Boundary States and Anomalous Symmetries of Fermionic Minimal Models
PPrepared for submission to JHEP
Boundary States and Anomalous Symmetries of Fermionic Minimal Models
Philip Boyle Smith
Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Cambridge, CB3 OWA, UK
E-mail: [email protected]
Abstract:
The fermionic minimal models are a recently-introduced family of two-dimensional spin conformal field theories. We determine all of their conformal boundarystates and potentially anomalous Z global symmetries. The latter task hinges uponon a conjecture about su (2) affine parities generalising an earlier result known to havean interpretation in terms of Fermat curves. Our results indicate a close connectionbetween several properties of the models, including the matching of the sizes of theSPT classes of boundary states, the existence of anomalous Z symmetries, and thevanishing of the Ramond-Ramond sector, for which we provide an explanation. a r X i v : . [ h e p - t h ] F e b ontents – 1 – Introduction
The fermionic minimal models are a family of two-dimensional conformal field theoriesrecently introduced in [1–3]. They are close relatives of the bosonic minimal modelsthat have been known for a long time [4–8], but differ in one key respect: instead ofdepending solely on a Riemann surface with metric, they also depend on a choice ofspin structure.The fact that these theories depend upon a spin structure opens up the possibilityof new fermionic phenomena that are not seen in the ordinary bosonic minimal models.For a start, there can be an issue with their boundary conditions. While in the bosonicminimal models, there exists a complete set of boundary conditions such that when oneplaces the theory on a spatial interval and imposes different boundary conditions atthe two ends, one always obtains a physically sensible theory [8], in fermionic theoriesthis no longer need be the case. Instead, one may find that for certain choices ofboundary conditions, the spectrum on an interval includes an unpaired Majorana zeromode, rendering the theory inconsistent [9]. As we review later in this introduction,the fact this issue arises only for fermionic theories is closely tied to the classificationof SPT phases in two dimensions, where it is reflected in the mathematical fact thatΩ
SO2 (pt) = 0 is trivial while Ω
Spin2 (pt) = Z is not [10, 11].Second, there is also the possibility of supporting symmetries with nontrivial ’tHooft anomalies. For the bosonic minimal models, this is not an option: all globalsymmetries are necessarily non-anomalous [12]. But for fermionic theories, it is apossibility. We will limit ourselves to Z global symmetries. In fermionic theories,such symmetries can carry a mod-8 valued anomaly. As was explained in [13, 14], thisanomaly is related to SPT phases in three dimensions, and the fact it arises only forfermionic theories is encoded in the facts Ω SO3 ( B Z ) = 0 while Ω Spin3 ( B Z ) = Z .The above two phenomena are easy to exhibit for the simplest fermionic minimalmodel, the Majorana fermion. Here, one can simply use free-field techniques to ex-plicitly show everything one could possibly want to show. But for the other fermionicminimal models, which are all interacting, things are not so transparent. Our goal is toextend the analysis of boundary conditions and anomalous symmetries to these remain-ing fermionic minimal models. We would like to see which features of the Majoranafermion generalise to the family as a whole, and which are artefacts of a free theory.Therefore, to set the scene, it will be useful to first review the basic facts about theMajorana fermion we want to generalise. A summary of our results follows, and afterthat the organisation of the paper. – 2 – .1 A Simple Illustration We begin with a motivating discussion about SPT phases. A Majorana fermion of mass m in d = 1 + 1 dimensions has action S = i (cid:90) dtdx χ + ∂ + χ + + χ − ∂ − χ − + mχ + χ − where ∂ ± = ∂ t ± ∂ x . When m (cid:54) = 0 the theory is gapped, and the two possible gappedphases with m > m < m → m ( x ) and consider a domain-wall profile for m ( x ),say one with m ( x = ±∞ ) = ± M for some unimportant constant M >
