n X λ,λ ′ =1 ¯ χ λ δ λλ ′ χ λ ′ = 12 n X λ =1 | χ λ | = n − X λ =1 | χ λ | (2.32)where we have used χ λ = − χ − λ = χ λ +2 n . Thus we see that Ω n when acting on therestricted space (1 ≤ λ ≤ n −
1) still operates as the identity matrix. Hence ( I n ) λλ ′ = δ λλ ′ ,where λ, λ ′ takes values from 1 to n −
1. Next we consider Ω . Here α = [2 , p ] = 1,– 14 –ince n = 2 p and p >
2. We need to chose ω = 1 mod 8 p . For this, we can chose ω = 2 p − n − ω = 4( p − p + 1 = 1 mod 8 p . The associatedmodular invariant is: Z = 12 n X λ,λ ′ =1 ¯ χ λ (Ω ) λλ ′ χ λ ′ = 12 n X λ,λ ′ =1 ¯ χ λ δ λ ′ , ( n − λ χ λ ′ = 12 n X λ =1 ¯ χ λ χ ( n − λ = 12 n X λ even ¯ χ λ χ ( n − λ + 12 n X λ odd ¯ χ λ χ ( n − λ (2.33)Now, if λ is even then in the first sum we can shift the subscript χ ( n − λ by some multipleof 2 n and bring it to χ − λ = − χ λ . Similarly, for the case of odd λ , we can shift the subscriptto bring it to χ n − λ . Therefore we have: Z = − n X λ even | χ λ | + 12 n X λ odd ¯ χ λ χ n − λ = − n − X λ even | χ λ | + n − X λ odd ¯ χ λ χ n − λ (2.34)So the matrix elements for I can be written:( I ) λλ ′ = ( − δ λλ ′ , λ even δ λ ′ ,n − λ , λ odd (2.35)Note that there are n − = p − n = p odd values for λ . Also, the onlydiagonal term for odd λ occurs at λ = n = p .With this information we are now equipped to compute the matrix d JK of inner prod-ucts defined in Eq. (2.18), where J, K take the two values 2 , n . We have:Tr( I n ) = 2 p − I n I ) = 2 − p Tr( I ) = 2 p − p − − p − p p − ! c n c ! = | Γ sub \ Γ | Tr( I n X seed )Tr( I X seed ) ! (2.37)– 15 –here we have taken P γ M † γ X seed M γ = c n I n + c I . Solving the above equation, we get: c n c ! = | Γ sub \ Γ | p − p − p − p − p − ! Tr( I n X seed )Tr( I X seed ) ! = | Γ sub \ Γ | p − (2 p − I n X seed ) + ( p − I X seed )( p − I n X seed ) + (2 p − I X seed ) ! (2.38)This is the general expression for the result of a Poincar´e sum for an arbitrary seed at level k = n − p − X seed and see how they explain thedata that we found in Table 2. We will use the fact that Z n = Z A and Z = Z D − Z A ,from which it follows that c A = c n − c , c D = c . • ( X seed ) λλ ′ = δ λ δ λ ′ . In this case, Tr( I n X seed ) = 1 and Tr( I X seed ) = 0. This meansthat c n = 2 p − c = p −
2. So the Poincar´e sum is proportional to:(2 p − Z n + ( p − Z = ( p + 1) Z A + ( p − Z D (2.39)The ratio of the two coefficients is c D c A = p − p +1 = n − n +2 = k − k +4 . In the limit p → ∞ , wehave c D c A →
1. One can also verify that all diagonal seeds of the form ( X seed ) λ,λ ′ = δ λ,λ δ λ ′ ,λ , where λ is odd but not equal to p , give the same result. • ( X seed ) λλ ′ = δ λ δ λ ′ . This gives Tr( I n X seed ) = 1 and Tr( I X seed ) = − c n = p + 1 and c = − ( p + 1). So the Poincar´e sum is proportional to:( p + 1) Z n − ( p + 1) Z = 2( p + 1) Z A − ( p + 1) Z D (2.40)One can also verify that all diagonal seeds of the form ( X seed ) λ,λ ′ = δ λ,λ δ λ ′ ,λ , where λ is even, give the same result. • ( X seed ) λλ ′ = δ λp δ λ ′ p . This gives Tr( I n X seed ) = 1 and Tr( I X seed ) = 1 and hence c n = 3( p −
1) and c = 3( p − p − Z n + (3 p − Z = (3 p − Z D (2.41)Thus for models with k + 2 = 2 p , if we start with the seed χ p then the Poincar´e sumgives us only the Z D invariant. • ( X seed ) λλ ′ = δ λ δ λ ′ , p − = ⇒ Tr( I n X seed ) = 0 and Tr( I X seed ) = 1. This means that c n = p + 1 and c = − ( p + 1). So, the Poincar´e sum is P , p − ∝ ( p − Z n + (2 p − Z = − ( p + 1) Z A + (2 p − Z D (2.42)this expression holds true for any odd λ ( = p ) because Tr( I n X seed ) = 0 and Tr( I X seed ) =1 still holds true. – 16 –ne can now verify that the above expressions exactly match the linear combinationsappearing in Table 2. In Table 3 we summarise the above results by listing the values of c A , c D in each category. Seed c A , c D λ = λ ′ = odd = p p + 1 , p − λ = λ ′ = even 2( p + 1) , − ( p + 1) λ = λ ′ = p (0 , p − λ = odd = p, λ ′ = 2 p − λ − ( p + 1) , p − Table 3 : General result, two modular invariants. The seeds are labelled by λ , λ ′ where( X seed ) λλ ′ = δ λλ δ λ ′ λ ′ So far we have worked with individual (and diagonal) seeds. More generally we canconsider linear combinations of different seeds. For future use, we would like to find theseed that will give us an arbitrary general linear combination αZ A + βZ D . Since thereare only two terms in the combination but more than two possible seeds, this can clearlyalways be done. Let us therefore find the most general seed Z seed ( α, β ) that satisfies: X γ Z seed ( α, β ) = αZ A + βZ D (2.43)The general solution (now also including possible non-diagonal seeds) is: Z seed ( α, β ) = a X λ odd , = p b λ | χ λ | + a X λ even c λ | χ λ | + a | χ p | + a p − X λ odd , =1 d λ ( χ λ χ p − λ +c . c . ) (2.44)where: X λ b λ = X λ c λ = X λ d λ = 1 a = (2 p − α + ( p + 1) β p − − ( a + a ) , a = a + 2 a − αp + 1 (2.45)It is easily verified that the Poincar´e sum over (2.44) gives αZ A + βZ D as desired. Also,special values of the coefficients a i , b λ reproduce the single-seed results: for example, choos-ing a = 1 , a = a = a = 0 and b λ = 1 for some odd λ reproduces the answer( p + 1) Z A + ( p − Z D , and similarly for the other cases. Three Physical Invariants
Finally we turn to the cases with three physical invariants, corresponding to σ ( n ) = 6 , k = 10 ,
16. Let us write Z A/D/E = χ Ω A/D/E χ . Then, inthe first case we have the matrices Ω , Ω , Ω from which we get Ω A = Ω , Ω D = Ω + Ω and Ω E = Ω + Ω + Ω , the last one corresponding to the E invariant. In the second– 17 –ase the matrices are Ω , Ω , Ω from which one gets Ω A = Ω , Ω D = Ω + Ω andΩ E = Ω + Ω + Ω , of which the last one corresponds to the E invariant. For k = 28 weinstead get four matrices Ω , Ω , Ω , Ω but the physical partition functions comes fromjust three matrices Ω A = Ω , Ω D = Ω + Ω and Ω E = Ω + Ω + Ω + Ω with the lastone being associated to the E invariant. However, since there are four invariants in allbut only three physical invariants, there is one independent unphysical linear combinationthat can appear in the Poincar´e sum. In Table 7 we see that this unphysical invariant doesmake an appearance in the Poincar´e sum, so this case cannot be interpreted as an averageover three RCFT’s. N ) k WZW Models
We now move on to discuss the case of SU( N ) k for N >
2. We will encounter bothsimilarities and differences with respect to the SU(2) k case analysed above. One of the keydifferences is that all representations of SU(2) are real or pseudo-real but for SU( N ), N > N ) k , using the general proceduredescribed earlier. The idea, as before, is to identify models that have only physical modularinvariants, then express the Poincar´e sums in terms of these invariants and attempt tointerpret the linear combination as an average over CFT’s.While modular invariants for SU(2) k were completely classified in [21, 22], the caseof SU(3) k is considerably more complicated and was carried out later in [28, 29]. Theclassification of modular invariants for the SU( N ) series was conjectured in [30] and provedin [31]. Some results for the general case of SU( N ) k can been found in [32]. In what follows,we will focus on the families SU( N ) and SU(3) k , for which the general classification iswell-understood and reasonably tractable. N ) The case of SU( N ) shares several features with SU(2) k WZW models. In particular,we will see below that the modular invariants can be written down in a similar way as forSU(2) k . In SU( N ) there are N allowed representations including the identity. Using (B.7),we see that the allowed Dykin labels are given by λ i = ρ i = 1 or λ i = ρ i + δ b,i = 1 + δ b,i for some b between 1 to N −
1. So, we can denote these as λ a where a ∈ { , , . . . , N − } and ( λ a ) i = 1 + δ a,i . The characters can now be labelled by the index a .The central charge and the conformal dimensions are: c = N − a = a ( N − a )2 N (2.47)For a > a and N − a are complex conjugates of eachother, due to which the characters labelled by a and N − a are identical. The fact that h a = h N − a is consistent with this identification. This leads to the multiplicity 2 in thepartition function that was referred to above.Following [30, 31], we now list the modular invariants that appear in these theories.We define the integer m by: m = ( N , if N is odd N , if N is even (2.48)This integer plays a somewhat similar (but not identical) role to the height n = k + 2for SU(2) k theories. Now for every divisor δ of m , define α ≡ [ δ, m/δ ] (the GCD of theintegers) and let ω ( δ ) = (cid:0) ρ mαδ + σ δα (cid:1) mod Nα , where we have chosen integers ρ, σ suchthat ρ mαδ − σ δα = 1. This is always possible because mαδ and δα are coprime. We note that ω ( m/δ ) = − ω ( δ ) mod Nα . Finally, we define the matrices Ω δ :(Ω δ ) aa ′ = ( , α ∤ a or α ∤ a ′ P α − ξ =0 δ a ′ , ω ( δ ) a + ξN/α , otherwise (2.49)The modular invariant partition functions are then: Z δ = N − X a =0 ¯ χ a (Ω δ ) aa ′ χ a ′ (2.50)Here the indices a, a ′ in the matrix elements and characters will always be understood asintegers modulo N . Not all the Z δ above are independent, in fact Z δ and Z m/δ are equal.This can be seen from the fact that:(Ω m/δ ) aa ′ = α − X ξ =0 δ a ′ , − ω ( δ ) a + ξN/α = α − X ξ ′ =0 δ a ′ , N − ω ( δ ) a − ξ ′ N/α ( α | a and α | a ′ ) (2.51)and the corresponding partition function is: Z m/δ = N − X a =0 α | a α − X ξ ′ =0 ¯ χ a χ N − ω ( δ ) a − ξ ′ N/α = N − X a =0 α | a α − X ξ ′ =0 ¯ χ a χ ω ( δ ) a + ξ ′ N/α = Z δ (2.52)In the above, we used ξ ′ = α − ξ and χ b = χ N − b . Thus we have:Number of linearly independent Z δ = ( σ ( m )2 , σ ( m ) is even σ ( m )+12 , σ ( m ) is odd (2.53)Remarkably, for SU( N ) all the Z δ are not just modular invariant but also physical – they– 19 –atisfy positive integrality of coefficients and non-degeneracy of the vacuum state. This is amajor difference from SU(2) k . Another difference is that while the latter can have at mostthree physical invariants namely Z A , Z D , Z E , there is no such restriction for SU( N ) whereone can have an arbitrarily large number of physical modular invariants for sufficientlylarge and suitably chosen N .We now perform a similar classification as before to find conditions for one invariant,two invariants and so on. We also derive the linear combinations of the invariants thatwill appear by inverting the matrix of inner products among the modular invariant basismatrices. To do this, we will use all the available Ω δ as the basis for a given model andthen compute the cofficients c δ corresponding to the modular invariant Z δ . At the end,we will impose the constraint that Z m/δ = Z δ which equates the label for a representationand its complex conjugate. One Physical Invariant
These models correspond to what we have called type (I). As can be seen from (2.53), wehave a unique invariant whenever σ ( m ) = 1 or 2. This is true when m = 1 or p , where p is a prime. So altogether we have N = 2 , p, p where p is prime. In Table 4 we list someexamples of models of this type.SU( N ) ( N, c, N ) Seed Primary ( χ λ ) Poincar´e sum (2 , ,
4) 0 , Z p (3 , ,
6) 0 6 Z p , Z p (4 , ,
8) 0 , Z p , Z p (5 , ,
10) 0 12 Z p , , , Z p (6 , ,
12) 0 , Z p , , , Z p (7 , ,
14) 0 18 Z p , , , , , Z p (10 , ,
20) 0 , Z p , , , , , , , Z p (19 , ,
38) 0 54 Z p , , · · · ,
18 27 Z p Table 4 : Poincar´e sums for SU( N ) WZW models with a unique modular invariant.Just like the SU(2) k examples with unique modular invariants listed in Table 1, the– 20 –ases in Table 4 provide candidate gravity partition functions dual to a unique SU( N ) WZW model.
Two physical invariants
Now, we consider the case where there are exactly two modular invariants. From (2.53),we see that here σ ( m ) = 3 or 4 which is true when m = p , p , pq , where p, q are primesand m >
5. So the relevant values of N are N = p , p , pq, p , p or 2 pq .All these values of N describe models which fall into the type (IIa) category. UnlikeSU(2) k , we do not need to restrict further to sub-families as all of them are physical. InTable 5 we list examples of models having m = p , i.e. N = p or 2 p , along with theresult of the Poincar´e sum for each seed primary. In these models the possible valuesof the integers δ that divide m are 1 , p, p and correspondingly there are three modularinvariants that we label Z , Z p , Z p . We have Z = Z p resulting from complex conjugation,as described above, so the results can only be linear combinations of Z p , Z p . SU ( N ) ( N, c, N ) Seed Primary ( χ λ ) Poincar´e sum (8 , ,
16) 0 , Z p + Z p )0 = mod 2 8(2 Z p − Z p )0 = mod 2 , = 0 , Z p (9 , ,
18) 0 6(3 Z p + Z p )0 = mod 3 4 . Z p − Z p )0 = mod 3 , = 0 1 . Z p + 7 Z p )(18 , ,
36) 0 , Z p + Z p )0 = mod 3 9(3 Z p − Z p )0 = mod 3 , = 0 , Z p + 7 Z p )(25 , ,
50) 0 12(5 Z p + 2 Z p )0 = mod 5 7 . Z p − Z p )0 = mod 5 , = 0 4 . Z p + 7 Z p )(49 , ,
98) 0 18(7 Z p + 3 Z p )0 = mod 7 10 . Z p − Z p )0 = mod 7 , = 0 1 . Z p + 43 Z p )(50 , , ,
25 24(5 Z p + 2 Z p )0 = mod 5 15(5 Z p − Z p )0 = mod 5 , = 0 ,
25 9(5 Z p + 7 Z p ) Table 5 : Poincar´e sums for SU ( N ) with two invariants, for N = p , p These Poincar´e sums provide candidate gravity partition functions where the dual– 21 –onsists of an average over the two CFTs. The precise linear combinations of the physicalCFT partition function that we obtain can be derived using methods similar to those for SU (2) k . We illustrate this for the same sub-families N = p and N = 2 p , that wereexplicitly studied above. One can easily generalise this derivation to any of the othertwo-invariant families. Two invariants: Coefficients for N = p Here, m = N = p and from the discussion of the available modular invariant partitionfunctions around (2.50), the relevant matrices are Ω p , Ω , and Ω p . Their matrix elementsare given by:(Ω p ) aa ′ = δ a ′ a , (Ω ) aa ′ = δ a ′ , − a , (Ω p ) aa ′ = ( , p ∤ a or p ∤ a ′ P p − ξ =0 δ a ′ ,ξp , otherwise (2.54)For the case of Ω p , we have used the fact that α ≡ [ p, p ] = p and so N/α = 1. Hence, ω ( p ) = 0. The modular invariant functions that we get from these matrices are: Z p = N − X a,a ′ =0 ¯ χ a (cid:0) Ω p (cid:1) aa ′ χ a ′ = N − X a,a ′ =0 ¯ χ a δ a ′ a χ a ′ = N − X a =0 | χ a | Z = N − X a,a ′ =0 ¯ χ a (Ω ) aa ′ χ a ′ = N − X a,a ′ =0 ¯ χ a δ a ′ , − a χ a ′ = N − X a =0 ¯ χ a χ − a Z p = N − X a,a ′ =0 ¯ χ a (Ω p ) aa ′ χ a ′ = N − X a,a ′ =0 p | a , p | a ′ p − X ξ =0 ¯ χ a δ a ′ ,ξp χ a ′ = N − X a =0 p | a ¯ χ a p − X ξ =0 χ ξp = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − X a =0 , p | a χ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.55)From the fact that χ − a = χ N − a = χ a , we see that Z = Z p . However, as explained above,we treat them as independent for now and identify them at the end.We see that the theories corresponding to Z p are holomorphically factorised, or mero-morphic, CFT’s. This can be understood as follows. The central charge of SU ( N ) with N = p is p −