0. As wasfamously shown by Jackiw and Rebbi [15], the spectrum of the theory then includes aquantum-mechanical zero mode localised on the domain wall, obeying χ = χ † { χ, χ } = 1This system exhibits an anomaly under fermion parity ( − F . (One way to see this isthat χ acquires a nonzero VEV on a periodic temporal circle.) The appearance of thisanomalous degree of freedom on the interface signals that the m > m < χ + = + χ − − : χ + = − χ − Consider a mass profile m ( x ) that interpolates from 0 to ± M . On the left side lives amassless Majorana fermion, while the right side is gapped at a scale M . At low energies,the massless fermion experiences this set-up as a boundary condition. Pictorially, themap from gapped phases to boundary conditions isSimilarly when the gapped phase sits on the left, one obtains a left boundary condition.However, due to an annoying technicality, the outcome is now reversed: ( (cid:63) )– 3 –he analog of the previous story is now that on a spatial interval with ++ boundaryconditions at both ends, the spectrum again includes an unpaired Majorana zero mode.A lattice version of this mechanism also exists, and was the subject of [21].In conformal field theories, boundary conditions are more naturally described asboundary states. It is these that will form the main focus of the paper. To each rightboundary condition, there is an associated boundary state, which lives in the NS sectorof the theory: χ + = ± χ − on right → |±(cid:105) Meanwhile, boundary conditions on the left are described by dual states. One mighthave thought that the correct dual state that describes boundary condition A on the leftis simply the dual of the state that describes boundary condition A on the right. Butthis isn’t quite right. Annoyingly, there is an extra sign flip, and the correct mappingof boundary conditions to dual states is actually χ + = ± χ − on left → (cid:104)∓| Because of the similar sign flip in ( (cid:63) ), however, a gapped phase m = ± M correspondsto the same boundary state (cid:104)±| or |±(cid:105) regardless of whether we sit at a left or a rightboundary. It is for this reason that boundary states more precisely correspond to SPTphases than boundary conditions. In any case, once the boundary states are known,there is a simple algebraic procedure to calculate the partition function for states on aninterval. Doing this for the boundary states (cid:104)−| and | + (cid:105) yields the partition function Z − + ( τ ) = √ χ / ( τ )where χ / ( τ ) is a Virasoro character. The most important feature of this partitionfunction is the overall factor of √
2. This signals the unpaired Majorana mode, hencethe distinctness of the SPT phases underlying (cid:104)−| and | + (cid:105) . For more details of thispoint of view on unpaired Majorana modes, see also [9, 22, 23].For the second half of our story, we will be interested in the Z global symmetryknown as chiral fermion parity. It acts by flipping the sign of only one of the fermions,which without loss of generality we can take to be the left-movers: Z : χ + → χ + χ − → − χ − Under this symmetry, the mass parameter m is odd. The symmetry therefore ex-changes the two SPT phases. One can check that the symmetry also exchanges thecorresponding boundary states Z : |±(cid:105) → |∓(cid:105) as expected. – 4 –mportantly, the symmetry also carries an anomaly whose strength is 1 mod 8. Tosee this, one places the theory on a background with a defect line for the Z symmetry.It will be sufficient for us to consider a torus. We will denote the partition function onsuch a background by Z (cid:104) τ ; PAP (cid:105) where τ is the modular parameter describing a choice of flat metric on the torus, thediagram labels the spin structure, and the dashed line, if present, labels the defect.Under the shift τ → τ + 2, we have the transformation law Z (cid:104) τ + 2; PAP (cid:105) = e πi/ Z (cid:104) τ ; PAP (cid:105)
The presence of the phase e πi/ indicates the anomaly. In general, it could be anyeighth root of unity e πik/ ; the fact that for the Majorana fermion k = 1 indicates thatthe strength of the anomaly is 1 mod 8, as claimed [24, 25]. The anomaly also manifestsitself in a far more obvious way, which becomes clear if we look at the partition function.It turns out to be Z (cid:104) τ ; PAP (cid:105) = √ χ + χ / )( τ ) χ / ( τ )Once again we find a notorious factor of √ Z symmetry defect, contains anunpaired Majorana zero mode. (This statement would also have been true of the NSsector.) We now consider generalising these facts to the other models in the family. Each modelshould have a complete set of conformal boundary states. We expect that all of thesestates will arise from a deformation to a gapped phase. If so, then each boundary statewill fall into one of two distinct classes depending on which SPT class its gapped phasesits in.Our first main claim is that this expectation is borne out. We determine, for eachfermionic minimal model, the complete list of conformal boundary states. We showhow these naturally fall into two classes. When boundary conditions a and b are takenfrom the same class, we find the partition function Z ab ( τ ) = (cid:88) i ∈ KT n iab χ i ( τ )– 5 – igure 1 . The classification of fermionic minimal models. Both the infinite series andexceptionals are labelled by a choice of integer m . where KT denotes the Kac table, as we review in Section 2. The coefficients n iab arenon-negative integers, so this is a manifestly sensible partition function. In contrast,when a and b are from different classes, we find Z ab ( τ ) = √ (cid:88) i ∈ KT n iab χ i ( τ )which contains a factor of √
2, signalling an unpaired Majorana mode, but other thanthat the partition function is perfectly sensible, with the coefficients n iab again given bynon-negative integers. (For a previous example of this phenomenon in a different familyof models, see also [9].) All partition functions are explicitly presented in Section 3.To describe these results in a little more detail, we recall that there are two kindsof fermionic minimal models: an infinite series, and two exceptionals. Both kinds arelabelled by an integer m ; for the infinite series, this integer takes the values m ≥ m = 11 ,
12. This situation is depicted inFigure 1. We explain our results for each kind of model in turn.
Infinite series
Here the boundary states are labelled by pairs ( r, s ) in the bottom-left quadrant of theKac table, defined by the inequalities r ≤ m/ s ≤ ( m + 1) / (cid:98) ( m − / (cid:99) class 2: (cid:98) m/ (cid:99) – 6 – = 3 m = 4 m = 5 m = 6 Figure 2 . Boundary states for the infinite series of models are labelled by points in thebottom-left quadrant of the Kac table. The two classes are shown in blue and red. Note thatfor m = 3 , Figure 3 . The models with an extra Z global symmetry, shown in red. Note that these arethe same models which had matching class sizes earlier, as well as the models with vanishingRR sector. Exceptionals