1. It is well-known that for any prime p , this number is divisible by 24. Now c being a multiple of 24 is a necessary condition to encounter meromorphic (one-character)CFT’s, and these are what the Z p are giving us. As an example, the Z p invariant for SU (25) is entry No. 67 in the table of [33], in which all meromorphic CFT with c = 24are classified.We can now compute the matrix d JK of inner products. Its elements are:Tr(Ω p ) = p , Tr(Ω ) = p , Tr(Ω p ) = p (2.56)Tr(Ω p Ω ) =Tr(Ω ) = 1 , Tr(Ω p Ω p ) = Tr(Ω p ) = p , Tr(Ω Ω p ) = p (2.57)– 22 –he linear combinations that appear in the Poincar´e sums can now be computed: c p c c p = | Γ sub \ Γ | p p p pp p p − Tr(Ω p X seed )Tr(Ω X seed )Tr(Ω p X seed ) = | Γ sub \ Γ | p ( p − p − p p − p − p − p p Tr(Ω p X seed )Tr(Ω X seed )Tr(Ω p X seed ) = | Γ sub \ Γ | p ( p − p Tr(Ω p X seed ) − p Tr(Ω p X seed ) p Tr(Ω X seed ) − p Tr(Ω p X seed ) − p Tr(Ω p X seed ) − p Tr(Ω X seed ) + (1 + p )Tr(Ω p X seed ) (2.58)Let us now consider different choices of seeds: • ( X seed ) a,a ′ = δ a, δ a ′ , . We have Tr(Ω p X seed ) = Tr(Ω X seed ) = Tr(Ω p X seed ) = 1 sothe Poincar´e sum is proportional to: p ( p − Z p + Z ) + ( p − Z p = 2( p − (cid:18) p Z p + ( p − Z p (cid:19) (2.59)To get the second equality we have used Z p = Z . • ( X seed ) a,a ′ = δ a, δ a ′ , . We have Tr(Ω p X seed ) = 1 , Tr(Ω X seed ) = 0 , Tr(Ω p X seed ) = 0and so the Poincar´e sum is proportional to: p Z p − pZ p = p ( p Z p − Z p ) (2.60)This is true for any seed primary | χ a | where p ∤ a . • ( X seed ) a,a ′ = δ a,p δ a ′ ,p . We have Tr(Ω p X seed ) = 1 , Tr(Ω X seed ) = 0 , Tr(Ω p X seed ) = 1so in this case the Poincar´e sum is proportional to:( p − p ) Z p − p Z + ( p − p + 1) Z p = p ( p − Z p + ( p − p + 1) Z p (2.61)This is true for any seed primary | χ a | where p | a unless a = 0.These results can now be compared with the explicit examples in Table 5 that correspondto N = p and we find perfect agreement.From this example we learn the general lesson that Poincar´e sums will mix the partitionfunction of a meromorphic CFT (as long as it has a Kac-Moody algebra) with the diagonalinvariant of the same Kac-Moody algebra. This potentially leads to many more cases thatcan be investigated, a point to which he hope to return in the future.– 23 – wo invariants: Coefficients for N = 2 p Here, m = N = p , so again the relevant matrices are Ω p , Ω , and Ω p . The matrixelements are:(Ω p ) aa ′ = δ a ′ a , (Ω ) aa ′ = δ a ′ , − a , (Ω p ) aa ′ = ( , p ∤ a or p ∤ a ′ P p − ξ =0 δ a ′ ,a +2 ξp , otherwise (2.62)For the case of Ω p , we have used the fact that α = [ p, p ] = p , N/α = 2 and so we take ω ( p ) = 1. The modular invariant functions that we get from these matrices are Z p = N − X a,a ′ =0 ¯ χ a (cid:0) Ω p (cid:1) aa ′ χ a ′ = N − X a,a ′ =0 ¯ χ a δ a ′ a χ a ′ = N − X a =0 | χ a | (2.63) Z = N − X a,a ′ =0 ¯ χ a (Ω ) aa ′ χ a ′ = N − X a,a ′ =0 ¯ χ a δ a ′ , − a χ a ′ = N − X a =0 ¯ χ a χ − a (2.64) Z p = N − X a,a ′ =0 ¯ χ a (Ω p ) aa ′ χ a ′ = N − X a,a ′ =0 p | a , p | a ′ p − X ξ =0 ¯ χ a δ a ′ ,a +2 ξp χ a ′ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − X l =0 χ lp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − X l =0 χ (2 l +1) p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.65)Again Z = Z p , but Z p is no longer a meromorphic CFT – rather, it is a two-characterCFT.The elements of the matrix of inner products d JK are:Tr(Ω p ) = 2 p , Tr(Ω ) = 2 p , Tr(Ω p ) = 2 p (2.66)Tr(Ω p Ω ) =Tr(Ω ) = 2 , Tr(Ω p Ω p ) = Tr(Ω p ) = 2 p , Tr(Ω Ω p ) = 2 p (2.67)and the linear combinations in the Poincar´e sums are: c p c c p = | Γ sub \ Γ | p p p p p p p − Tr(Ω p X seed )Tr(Ω X seed )Tr(Ω p X seed ) = | Γ sub \ Γ | p ( p − p − p p − p − p − p p Tr(Ω p X seed )Tr(Ω X seed )Tr(Ω p X seed ) = | Γ sub \ Γ | p ( p − p Tr(Ω p X seed ) − p Tr(Ω p X seed ) p Tr(Ω X seed ) − p Tr(Ω p X seed ) − p Tr(Ω p X seed ) − p Tr(Ω X seed ) + (1 + p )Tr(Ω p X seed ) (2.68)For the various different seeds, this gives: – 24 – ( X seed ) aa ′ = δ a δ a ′ . We have Tr(Ω p X seed ) = Tr(Ω X seed ) = Tr(Ω p X seed ) = 1 andthe Poincar´e sum is proportional to: p ( p − Z p + Z ) + ( p − Z p = 2( p − (cid:18) p Z p + ( p − Z p (cid:19) (2.69)where we again used Z p = Z . This will also be true of the seed (cid:12)(cid:12) χ p (cid:12)(cid:12) . • ( X seed ) aa ′ = δ a δ a ′ . We have Tr(Ω p X seed ) = 1 , Tr(Ω X seed ) = 0 , Tr(Ω p X seed ) = 0and the Poincar´e sum is proportional to: p Z p − pZ p = p ( p Z p − Z p ) (2.70)This is true for any seed primary | χ a | where p ∤ a . • ( X seed ) aa ′ = δ ap δ a ′ p . We have Tr(Ω p X seed ) = 1 , Tr(Ω X seed ) = 0 , Tr(Ω p X seed ) = 1and the Poincar´e sum is proportional to( p − p ) Z p − p Z + ( p − p + 1) Z p = p ( p − Z p + ( p − p + 1) Z p (2.71)This is true for any seed primary | χ a | where p | a unless a = 0 , p .These predictions agree with the results of explicit computations, of which a few cases aredisplayed in Table 5. k Next we turn to the case of SU(3) k . These WZW models are considerably more complicatedthan those for SU(2) k and SU( N ) . The classification of modular invariants in this case hasbeen carried out by Gannon [28, 29] but not quite with the same methods as those employedin the classifications for SU(2) k and SU( N ) . In particular we do not have the analogueof the matrices Ω δ discussed above in Eqs.(2.23), (2.49). This makes it more difficult tomake general predictions about the result of the Poincar´e sums, though evaluating thesesums for any specific case is straightforward.The classification of [28, 29] provides a list of physical invariants that appear at differentlevels. It is found that two invariants called Z A , Z D exist for all heights n = k + 3 (with k > SU (2) case where Z D existed only for even n . There are alsofour exceptional invariants denoted Z E which appear at n = 5 , ,
21 (with two distinctinvariants appearing at n = 9).We calculated the Poincar´e sums for the seed corresponding to each individual char-acter for the values k = 1 , , . . . ,
16 using SageMath. As in the SU (2) k case, we found thatthe Poincar´e sums are generally linear combinations of multiple modular invariants. More-over these combinations often contain unphysical invariants that we have denoted Z new .The results up to k = 8 are given in Table 6. Level ( n, c, N ) Seed primary ( λ λ ) Poincar´e sum – 25 –4 , ,
12) (11) 24 Z A (12) 12 Z A (5 , ,
15) (11) , (22) 24 Z A (12) , (13) 12 Z A (6 , ,
18) (11) 18 Z A (12) , (13) , (23) 4 . Z A − Z D )(14) 4 . Z A + Z D )(22) 18 Z D (7 , ,
21) (11) , (22) , (33) 12( Z A + Z D )(12) , (24) 3(9 Z A − Z D )(13) , (14) , (15) , (23) 6( Z A + Z D )(8 , ,
24) (11) , (33) 4(2 Z A + 2 Z D + Z E )(12) , (15) 4(4 Z A − Z D − Z E )(13) , (34) 6(2 Z A − Z D + Z E )(14) 12 Z E (16) , (23) 2(2 Z A + 2 Z D + Z E )(22) 16( Z A + Z D − Z E )(24) , (25) 8( Z A + Z D − Z E )(9 , ,
27) (11) , (22) , (44) 3(6 Z A + Z D )(12) , (13) , (15) , (16) , (23) , . Z A − Z D )(24) , (26) , (34) , (35)(14) , (17) , (25) 1 . Z A + 5 Z D )(33) 27 Z D (10 , ,
30) (11) , (33) 4(3 Z A + 2 Z D + 2 Z new )(12) , (13) , (16) , (27) 0 . Z A − Z D − Z new )(14) , (25) 3(3 Z A + Z D + 4 Z new )(15) , (23) , (35) , (45) 0 . Z A − Z D + 6 Z new )(17) , (36) 3 Z A + 5 Z D − Z new (18) , (34) 2(3 Z A + 2 Z D + 2 Z new )(22) , (44) 12( Z D − Z new )(24) , (26) 6( Z D − Z new )(11 , ,
33) (11) , (22) , (33) , (44) , (55) 12( Z A + Z D )(12) , (13) , (16) , (18) , (23) , (24) , Z A − Z D )(26) , (37) , (45) , (46)(14) , (15) , (17) , (19) , (25) , (27) , . Z A + 5 Z D )– 26 –28) , (34) , (35) , (36) Table 6 : Poincar´e sums for SU (3) k .Here at height n = 10, Z new is an unphysical invariant given by: Z new = −| χ | − | χ | + | χ | − | χ | − | χ | − | χ | − | χ | + | χ | − | χ | − | χ | − | χ | − | χ | {− χ χ + χ χ − χ χ + χ χ − χ χ + χ χ + χ χ + χ χ + c . c . } (2.72)Based on the table we make the following observations: • Within a given model, several distinct seeds give the same linear combination ofmodular invariants. • The vacuum seed always gives a linear combination involving non-negative weights. • Unlike the SU (2) case, we have observed that the linear combination for all n > • However whenever the height n is a prime p >
5, the linear combination from thevacuum seed is proportional to Z A + Z D and moreover does not contain any unphysicalinvariants.It would be interesting to prove the above statements in general. We will not do sohere, instead we will derive the coefficients for the specific case of k = 4 and leave a moredetailed analysis for future work. From the table we see that at k = 4, we only find the Z A and Z D invariant. These invariants are given as: Z A = X λ | χ λ | (2.73) Z D = X λ ¯ χ A kt ( λ ) λ χ λ (2.74)where t ( λ ) = λ − λ and A λ = ( n − λ − λ , λ ). The operator A is of order three: A = 1.We also have the corresponding conjugate versions of these given by: Z CA = X λ ¯ χ λ χ Cλ (2.75) Z CD = X λ ¯ χ A kt ( λ ) λ χ Cλ (2.76)where Cλ is the complex conjugate representation to λ . This permits us to work out thecorresponding Ω matrices, which turn out to be:(Ω A ) λ,λ ′ = δ λ,λ ′ , (Ω CA ) λ,λ ′ = δ Cλ,λ ′ , (Ω D ) λ,λ ′ = δ λ,A kt ( λ ′ ) λ ′ , (Ω CD ) λ,λ ′ = δ Cλ,A kt ( λ ′ ) λ ′ (2.77)– 27 –he relevant traces of products of these matrices are then:Tr(Ω A ) = Tr(Ω CA ) = Tr(Ω D ) = Tr(Ω CD ) = 15Tr(Ω A Ω CA ) = 3 , Tr(Ω A Ω D ) = 5 , Tr(Ω A Ω CD ) = 9 , Tr(Ω CA Ω D ) = 9 , Tr(Ω CA Ω CD ) = 5 , Tr(Ω D Ω CD ) = 3 (2.78)Following the usual strategy, we now invert the relevant 4 matrix to get the linear combi-nations: c A c CA c D c CD = | Γ sub \ Γ | a + 3 b − c − d a + 15 b − c − d − a − b + 15 c + 3 d − a − b + 3 c + 15 d (2.79)where for compactness we have introduced a = Tr(Ω A X seed ) , b = Tr(Ω CA X seed ) , c =Tr(Ω D X seed ) , d = Tr(Ω CD X seed ).Let us now consider different choices of seeds: • ( X seed ) λλ ′ = δ λ, (1 , δ λ ′ , (1 , . We have then a = b = c = d = 1 and so the Poincar´esum is proportional to 4( Z A + Z CA + Z D + Z CD ) = 8( Z A + Z D ) (2.80)where we got the second form after using Z i = Z Ci . • ( X seed ) λλ ′ = δ λ, (1 , δ λ ′ , (1 , . We have a = 1 , b = c = d = 0 and so the Poincar´e sum isproportional to: 15 Z A + 3 Z CA − Z D − Z CD = 2(9 Z A − Z D ) (2.81) • ( X seed ) λλ ′ = δ λ, (1 , δ λ ′ , (1 , . We have a = d = 1 , b = c = 0 and so the Poincar´e sum isproportional to: 6 Z A − Z CA − Z D + 6 Z CD = 4( Z A + Z D ) (2.82)These results match precisely with the output of the SageMath calculations presented inthe table for this case. Poincar´e sums for minimal models were first discussed in [7]. In this case too, all thepossible modular invariants are classified [21, 22]. Indeed the results are very similar tothose for SU(2) k that we have already encountered.The characters for minimal models are labelled χ r,s , the range of these being definedin Appendix B. The identity character is χ , . Modular invariants for the ( m, m ′ ) minimal– 28 –odel, denoted Z δδ ′ , are defined in terms of a matrix Ω δδ ′ by: Z δδ ′ = 18 m ′ X r ,r =1 2 m X s ,s =1 ¯ χ r ,s [Ω δδ ′ ] ( r ,s ) , ( r ,s ) χ r ,s (2.83)where δ | m and δ ′ | m ′ . The parameters m and m ′ act as the analogues of the height n for SU(2) k , while the matrices Ω δδ ′ are simply tensor products of two of the Ω δ that weencountered in the SU (2) k case:[Ω δδ ′ ] ( r ,s ) , ( r ,s ) = (Ω δ ) s ,s (Ω δ ′ ) r ,r (2.84)The characters are given as: χ r,s ( τ ) = 1 η ( τ ) ∞ X t = −∞ (cid:18) q (2 tmm ′ + rm − sm ′ )24 mm ′ − q (2 tmm ′ + rm + sm ′ )24 mm ′ (cid:19) (2.85)from which it follows that χ r,s = − χ − r,s = χ r +2 m ′ ,s and similarly for the shifts of s . Withthis, many of the results for SU(2) k can be adapted. One physical invariant
This is only possible when σ ( m ) and σ ( m ′ ) are each individually equal to 2 or 3. Thiscondition is satisfied only if m and m ′ are either a prime or the square of a prime. Thisleads to four possibilities: ( m, m ′ ) = ( p, q ) , ( p , q ) , ( p, q ) , ( p , q ), where p, q are distinctprimes. Since the diagonal invariant always exists and is physical, it follows that wheneverthere is a unique modular invariant, it is physical. The corresponding invariant is labelled Z AA . However only a couple of unitary minimal models belong in the families listed above,namely ( m, m + 1) = (3 ,
4) and (4 , Two physical invariants
There will be exactly two modular invariants when one of m or m ′ is of the form p or p ,while the other one is qr, q , q , where p, q, r are primes. However this does not ensure thatboth are physical. From our previous discussion we know that physical invariants only occurif Ω contributes, which in turn requires m ′ ∈ { q, , } . Meanwile m = p, p as before.Thus the models that contain precisely two physical invariants are those with ( m, m ′ ) =( p, q ) , ( p , q ) where p, q >
2, along with ( m, m ′ ) = ( p, , ( p , , ( p, , ( p , Z AA , Z AD . Due to the symmetry between m and m ′ ,we identify models where the values of these integers are interchanged. As already notedin Appendix of [7], there are some unitary models in this set – namely those with m = 2 p where p is a prime such that 2 p + 1 is also prime – these are known as Sophie Germainprimes.In Table 8 one finds that for the unitary minimal models with ( m, m + 1) ranging from(5 ,
6) to (10 , Z AA and Z AD appear. From the discussionabove there are infinitely many more such examples, both unitary and non-unitary.– 29 –et us see how to derive the linear combination of the two invariants that appearswhen we perform the Poincar´e sum for the minimal models of type ( m, m ′ ) = ( m, q ) where m = p or p . For this we consider the matrices Ω mm ′ and Ω m acting on the characters χ r,s where 1 ≤ r ≤ m ′ − ≤ s ≤ m −
1. Recall that in the formula Eq. (2.83) the rangeof these indices is temporarily doubled, with a constraint that ultimately reduces them totheir physical range. For, the case of Ω mm ′ , we have: Z mm ′ = 18 m ′ X r ,r =1 2 m X s ,s =1 ¯ χ r ,s [Ω mm ′ ] ( r ,s ) , ( r ,s ) χ r ,s = 18 m ′ X r ,r =1 2 m X s ,s =1 ¯ χ r ,s δ s ,s δ r ,r χ r ,s = 12 m ′ − X r =1 m − X s =1 | χ r ,s | (2.86)Thus when acting on χ r,s , the matrix ( I mm ′ ) ( r ,s ) , ( r ,s ) = δ r ,r δ s ,s . Next, we look at thecase of Ω m : Z m = 18 m ′ X r ,r =1 2 m X s ,s =1 ¯ χ r ,s [Ω m ] ( r ,s ) , ( r ,s ) χ r ,s = 14 m − X s =1 2 m ′ X r ¯ χ r ,s χ ( m ′ − r ,s = 14 m − X s =1 − m ′ X r even=1 | χ r ,s | + m ′ X r odd=1 ¯ χ r ,s χ m ′ − r ,s (2.87)So the matrix I m is:[ I m ] ( r ,s ) , ( r ,s ) = δ s ,s × ( − δ r ,r , r even δ r ,m ′ − r , r odd (2.88)Now we compute the inner products for these two matrices to get, for m ′ = 2 q :Tr( I mm ′ ) = ( m − q − I mm ′ I m ) = ( m − − q )Tr( I m ) = ( m − q −
1) (2.89)We see that all the inner products are the same as in (2.36) upto an overall factor of( m −
1) and so we see that the coefficients will be given by solving the same equation asin (2.37). Due to the relation ( r, s ) ∼ ( m ′ − r, m − r ), at first it might seem that there issome ambiguity in choosing the X seed . However this is not the case. If we want the seedto be | χ r ,s | , it is easily verified that the two choices:( X seed ) ( r,s ) , ( r ′ ,s ′ ) = δ r,r δ r ′ ,r δ s,s δ s ′ ,s ( ˜ X seed ) ( r,s ) , ( r ′ ,s ′ ) = δ r,m ′ − r δ r ′ ,m ′ − r δ s,m − s δ s ′ ,m − s (2.90)– 30 –ive the same result.The result of this computation is that the linear combinations of physical invariantsappearing in the Poincar´e sums for ( m, q ) minimal models (where m = p or p ) are exactlythe same as those that were obtained in the computations of Poincar´e sums for the case of SU (2) k where k + 2 = n = 2 q . This can be verified for the examples in Table 8. Three physical invariants
Minimal models can have at most three physical invariants. This happens when the con-ditions are satisfied to admit an exceptional invariant, and requires m to be 12,18 or 30and m ′ is any odd number (or the equivalent with m ↔ m ′ ). However almost all suchcases have additional unphysical invariants. To have precisely three physical and no un-physical invariants, the total number of invariants should also be 3. This will happen when m = 12 ,
18 and m ′ = p, p for prime p . While there is an exceptional invariant at m = 30,it is easy to verify that the total number of invariants in that case is at least 4.Applying this to the unitary models, we find that the cases (11 ,
12) and (12 ,
13) haveprecisely three physical invariants. At (13 ,
14) we again have only two invariants. Startingwith (14 ,
15) we start to encounter three invariants but only for (17 , , (18 ,
19) are thereno more. So for m ≥ , m = 17 ,
18, and other than the cases discussed in the previoussubsection, there is at least one unphysical invariant. This agrees with the results foundin [7] and provides a more complete picture.