In this case the labelling of the boundary states is a little more complicated and will beleft to Section 3. However, the counting is straightforward. Now both classes have thesame size. For the m = 11 exceptional, both classes have size 10, while for the m = 12exceptional, both have size 12.Our second main result is a classification of which fermionic minimal models havea global Z symmetry, modulo ( − F . We find that the first two models in the infi-nite series and both exceptionals have a unique such symmetry, generalising the chiralfermion parity of the Majorana fermion, while the remaining models have none. Thissituation is depicted in Figure 3.In the four models where the symmetry exists, we also find that the strength of theanomaly takes the same value 1 mod 8 for all models; that the twisted-sector partitionfunctions contain a Majorana zero mode whenever there is a symmetry defect crossingthe equal-time contour; and that the symmetry exchanges the two classes of boundarystates. This forces them to have equal sizes, which is indeed what we found earlier. Wealso notice that these models are the same as the ones that have a vanishing RR sectorpartition function. – 7 –n view of the above results, the four special models form a close generalisation ofthe Majorana fermion, with all the facts we saw in Section 1.1 continuing to hold. Theremaining models, on the other hand, do not form such a close generalisation, and theonly fact that survives is the existence of two incompatible classes of boundary states.Finally we mention earlier related research. The ‘chiral fermion parity’ in the m = 3 , In Section 2 we review a handful of facts about the minimal models and their fermioniccounterparts that we will need to use, for the purposes of self-containment. In Section 3,we write down the two classes of boundary states, and explicitly compute all intervalpartition functions. In Section 4, we determine all Z global symmetries, and in doingso compute their anomalies and all twisted-sector partition functions. We also computetheir action on the boundary states found earlier. Finally, in Section 5, we give somebrief arguments showing various implications between our results. In this section we review the barest essentials from the theory of minimal models thatwe will need to use in this paper. We will not review this material from scratch; instead,for reviews of minimal models see for example [7], whose conventions we have strivedto match, while for those of the fermionic ones see [2, 3].The fermionic minimal models are defined to be the set of all unitary spin-CFTsthat are rational with respect to the Virasoro algebra. Their classification has beencarried out, and the results are as in Figure 1. The classification consists of • An infinite series, with m ≥ • Two exceptionals, with m = 11 , m dictates the chiral data of the theory. For example, the central chargeis determined via c = 1 − m ( m + 1)– 8 –f special importance is the Kac table, which is defined as a set of pairs of integersmodulo an equivalence relation,KT = (cid:8) ( r, s ) : 1 ≤ r ≤ m − , ≤ s ≤ m (cid:9) ( r, s ) ∼ ( m − r, m + 1 − s )The relevance of the Kac table is that it labels the available Virasoro characters at thiscentral charge. We will denote them by χ r,s ( τ ), and their conformal dimensions by h r,s .Sometimes, we will find it economical to compress an element of KT down to a singlecomposite index, which we will denote by a letter such as i, j, k, . . . .For the special values m = 11 ,
12, one also faces a dichotomy between the infiniteseries and the exceptionals. This choice affects the way the characters are combined inthe partition function. We will follow the presentation of [2]. We arrange the states ofeven/odd fermion parity on an antiperiodic/periodic circle into a 2 × even oddAP (cid:80) ( m +1) r + ms odd1 ≤ s ≤ r 2. After performing this rescaling, we arrive atour final result | i (cid:105) F = (cid:40) | i (cid:105) B i = i (cid:48)| i (cid:105) B + | i (cid:48) (cid:105) B √ i (cid:54) = i (cid:48) – 12 –ot surprisingly, the two cases will turn out to correspond to the two classes of boundarystates. We also take the opportunity to remark that the two cases have a very simplegraphical interpretation: the first corresponds to the border of the quadrant, the secondto