In this Section we consider RCFT’s on 3-manifolds with multiple genus-1 boundaries. Thisis crucial for the interpretation that a bulk gravity is dual to an average over CFT’s.Suppose we have N B > τ A , A = 1 , , · · · , N B . A generic seed would be made up of linear combinations of termslike: Z i j ( τ ) Z i j ( τ ) · · · Z i NB j NB ( τ N B ) (3.1)with each factor defined as in Eqs. (2.5) or (2.6). Thereafter the Poincar´e sum is carriedout over independent modular transforms of each τ A . Since we allow linear combinationsof seeds as above, the final Poincar´e sum is not a simple product of Poincar´e sums overthe contribution from each boundary. Now, once we have determined the measure foraveraging over CFT’s in the single-boundary case, the same measure should apply in themulti-boundary case. This requires that: Z ( τ , τ , · · · , τ N B ) = X J c J Z J ( τ ) Z J ( τ ) · · · Z J ( τ N B ) (3.2) Alternatively one could identify the modular parameters of all the boundaries and carry out a singlePoincar´e sum over them. We do not consider this case here. – 31 –or every N B , where the LHS is the result of the multiple Poincar´e sum described aboveand the RHS has the same coefficients c J that appeared in Eq. (2.8). As emphasised in adifferent class of examples [11], it is not at all obvious that this can hold for all N B > ′ of genus g and g ′ ,each having its own moduli. If the dual of the bulk gravity theory is a single CFT, thenwe expect that the one- and two-boundary partition functions should be: Z grav (Σ) = Z CFT (Σ) , Z grav (Σ , Σ ′ ) = Z CFT (Σ) Z CFT (Σ ′ ) (3.3)If the dual of the bulk theory is not a single CFT but rather an ensemble with a probabilitydistribution over physical CFTs, then the gravity partition function will be an average overdifferent CFT partition functions. For one boundary we have seen that: Z grav (Σ) = h Z CFT (Σ) i = X J c J Z J (Σ) , X J c J = 1 (3.4)where the sum is over physical CFTs. The J ’th CFT occurs with probability c J . Oncewe have the probability distribution { c J } , the two-boundary partition function should be: Z grav (Σ , Σ ′ ) = h Z CFT (Σ , Σ ′ ) i = X i c J Z J (Σ) Z J (Σ ′ ) (3.5)with the same probabilities c J as in (3.4). The above equation (3.5) has an obvious gener-alisation to an arbitrary number of boundaries, once we know the c J and Z J (Σ). To showthat indeed the dual of bulk gravity theory is an ensemble of CFTs one should be able tocompute the LHS and RHS of (3.5) independently and show that they match.Computation of the LHS is complicated for the following reason: one class of multi-boundary manifolds is simply the sum of disconnected single-boundary manifolds. Forthis choice the Poincar´e sum is simply the product of independent Poincar´e sums over thedisconnected components, leading to the result ( P J c J Z J (Σ)) . This is of course not thedesired answer. Extra contributions will come from manifolds that connect two or moreboundaries in all possible ways. These can be thought of as wormholes. If we knew theanalogue of a “vacuum” wormhole (corresponding to some particular seed for the Poincar´esum) we would know how to compute these contributions and could verify Eq. (3.5)and its N -boundary generalisations.Lacking this information, we will take a different approach. We will assume thatthe contribution of the fully disconnected 3-manifold, each with one genus-1 boundary,is described by a product of independent Poincar´e sums that start with the identity seed,following results in the previous sections. Now we seek a rule for the wormhole contributioncorresponding to manifolds with all numbers of boundaries >
1. We then combine these inall possible ways to generate the full N B -boundary result. Then we try to find a seed such Here the c J are normalised such that their sum is unity, though we have not changed the notation toreflect this. It should be evident that the normalised c J are equal to the un-normalised ones divided by P J c J . – 32 –hat the Poincar´e sum over it reproduces the desired contribution. We now apply this tothe case of SU(2) k . k WZW Models
We have seen that the SU(2) k WZW models with n = k + 2 = 2 p (with prime p > k , this procedure works in the same way for all models with two invariants.Let us label each boundary torus with its corresponding independent modulus τ ( i ) .The full partition function should be modular invariant with respect to each of the moduli.To perform the gravity calculation, we compute a Poincar´e sum analogous to the toruspartition function: Z grav ( τ (1) , τ (2) , . . . , τ ( N ) ) = X γ (1) ,...,γ ( N ) Z seed ( γ (1) τ (1) , γ (2) τ (2) , . . . , γ ( N ) τ ( N ) ) (3.6)From Eq. (2.39) we find that the normalised probabilities, starting with the identityseed, are: c A = p + 12 p − , c D = p − p − Z grav ( τ (1) , τ (2) , . . . , τ ( N ) ) = h Z CFT ( τ (1) , τ (2) , . . . , τ ( N ) ) i = c A N Y i =1 Z A ( τ ( i ) ) + c D N Y i =1 Z D ( τ ( i ) ) (3.8)To simplify the notation we will write Z A/D ( τ ( i ) ) as Z ( i ) A/D in what follows. Also for anyfunction that depends on all the τ ( i ) , we will not write the argument explicitly.Let us start with the case of two boundaries. Then the postulated answer: Z grav = c A Z (1) A Z (2) A + c D Z (1) D Z (2) D (3.9)should come from a sum over manifolds with the two genus-one surfaces as their boundaries.In this case there can be two distinct types of manifold: a pair of disconnected manifoldseach with a torus boundary, and a single connected manifold with two disjoint genus-1boundaries: Z grav = Z disconn . grav + Z conn . grav (3.10)From our previous analysis of the single-boundary case, we know that: Z disconn . grav = (cid:16) c A Z (1) A + c D Z (1) D (cid:17) (cid:16) c A Z (2) A + c D Z (2) D (cid:17) (3.11)– 33 –t follows that the connected contribution should be: Z conn . grav = (cid:16) c A Z (1) A Z (2) A + c D Z (1) D Z (2) D (cid:17) − (cid:16) c A Z (1) A + c D Z (1) D (cid:17) (cid:16) c A Z (2) A + c D Z (2) D (cid:17) = c A c D ( Z (1) A − Z (1) D )( Z (2) A − Z (2) D ) (3.12)Thus the connected part is a product of terms of the form Z A − Z D with an overall factordepending on the probabilities. We will see that this is the general form for the connectedcontribution linking any number of boundaries.Before going to the general case, let us extend the above calculation to three genus-oneboundaries. Here we expect that: Z grav = c A Y i =1 Z ( i ) A + c D Y i =1 Z ( i ) D (3.13)Now we have three distinct types of manifolds to sum over: (i) a sum of three disconnectedpieces each with one boundary, (ii) a manifold that connects any two of the three bound-aries, together with one that has a single boundary, (iii) manifolds which connect all threeboundaries. These cases are associated, respectively, to the following partitions of N = 3: { , , } , { , } , { } . Henceforth we will label different contributions by partitions of N .Thus we can write: Z grav = Z { } grav + Z { } grav + Z { } grav (3.14)Note that some of the partitions occur in multiple ways, for example there are three waysto realise { , } depending on the choice of boundary that remains disconnected. The firstterm on the RHS will be a product over three copies of (2.39), while the second term willbe a sum of products of one copy of (3.12) and one copy of (2.39), in three different ways.Then the last term should be: Z { } grav = Z grav − Z { } grav − Z { } grav = c A Y i =1 Z ( i ) A + c D Y i =1 Z ( i ) D − Y i =1 ( c A Z ( i ) A + c D Z ( i ) D ) − c A c D ( Z (1) A − Z (1) D )( Z (2) A − Z (2) D )( c A Z (3) A + c D Z (3) D ) − c A c D ( Z (1) A − Z (1) D )( Z (3) A − Z (3) D )( c A Z (2) A + c D Z (2) D ) − c A c D ( Z (2) A − Z (2) D )( Z (3) A − Z (3) D )( c A Z (1) A + c D Z (1) D ) (3.15)where the contribution of Z { } grav contains three terms, one for each pair of boundaries thatare connected. Upon simplifying we have: Z { } grav = c A c D ( c D − c A )( Z (1) A − Z (1) D )( Z (2) A − Z (2) D )( Z (3) A − Z (3) D ) (3.16)so, we see again that the completely connected part is given as product of terms of the– 34 –orm Z A − Z D . This is now easily generalised to N B boundaries. We find: Z { N B } grav = f ( c A , N B ) N B Y i =1 ( Z ( i ) A − Z ( i ) D ) , N B > f ( c A , N B ) is a function of the number of boundaries and the probability c A , whichwe will compute below (we have used c D = 1 − c A ).These observations tells us that if we can find a seed for generating Z { N B } grav then we canfind the seed for the full N B -boundary gravity partition function. For example, Z N B =1seed = Z { } seed ( τ (1) ) Z N B =2seed = Y i =1 Z { } seed ( τ ( i ) ) + Z { } seed ( τ , τ (2) ) Z N B =3seed = Y i =1 Z { } seed ( τ ( i ) ) + Z { } seed ( τ (1) ) Z { } seed ( τ (2) , τ (3) ) + Z { } seed ( τ (2) ) Z { } seed ( τ (1) , τ (3) )+ Z { } seed ( τ (3) ) Z { } seed ( τ (1) , τ (2) ) + Z { } seed ( τ (1) , τ (2) , τ (3) ) (3.18)One possible choice for Z { } seed , as we have seen, is: Z { } seed ∼ | χ | (3.19)with a suitable normalisation. But as we have seen in Eqs. (2.44),(2.45), there are moregeneral choices for this seed. Inserting the normalised values of c A , c D for the identity seed,namely p +12 p − , p − p − , we find the general answer to be given by: Z { } seed = a X λ odd , =p b λ | χ λ | + a X λ even c λ | χ λ | + a | χ p | + a p − X λ odd , =1 d λ ( χ λ χ p − λ + c . c . ) (3.20)with: a = 12 p − − a − a , a = a + 2 a − p − , X b λ = X c λ = X d λ = 1 (3.21)Clearly a = p − , b = 1 with all other a i , b λ = 0 is a possible solution in this case, leadingto Eq. (3.19). But far more general solutions are allowed.Let us now try to find a seed that will give us the linear combination Z A − Z D . Thisis given by Eqs.(2.44),(2.45) with α = 1 , β = −
1. Then Eq. (2.45) leads to: a = p − p − − a − a , a = a + 2 a − p + 1 (3.22)This time there is no solution with all a i = 0 other than a . One of the simplest solutions– 35 –s: a = 1 p + 1 , a = 1 − p p − , a = a = 0 , b = 1 (3.23)However as we will shortly see, there are other solutions that are of interest.We can now write the connected contribution for N B boundaries: Z conn . grav ( τ (1) , . . . , τ ( N B ) ) = X γ ( i ) ∈ Γ( N ) \ Γ Z conn . seed ( { γ ( i ) τ ( i ) } ) (3.24)where: Z conn . seed ( { τ ( i ) } ) = f ( c A , N B ) N B Y i =1 Z A − D seed ( τ ( i ) ) (3.25)Finally, using (3.24) in expressions of the form (3.18) gives the general seed for the partitionfunction with N B genus-one boundaries.It remains to find the form of the function f ( c A , N B ). We provide a recursive algorithmto generate this function, where the answer for a given N B is given in terms of the answerfor all numbers of boundaries < N B . To do this we first introduce a generating functionfor computing the moments of a random variable (say x ): P ( k ) = h e kx i = X n ≥ k n n ! h x n i and ln P ( k ) := X n ≥ k n n ! h x n i c (3.26)where h x n i are the usual moments and h x n i c denote the connected moments. We can usethe above equations to find relations between the two: X n ≥ k n n ! h x n i = exp X n ≥ k n n ! h x n i c = Y n ≥ X l ≥ k nl l ! (cid:18) h x n i c n ! (cid:19) l (3.27)which implies that: h x m i = m ! X { l n } m Y n ≥ l n ! (cid:18) h x n i c n ! (cid:19) l n (3.28)The sum is over the set of { l n } such that P n nl n = m , which are just the partitions of theinteger m .To apply this to the present case we replace x by Z and find: h Z m i = m ! X { l n } m Y n ≥ l n ! (cid:18) h Z n i c n ! (cid:19) l n (3.29)Replacing the LHS by the desired average over products of CFT’s, and separating twoterms in the RHS from the rest in an obvious way, we get:( c A Z mA + c D Z mD ) = h Z i m + m ! ′ X { l n } m − Y n ≥ l n ! (cid:18) h Z n i c n ! (cid:19) l n + h Z m i c (3.30)– 36 –his can now be solved to get the connected part: h Z m i c = ( c A Z mA + c D Z mD ) − ( c A Z A + c D Z D ) m − m ! ′ X { l n } m − Y n ≥ l n ! (cid:18) h Z n i c n ! (cid:19) l n (3.31)Now using the fact that h Z m i c , ( n >
1) is of the form f ( c A , m )( Z A − Z D ) m and comparingthe coefficients of Z mA , we finally get: f ( c A , m ) = c A (1 − c m − A ) − m ! ′ X { l n } ( c A ) l l ! m − Y n ≥ l n ! (cid:18) f ( c A , n ) n ! (cid:19) l n (3.32)This is the desired recursive formula for the coefficients in Eq. (3.25). Some examples ofthe above formula are as follows: m = 2 : f ( c A ,
2) = c A (1 − c A ) m = 3 : f ( c A ,
3) = c A (1 − c A ) − c A ) (1 − c A ) = c A (1 − c A )(1 − c A ) m = 4 : f ( c A ,
4) = c A (1 − c A )(1 − c A + 6 c A ) (3.33)The fact that such a rule exists at all, and has nice properties, is very encouraging. Inprinciple it could be checked if we can have a better understanding of the wormhole-typemanifolds, though – as already emphasised here and in [11], the semi-classical notion ofmanifold does not really apply in the RCFT context.We have shown how to write down a general seed for the partition function with N B genus-1 boundaries. This expressions involved free parameters and hence give an infinitefamily of possible seeds. To conclude this section we now give an example of a particularchoice of these parameters which allows us to write down the full N B -boundary seed in asimple compact form. We are not sure whether there is any physical interpretation for thischoice, but mathematically it is appealing.We begin with the general form of Z { } seed , given in Eq. (3.20). Now we set b = c = 1and all the remaining b λ = c λ = 0. We also demand that a = 0. From Eq. (3.21), thisleads to the constraint a + 2 a = p − , so the disconnected seed can be written: Z { } seed = (cid:18) p − − a (cid:19) | χ | + a | χ | + a | χ p | (3.34)For the case of Z A − D seed , we make a similar demand. Here we will denote the parameters witha prime to distinguish from the above. We set b ′ , c ′ = 1 with the remaining b ′ λ = c ′ i = 0.We also set a ′ = 0. This implies a ′ + 2 a ′ = p +1 . Then the corresponding seed is: Z A − D seed = (cid:18) p + 1 − a ′ (cid:19) | χ | + a ′ | χ | + (cid:18) a ′ − p − p − (cid:19) | χ p | (3.35)Thus we have found the required seeds to set up the recursion relation, as functions oftwo arbitrary parameters a , a ′ . We now show that for special values of these parameters,– 37 –hings simplify considerably. Let us choose: a = p − p − p − , a ′ = p − p − p + 1) (3.36)Now using (3.18), (3.34) and (3.36) we find for one boundary: Z N B =1seed = Z { } seed = p + 1(2 p − p − (cid:18) | χ | + p − p + 1 | χ | + p − p + 1 | χ p | (cid:19) (3.37)while for two boundaries, we get: Z N B =2seed = Y i =1 Z { } seed ( τ ( i ) ) + p A (1 − p A ) Z A − D seed ( τ (1) , τ (2) )= p + 1(2 p − p − Y i =1 (cid:18)(cid:12)(cid:12)(cid:12) χ ( i )1 (cid:12)(cid:12)(cid:12) + (cid:18) p − p + 1 (cid:19) (cid:12)(cid:12)(cid:12) χ ( i )2 (cid:12)(cid:12)(cid:12) (cid:19) + (cid:18) p − p + 1 (cid:19) Y i =1 (cid:12)(cid:12)(cid:12) χ ( i ) p (cid:12)(cid:12)(cid:12) ! (3.38)From the above, we see a pattern emerging. So one may conjecture that for N B boundaries, Z seed is: Z N B seed = p + 1(2 p − p − N B N Y i =1 (cid:18)(cid:12)(cid:12)(cid:12) χ ( i )1 (cid:12)(cid:12)(cid:12) + (cid:18) p − p + 1 (cid:19) (cid:12)(cid:12)(cid:12) χ ( i )2 (cid:12)(cid:12)(cid:12) (cid:19) + (cid:18) p − p + 1 (cid:19) N B Y i =1 (cid:12)(cid:12)(cid:12) χ ( i ) p (cid:12)(cid:12)(cid:12) ! (3.39)Now one proves the conjecture by directly evaluating the Poincar´e sum over this seed andfinding that it gives the desired N B -boundary answer. We get: X γ Z N B seed = p + 1(2 p − p − N B N B Y i =1 (3 p − Z ( i ) A + (cid:18) p − p + 1 (cid:19) N B Y i =1 (3 p − Z ( i ) D ! = 12 p − ( p + 1) N B Y i =1 Z ( i ) A + ( p − N B Y i =1 Z ( i ) D ! = h Z ( τ (1) ) · · · Z ( τ ( N B ) ) i (3.40)Thus we see that (3.39) provides a simple, compact choice of seed which reproduces exactlythe desired result for the N B -boundary partition function. However, as explained above, itis not unique and one can have more complicated expressions by choosing the parametersdifferently. It would be worth investigating whether some additional physical criteriondetermines the free parameters to produce this compact result. We have attempted to resurrect the original attempt by [7] of finding “non-semi-classical”3d gravity duals to 2d rational conformal field theories. While doing so, we have generalisedtheir considerations to SU( N ) k WZW models, and also extended their results on Virasoro– 38 –inimal models. Most important, we have explored the idea that the dual of AdS gravitycan be an average of an ensemble of CFT’s rather than a single CFT. The ensembles wehave studied are extremely small, in most cases having just two members, which makes theproblem highly tractable. In the present work, our considerations have been limited to 3dspacetimes with one or more genus-1 boundaries.Our principal results are: (i) there are infinitely many unitary WZW models havingsingle CFT duals, whose partition functions are correctly reproduced by a Poincar´e sum.This is in contrast to the minimal models, for which the same considerations only lead totwo unitary cases with single duals, namely ( m, m + 1) = (3 ,
4) and (4 , N ) , where the dual can be anaverage over arbitrarily many CFT’s.Our analysis has been most detailed for SU(2) k models. However the results for min-imal models are quite similar. This seems to arise from a kind of “doubling” relationbetween the two classes of models, which is well-known from the classic papers of Cappelli,Itzykson and Zuber [21, 22]. We hope to return to more general classes of RCFT for whicha given set of characters admits more than one modular invariant.In the early studies of Poincar´e series for gravity [1], it was assumed that the “seed” forthe series is the identity module, corresponding to thermal AdS on the gravity side. Laterstudies, notably [6], allowed for more general primary seeds and studied their contribution.The recent work [18] also invokes additional seeds associated to conical singularities. Thusthere is no real consensus on what should be the seed for a Poincar´e series. In the presentwork we have similarly taken an agnostic view on this. However, as seen in many of our ta-bles, the identity seed often seems to return a positive linear combination of CFT partitionfunctions, while other primary seeds often do not . Related to this is the question of addi-tional seeds that must be added to account for analogue wormholes in the multi-boundarycase. We have found candidates that work correctly, which we think is remarkable, but wedo not have a first-principles reason why the wormhole seeds should be what we propose.Progress on this issue would be most helpful and could illuminate the more general case ofpure AdS gravity.Our considerations have been based on considering Euclidean AdS with one or moregenus-1 boundaries. For minimal models, it has been argued [8] that stronger constraintsare obtained by considering, for example, a single genus-2 boundary. It would be worthwhile Amusingly this seems to be the opposite of what is found in pure AdS gravity, where the identity seedleads to negative coefficients and other primary seeds are added to try and cure this. – 39 –o know what constraints can be put on the dualities described in the present work byconsidering genus-2 or higher-genus boundaries.In the context of Narain lattices which have U(1) D Kac-Moody algebras, [11] proposedthat the gravity dual is a topological U(1) D Chern-Simons theory. In the same spirit, wehave suggested that the gravity dual for WZW theories based on non-abelian SU( N ) k × SU( N ) k Kac-Moody algebras is a topological SU( N ) k × SU( N ) k Chern-Simons theory.However we did not really make use of this in the present work. Perhaps one can obtainmore information about the dualities discussed by invoking properties of the “gravity” side.Among possible future directions, one would be to investigate SU( N ) WZW modelsin greater detail. Since all the modular invariants are physical, we do not need to restrict toany special class as was the case for SU(2) k . Also the number of invariants is unbounded.Moreover, for any given number of invariants there will, in general, be an infinite familyof models within this category. Another interesting point is that for each such family wecan consider the large- N (large- c ) limit, which may permit considerations that are closerto being semi-classical. We hope to return to this in future.Another fascinating direction is to start with meromorphic RCFT. As is well-known,these correspond to free bosons on an even unimodular lattice, orbifolds thereof and somegeneralisations [33, 34]. Such CFT’s typically have Kac-Moody algebras and their singlecharacter is just an integral linear combination of Kac-Moody characters. This automati-cally gives us (at least) a pair of CFT’s: the meromorphic one and the diagonal invariant.Thus when computing Poincar´e sums, one expects to find linear combinations of both par-tition functions. This has already been verified in one example, namely SU(25) , in thepresent paper – see the discussion following Eq. (2.55). A large fraction of meromorphicCFT’s are based on non-simple Kac-Moody algebras which renders the analysis more novelas well as more complicated.Recently, the considerations of [7] have been extended to the case of boundary minimalmodels [35]. This requires the introduction of Randall-Sundrum branes [36] associated toCardy states [37]. It should be possible to generalise these ideas to the WZW models thatwe have studied here, and even take the large- c limit as suggested above.Finally, the MLDE approach to classifying rational CFT [38, 39] might be fruitfullycombined with the Poincar´e sum method to generate new examples of RCFT with multiplemodular invariants. Acknowledgements
We thank Matthias Gaberdiel, Alex Maloney and Rahul Poddar for helpful discussions,and Tushar Gopalka for collaboration at the early stages of this project. VM would liketo acknowledge the INSPIRE Scholarship for Higher Education, Government of India.PS would like to acknowledge support from the Clarendon Fund and the MathematicalInstitute, University of Oxford. All of us are grateful for support from a grant by PrecisionWires India Ltd. for String Theory and Quantum Gravity research at IISER Pune.– 40 – ppendicesA Congruence subgroups
Here we collect a number of definitions and properties of congruence subgroups of Γ =SL(2, Z ). We have: Γ( N ) := ( a bc d ! ∈ Γ (cid:12)(cid:12)(cid:12)(cid:12) a, d ≡ b, c ≡ N ) Γ ( N ) := ( a bc d ! ∈ Γ (cid:12)(cid:12)(cid:12)(cid:12) a, d ≡ c ≡ N ) Γ ( N ) := ( a bc d ! ∈ Γ (cid:12)(cid:12)(cid:12)(cid:12) c ≡ N ) (A.1)We have the inclusions: Γ( N ) ⊂ Γ ( N ) ⊂ Γ ( N ) ⊂ Γ. Also, Γ( nN ) ⊂ Γ( N ) for any integer n >
1, and similarly for the others. We also have:Γ ( N ) := ( a bc d ! (cid:12)(cid:12)(cid:12)(cid:12) b ≡ N ) (A.2)Γ( N ) is called the principal congruence subgroup, and is a normal subgroup of Γ. Anysubgroup of Γ that contains Γ( N ) for some N is called a congruence subgroup, but will ingeneral not be normal. The smallest such N (corresponing to the largest Γ( N )) is calledthe level of the congruence subgroup.We can write down the indices of various congruence subgroups in Γ:[Γ : Γ( N )] = N Y p | N (cid:18) − p (cid:19) [Γ : Γ ( N )] = N Y p | N (cid:18) − p (cid:19) [Γ : Γ ( N )] = N Y p | N (cid:18) p (cid:19) (A.3)from which we find the useful relation:[Γ ( N ) : Γ( N )] = N (A.4)– 41 – Basic formulae for WZW and Virasoro minimal models
SU(2) k WZW models
We will restrict our attention to the unitary series of SU(2) k models. An integer level k characterises the values of c , h i in the model. Defining the “height” n = k + 2, we have: c = 3 − n , ( n integer ≥
3) (B.1)The primaries can be labelled by λ = 2 j + 1 where j is the isospin, with 1 ≤ λ ≤ n − h λ = λ − n (B.2)Under modular transformations T and S , the characters χ λ ( τ ) go into linear combinationsof themselves as follows: T λ,λ ′ = δ λ,λ ′ exp (cid:16) πi (cid:16) h λ − c (cid:17)(cid:17) S λ,λ ′ = r n sin (cid:18) πλλ ′ n (cid:19) (B.3) SU(3) k WZW models
For SU(3) k the representations are labelled by a pair of integers λ , λ ≥
1. In terms ofthese and the height n = k + 3, we have the constraint λ + λ < n . The central chargeand conformal dimensions are: c = 8 − n , h λ = λ + λ + λ λ − n (B.4)It follows from the expression for the conformal dimensions that the semi-conductor in thiscase has the simple form, N = 3 n , whereas the total number of characters that appear at– 42 –evel k is n ( n − . Finally, the modular T and S matrices can be expressed as: T λ,µ = δ λ,µ exp (cid:16) πi (cid:16) h λ − c (cid:17)(cid:17) S λ,µ = − in √ (cid:20) exp (cid:18) πi n (2 λ µ + λ µ + λ µ + 2 λ µ ) (cid:19) + exp (cid:18) πi n ( − λ µ − λ µ + λ µ − λ µ ) (cid:19) + exp (cid:18) πi n ( − λ µ + λ µ − λ µ − λ µ ) (cid:19) − exp (cid:18) πi n ( − λ µ − λ µ − λ µ − λ µ ) (cid:19) − exp (cid:18) πi n (2 λ µ + λ µ + λ µ − λ µ ) (cid:19) − exp (cid:18) πi n ( − λ µ + λ µ + λ µ + 2 λ µ ) (cid:19)(cid:21) (B.5) SU( N ) k WZW models