the interior.We are now in a position where we can list the two classes of boundary states, anddemonstrate the consistency of their interval partition functions. • Class 1 consists of the points ( r, s ) in the interior of the bottom-left quadrant ofthe Kac Table, defined by the inequalities 1 ≤ r < m/ ≤ s < ( m + 1) / | ( r, s ) (cid:105) F = √ (cid:88) ≤ s (cid:48) ≤ r (cid:48) 11 and r odd. It follows that the admissibleboundary states of the fermionic model must lie in the span | a (cid:105) ∈ span (cid:110) (cid:107) ( r, (cid:105)(cid:105) , (cid:107) ( r, (cid:105)(cid:105) , (cid:107) ( r, (cid:105)(cid:105) , (cid:107) ( r, (cid:105)(cid:105) : r odd (cid:111) (3.5)Our first goal will be to write down the simplest consistent boundary state obeying thisproperty. To do this, we assume that we already know in advance what the coefficients n iaa will be. We can then reverse-engineer a boundary state that does the job: a i = (cid:113) S ij n jaa For most choices of the coefficients n iaa , the resulting boundary state | a (cid:105) will not beallowed, due to containing extra states (cid:107) i (cid:105)(cid:105) not in the span (3.5). We would like tomake the simplest choice of n iaa such that it is allowed. Our claim is that this choice is n iaa = δ i, (1 , + δ i, (1 , + δ i, (1 , + δ i, (1 , Proof To demonstrate the above assertion, we need to show that S ( r,s ) , (1 , + S ( r,s ) , (1 , + S ( r,s ) , (1 , + S ( r,s ) , (1 , = 0 for s (cid:54)∈ { , , , } For this we invoke the explicit formula for the modular S -matrix S ( r,s ) , ( r (cid:48) ,s (cid:48) ) = (cid:115) m ( m + 1) sin (cid:18) πrr (cid:48) m (cid:19) sin (cid:18) πss (cid:48) m + 1 (cid:19) ( − ( r + s )( r (cid:48) + s (cid:48) ) and well as the trigonometric identity (cid:88) s (cid:48) =1 , , , sin (cid:18) πss (cid:48) (cid:19) = (cid:40) √ s ∈ { , , , } (cid:3) So far we have obtained a single consistent seed state | a (cid:105) with coefficients a i = (cid:112) S i, (1 , + S i, (1 , + S i, (1 , + S i, (1 , – 14 –ext we cook up a second seed state | b (cid:105) by making a different set of sign choices in thecoefficients. We choose b ( r,s ) = a ( r,s ) (cid:40) +1 s = 1 , − s = 5 , r isodd and s = 1 , , , 11. Our next claim is that the state | b (cid:105) gives rise to the intervalpartition functions n ibb = n iaa and n iab = √ δ i, (1 , + δ i, (1 , ) (3.6)This is telling us that the two seed states | a (cid:105) , | b (cid:105) lie in different SPT classes, but areotherwise consistent. Proof The first equation in (3.6) is trivial since a i and b i differ only by signs. Thecomputation of n iab is somewhat trickier. This time the key fact we need is( − [ s =5 , (cid:88) s (cid:48) =1 , , , sin (cid:18) πss (cid:48) (cid:19) = √ (cid:88) s (cid:48) =4 , sin (cid:18) πss (cid:48) (cid:19) Using the above identity, we compute a ( r,s ) b ( r,s ) = ( − [ s =5 , (cid:88) s (cid:48) =1 , , , S ( r,s ) , (1 ,s (cid:48) ) = √ (cid:88) s (cid:48) =4 , S ( r,s ) , (1 ,s (cid:48) ) By (3.1) the claimed result for n iab immediately follows. (cid:3) With the two seed states in hand, all remaining boundary states can now be gener-ated by fusion [28, 29]. This is a recipe which takes as input a boundary state, whichfor us will be either | a (cid:105) or | b (cid:105) , as well as a primary operator i ∈ KT, and produces anew consistent boundary state. The recipe for the fused boundary states is | a ; i (cid:105) = (cid:88) j ∈ KT M jj =1 a j S ij S j (cid:107) j (cid:105)(cid:105) | b ; i (cid:105) = (cid:88) j ∈ KT M jj =1 b j S ij S j (cid:107) j (cid:105)(cid:105) which reduce back to the seed states when i = 0. Our final claim is that to obtain acomplete basis of fundamental boundary states, we should let i range over i = ( r, s ) where r = 1 , , , , s = 1 , · · (cid:107) j (cid:105)(cid:105) , as these correspond to j = ( r, s ) with