As in the case of SU(2) k , the level k characterises the values of c , h i in the model. Definingthe height n = k + N , we have: c = k ( N − N + k = ( N − (cid:18) − Nn (cid:19) (B.6)For any given level k only a finite number of representations are allowed, which are selectedby the constraint: N − X i =1 λ i < n (B.7)where λ ≡ ( λ , λ , . . . , λ N − ) is the Dynkin label for SU( N ) , which labels the characters.The number of such representations is given by: k ) = ( N + k − k !( N − λ can be expressed as: h λ = ( λ − ρ, λ + ρ )2 n (B.9) Note that we are following the convention where the Dynkin label of the trivial representation is theunit vector (1 , , . . . , – 43 –here ρ = (1 , , . . . , x, y ) = x i κ ij y j , where κ is the quadraticform matrix of SU( N ): κ ≡ N N − N − N − · · · N − N −
2) 2( N − · · · N − N −
3) 3( N − · · · · · · N − N −
21 2 3 · · · N − N − (B.10)The action of S and T modular transformations on the characters are given as: T λλ ′ = δ λ,λ ′ exp (cid:16) πi (cid:16) h λ − c (cid:17)(cid:17) (B.11) S λλ ′ = i N ( N − / √ N n N − X w ∈ W det( w ) exp (cid:18) − πi (cid:18) λ · w ( λ ′ ) n (cid:19)(cid:19) (B.12)where W is the Weyl group of SU ( N ). Virasoro minimal models
These models are labeled by two co-prime integers ( m, m ′ ) with m ′ > m >
2. The unitaryminimal models are the sub-series obtained by setting m ′ = m + 1. The central charge interms of these parameters is: c ( m, m ′ ) = 1 − m − m ′ ) mm ′ (B.13)The primaries for a given model are labeled by a pair of integers ( r, s ) where: h r,s = ( mr − m ′ s ) − ( m − m ′ ) mm ′ (B.14)1 ≤ r ≤ m ′ − , ≤ s ≤ m − , ( r, s ) ∼ ( m ′ − r, m − s )Due to the above identifications we have a total of ( m − m ′ − / Z ) representation which will act on thecharacters χ r,s .The T, S matrices in this representation are given as: T rs,vw = δ r,v δ s,w exp (cid:16) πi (cid:16) h r,s − c (cid:17)(cid:17) S rs,vw = r mm ′ ( − sv + rw sin (cid:16) πrvmm ′ (cid:17) sin (cid:18) πswm ′ m (cid:19) (B.15) C Sage calculations for SU(2) k – 44 – evel ( k, c, N ) Seed primary label λ Poincar´e sum (1 , ,
4) 1 , Z A (2 , ,
16) 1 , , Z A (3 , ,
20) 1 , , , Z A (4 , ,
20) odd , = 3 8 (4 Z A + Z D )even 32 (2 Z A − Z D )3 48 Z D (5 , ,
28) 1 , . . . , Z A (6 , ,
32) 1 , , , , Z A + Z D )2 , Z A − Z D )(7 , ,
36) 1 , . . . , Z A (8 , ,
40) odd , = 5 24 (2 Z A + Z D )even 48 (2 Z A − Z D )5 96 Z D (9 , ,
44) 1 , . . . ,
10 54 Z A (10 , ,
48) 1 , , ,
11 32( Z A + Z D + Z (10) E )2 ,
10 32(4 Z A − Z D − Z (10) E )3 , , Z A + Z D − Z (10) E )4 , Z A − Z D + 2 Z (10) E )(11 , ,
52) 1 , . . . ,
12 84 Z A (12 , ,
56) odd , = 7 8 (8 Z A + 5 Z D )even 64 (2 Z A − Z D )7 144 Z D (13 , ,
60) 1 , , , , , , ,
14 8 (6 Z A + Z (13) new )3 , , ,
12 72 (2 Z A − Z (13) new )5 ,
10 96 Z (13) new (14 , ,
64) 1 , , , , , , , ,
15 64 ( Z A + Z D )2 , , , , ,
14 32 (5 Z A − Z D )(15 , ,
68) 1 , . . . ,
16 108 Z A (16 , ,
72) 1 , , , , ,
17 72 ( Z A + Z E )2 , , , , , , ,
16 72 (2 Z A − Z D )3 ,
15 24 (3 Z A + 7 Z D − Z E )9 48 (5 Z D − Z E )– 45 –17 , ,
76) 1 , . . . ,
18 120 Z A (18 , ,
80) 1 , , , , , , ,
19 32 (2 Z A + 2 Z D + Z (18) new )2 , , ,
18 16 (13 Z A − Z D − Z (18) new )4 , , ,
16 48 (3 Z A − Z D + 2 Z (18) new )5 , ,
15 128 ( Z A + Z D − Z (18) new )(19 , , = 0 mod 3 , Z A + Z (19) new )= 0 mod 3 96 (2 Z A − Z (19) new )= 0 mod 7 144 Z (19) new (20 , ,
88) odd , = 11 24 (4 Z A + 3 Z D )even 96 (2 Z A − Z D )11 240 Z D (21 , ,
92) 1 , . . . ,
22 144 Z A (22 , ,
96) 1 , , , , , , ,
23 32 (2 Z A + 2 Z D + Z (22) new )2 , , ,
22 16 (10 Z A − Z D + Z (22) new + Z (22) new )3 , , , ,
21 64 (2 Z A + 2 Z D − Z (22) new )4 ,
20 32 (10 Z A − Z D − Z (22) new − Z (2) new )6 ,
18 32 (2 Z A + 2 Z D − Z (22) new + 3 Z (2) new )8 ,
16 64 (4 Z A − Z D + Z (22) new − Z (2) new )(23 , , , . . . ,
24 150 Z A (24 , , , = 13 8 (14 Z A + 11 Z D )even 112 (2 Z A − Z D )13 288 Z D (25 , , = 0 mod 3 54 (3 Z A − Z (25) new )= 0 mod 3 36 (3 Z A + 5 Z (25) new )(26 , , = 0 mod 2 , Z A + 3 Z D + Z (26) new )= 2 mod 4 , = 14 96 (3 Z A − Z D − Z (26) new )= 0 mod 4 64 (3 Z A − Z D + 2 Z (26) new )= 0 mod 7 192 ( Z A + Z D − Z (26) new )(27 , , , . . . ,
28 180 Z A (28 , , , , , , , , ,
29 8 (12 Z A + 8 Z D + 11 Z E + Z (28) new )2 , , , , , , ,
28 16 (12 Z A − Z D − Z E + Z (28) new )3 , , ,
27 48 (4 Z A − Z E − Z (28) new )5 ,
25 32 (10 Z D − Z E + 2 Z (28) new )6 , , ,
24 96 (4 Z A − Z D + Z E − Z (28) new )10 ,
20 128 (2 Z D − Z E + Z (28) new )– 46 –5 192 (2 Z D − Z E )(29 , , , . . . ,
30 192 Z A (30 , , Z A + Z D )even, = 4 , , , ,
28 64 (5 Z A − Z D − Z (30) new )4 , , ,
28 64 (3 Z A − Z D + 5 Z (30) new )(31 , , = 0 mod 3 ,
11 48 (3 Z A + Z (31) new )= 0 mod 3 144 (2 Z A − Z (31) new )= 0 mod 11 240 Z (31) new (32 , , , = 17 24 (6 Z A + 5 Z D )even 144 (2 Z A − Z D )17 384 Z D (33 , , = 0 mod 5 , Z A + Z (33) new )= 0 mod 5 192 (2 Z A − Z (33) new )= 0 mod 7 216 Z (33) new (34 , , = 3 , , , , ,
33 48 (3 Z A + 3 Z D − Z (34) new )2 , , , , ,
34 24 (9 Z A − Z D − Z (34) new + 2 Z (34) new )3 , , ,
33 16 (9 Z A + 9 Z D + 12 Z (34) new − Z (34) new )4 , , , , , , ,
32 24 (15 Z A − Z D − Z (34) new − Z (34) new )6 ,
30 8 (27 Z A − Z D + 30 Z (34) new + 2 Z (34) new )9 , ,
27 64 (3 Z (34) new + Z (34) new )(35 , , , . . . ,
36 228 Z A (36 , , , = 19 8 (20 Z A + 17 Z D )even 160 (2 Z A − Z D )19 432 Z D (37 , , = 0 mod 3 ,
13 12 (14 Z A + 5 Z (37) new )= 0 mod 3 168 (2 Z A − Z (37) new )= 0 mod 13 288 Z (37) new (38 , , = 5 , , ,
35 32 (4 Z A + 4 Z D + Z (38) new )2 , , , , , , ,
38 16 (16 Z A − Z D + Z (38) new + 6 Z (38) new )4 , , ,
36 64 (8 Z A − Z D − Z (38) new − Z (38) new )5 , , , ,
35 128 (2 Z A + 2 Z D − Z (38) new )8 , , ,
32 96 (4 Z A − Z D + Z (38) new − Z (38) new )10 ,
30 64 (2 Z A + 2 Z D − Z (38) new + 6 Z (38) new )(39 , , , . . . ,
40 252 Z A (40 , , = 3 mod 6 , = 7 ,
35 8 (16 Z A + 40 Z D − Z (40) new + 2 Z (40) new )– 47 –ven, = 0 mod 6 , = 14 ,
28 32 (8 Z A − Z D + Z (40) new )3 , , , , ,
39 16 (16 Z A − Z D + 9 Z (40) new − Z (40) new )6 , , , , ,
36 128 (4 Z A − Z D − Z (40) new )7 ,
35 48 (3 Z (40) new + 2 Z (40) new )14 ,
28 192 Z (40) new
21 288 Z (40) new (41 , , , . . . ,
42 264 Z A Table 7 : Poincar´e sums for SU(2) k In the above table, terms labelled Z new are unphysical modular invariants. We list a fewof them below. Notice that for k = 22 there are two unphysical invariants. This numbergrows for generic k . Z (13) new = | χ − χ | + | χ + χ | + | χ − χ | + | χ + χ | + 2 | χ | + 2 | χ | Z (19) new = | χ + χ | + | χ − χ | + | χ + χ | + | χ − χ | + | χ + χ | + | χ + χ | + 2 | χ | + 2 | χ | Z (18) new = | χ − χ | + | χ + χ | + | χ + χ | + | χ + χ | + | χ + χ | + | χ − χ | Z (22) new = 2 (cid:16) | χ − χ | + | χ + χ | + | χ − χ | + | χ + χ | + | χ + χ | (cid:17) + | χ + χ + χ + χ | Z (22) new = 4 | χ − χ | + 2 | χ + χ − χ − χ | (C.1) D Sage calculations for unitary Virasoro minimal models
Minimal Model ( m, m ′ , c, N ) Seed Primary ( χ r,s ) Dimension ( h r,s ) Partition function
Ising χ , , χ , , Z AA (3 , , , χ , Z AA Tricritical Ising χ , , χ , , χ , , χ , , , , Z AA (4 , , , χ , , χ , , Z AA χ , , χ , , χ , , χ , , , , Z AA + Z AD )(5 , , , χ , , χ , , Z AD χ , , χ , , χ , , χ , , , , Z AA − Z AD )– 48 –ricritical Potts χ , Z AA + Z AD )(6 , , , χ , Z AD χ , Z AA − Z AD )Name χ , Z AA + Z AD )(7 , , , χ , Z AA + Z AD ) χ , Z AA − Z AD )Name χ , Z AA + Z AD )(8 , , , χ , Z AA + Z AD ) χ , Z AA − Z AD ) χ , Z AA + Z AD ) χ , Z AA + Z AD ) χ , Z AA − Z AD )Name χ , Z AA + Z AD )(9 , , , χ , Z AA − Z AD ) χ , Z AD χ , Z AA + Z AD ) χ , Z AA − Z AD ) χ , Z AD Name χ , Z AA + Z AD )(10 , , , χ , Z AA − Z AD ) χ , Z AD Name χ , Z AA + Z AD + Z (12) )(11 , , , χ , Z AA − Z AD − Z (12) ) χ , Z AA + Z AD − Z (12) ) χ , Z AA − Z AD + 2 Z (12) ) χ , Z AA + Z AD − Z (12) ) Table 8 : Poincar´e Sums for unitary Virasoro minimal models
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