r = 1 , , , , s = 1 , , , m = 11 exceptional model, andtheir interval partition functions: • Both classes are labelled by pairs ( r, s ) with r = 1 , , , , s = 1 , 2. Regardedas elements of the Kac Table, these are all distinct elements. The two classes ofstates, after some algebra, are | ( r, s ) (cid:105) = (cid:88) r (cid:48) =1 , , , , s (cid:48) =1 , , , (cid:40) α : s (cid:48) =1 β : s (cid:48) =5 β : s (cid:48) =7 α : s (cid:48) =11 (cid:41) S ( r,s ) , ( r (cid:48) ,s (cid:48) ) (cid:112) S (1 , , ( r (cid:48) ,s (cid:48) ) (cid:107) ( r (cid:48) , s (cid:48) ) (cid:105)(cid:105)| ( r, s ) (cid:105) = (cid:88) r (cid:48) =1 , , , , s (cid:48) =1 , , , (cid:40) α : s (cid:48) =1 − β : s (cid:48) =5 β : s (cid:48) =7 − α : s (cid:48) =11 (cid:41) S ( r,s ) , ( r (cid:48) ,s (cid:48) ) (cid:112) S (1 , , ( r (cid:48) ,s (cid:48) ) (cid:107) ( r (cid:48) , s (cid:48) ) (cid:105)(cid:105) where α = (cid:113) √ 3) and β = (cid:113) − √ • The interval partition functions are1–1 or 2–2: n kij = N xij [ N kx, (1 , + N kx, (1 , + N kx, (1 , + N kx, (1 , ]1–2 or 2–1: n kij = √ N xij [ N kx, (1 , + N kx, (1 , ]where the appearance of the fusion numbers is from Verlinde’s formula [30].Before we go on, we return to an earlier point and verify the completeness of the states.Using the above formulas, the multiplicity of the identity module is1–1 or 2–2: n ij = δ ij + N (1 , ij + N (1 , ij + N (1 , ij n ij = √ N (1 , ij + N (1 , ij ]One can show that for the range of values of i and j allowed, all the fusion numbersin the above expression vanish identically, and only the δ ij term survives. This showsthat n ab = δ ab between all pairs of states a and b , establishing completeness.We now briefly turn to the m = 12 exceptional. Here, the results are virtuallyidentical, except with the roles of r and s swapped around:– 16 – Both classes are labelled by pairs ( r, s ) with r = 1 , s = 1 , , , , , | ( r, s ) (cid:105) = (cid:88) r (cid:48) =1 , , , s (cid:48) =1 , , , , , (cid:40) α : r (cid:48) =1 β : r (cid:48) =5 β : r (cid:48) =7 α : r (cid:48) =11 (cid:41) S ( r,s ) , ( r (cid:48) ,s (cid:48) ) (cid:112) S (1 , , ( r (cid:48) ,s (cid:48) ) (cid:107) ( r (cid:48) , s (cid:48) ) (cid:105)(cid:105)| ( r, s ) (cid:105) = (cid:88) r (cid:48) =1 , , , s (cid:48) =1 , , , , , (cid:40) α : r (cid:48) =1 − β : r (cid:48) =5 β : r (cid:48) =7 − α : r (cid:48) =11 (cid:41) S ( r,s ) , ( r (cid:48) ,s (cid:48) ) (cid:112) S (1 , , ( r (cid:48) ,s (cid:48) ) (cid:107) ( r (cid:48) , s (cid:48) ) (cid:105)(cid:105) with the same constants α and β as before. • The interval partition functions are1–1 or 2–2: n kij = N xij [ N kx, (1 , + N kx, (5 , + N kx, (7 , + N kx, (11 , ]1–2 or 2–1: n kij = √ N xij [ N kx, (4 , + N kx, (8 , ] Here we turn to the second main goal of the paper, which is to list all Z globalsymmetries of the fermionic minimal models – including potentially those with ananomaly. The motivation for performing this task comes from looking at the boundarystates of the four special models for which the classes have equal sizes. In Section 3,we found these states to be • Infinite series, m = 3: (cid:107) (cid:105)(cid:105) ± (cid:107) (cid:105)(cid:105) (Here and in the next example, we label Ishibashi states by conformal dimensionsrather than elements of the Kac Table.) • Infinite series, m = 4: (5 −√ / / (cid:16) (cid:107) (cid:105)(cid:105) ± (cid:107) (cid:105)(cid:105) (cid:17) + (5+ √ / / (cid:16) (cid:107) (cid:105)(cid:105) ± (cid:107) (cid:105)(cid:105) (cid:17) (5+ √ / / √ (cid:16) (cid:107) (cid:105)(cid:105) ± (cid:107) (cid:105)(cid:105) (cid:17) − (5 −√ / / √ (cid:16) (cid:107) (cid:105)(cid:105) ± (cid:107) (cid:105)(cid:105) (cid:17) • Exceptional, m = 11: (cid:88) r (cid:48) =1 ... , odd s (cid:48) =1 , , , (cid:40) α : s (cid:48) =1 ± β : s (cid:48) =5 β : s (cid:48) =7 ± α : s (cid:48) =11 (cid:41) S ( r,s ) , ( r (cid:48) ,s (cid:48) ) (cid:112) S (1 , , ( r (cid:48) ,s (cid:48) ) (cid:107) ( r (cid:48) , s (cid:48) ) (cid:105)(cid:105) – 17 – Exceptional, m = 12: (cid:88) r (cid:48) =1 , , , s (cid:48) =1 ... , odd (cid:40) α : r (cid:48) =1 ± β : r (cid:48) =5 β : r (cid:48) =7 ± α : r (cid:48) =11 (cid:41) S ( r,s ) , ( r (cid:48) ,s (cid:48) ) (cid:112) S (1 , , ( r (cid:48) ,s (cid:48) ) (cid:107) ( r (cid:48) , s (cid:48) ) (cid:105)(cid:105) where, in all cases, the upper choice of sign for the ± corresponds to the first classand the lower choice to the second. The above formulas give rise to an importantobservation: the two classes differ only by flipping the sign of a certain subset of theIshibashi states. This strongly suggests that the classes are related by the action ofa Z global symmetry. Our goal in this section will be to show that these models doindeed have a unique Z symmetry that exchanges the classes, while the remainingmodels have no such symmetry. Anomalous Z Symmetries We start by reviewing some basic formalism about Z symmetries. Our perspective issimilar to that described in [31]. The Hilbert space in the AP sector is H AP = H evenAP ⊕ H oddAP = (cid:77) i,j ∈ KT N ij V i ⊗ V j = (cid:77) i,j ∈ KT N ij =1 V i ⊗ V j where the multiplicities N ij can be read off from the partition function tables in Sec-tion 2. Again, these only take the values N ij = 0 , 1, allowing us to make the final step.A Z symmetry must act on each V i ⊗ V j by a sign, which we will denote as s ij = ± Z (cid:104) τ ; APAP (cid:105) = (cid:88) i,j ∈ KT N ij =1 s ij χ i ( τ ) χ j ( τ )Knowledge of this one partition function then allows further partition functions to bedetermined. This is because the partition functions on various backgrounds are relatedamong each other by acting on τ with modular transformations S ( τ ) = − /τ and T ( τ ) = τ + 1. The relationships we need are encoded by the diagram Z (cid:104) τ ; APAP (cid:105) Z (cid:104) τ ; APAP (cid:105) Z (cid:104) τ ; APP (cid:105) Z (cid:104) τ ; APP (cid:105) Z (cid:104) τ ; PAP (cid:105) Z (cid:104) τ ; PAP (cid:105) T S T SS T – 18 –hich allows all partition functions to be determined from the one on the top-left. Aconsistent Z symmetry is one for which all these partition functions admit a sensibleexpansion into Virasoro characters. For most of the backgrounds, APAP APP APP PAP this means a sum weighted by integers. However for the two special backgrounds APAP PAP the partition function is untwisted by any symmetries, and so we impose the strongerconstraint that the weights be nonnegative integers.Actually, what we have described is not quite correct for anomalous symmetries. Inthis case we must weaken the above requirements in several ways. First, the diagramneed only hold projectively, meaning some of the relationships expressed by the edgesare violated by a phase. Second, the partition functions themselves are ambiguouslydefined up to a phase. By adjusting these phases if necessary, which amounts to makinga choice of gauge for the diagram, we can always cast the diagram into the form Z (cid:104) τ ; APAP (cid:105) Z (cid:104) τ ; APAP (cid:105) Z (cid:104) τ ; APP (cid:105) Z (cid:104) τ ; APP (cid:105) Z (cid:104) τ ; PAP (cid:105) Z (cid:104) τ ; PAP (cid:105) T S T = e + πik/ SS T = e − πik/ (4.1)where the phase violations are expressed by the notation Z [ τ ; A ] T = e iθ −−−−− Z [ τ ; B ], whichmeans that Z [ τ + 1; A ] = e iθ Z [ τ ; B ] and vice-versa. The integer k is the strength ofthe anomaly, and is valued mod 8. To use the diagram, we start with the top-left partition function, and determine theother partition functions and the integer k by insisting that the diagram commutes. If The argument for why this pattern of phases is universal is that the holonomy of a closed loop inthe diagram is the value of exp( − iπ η ( D )) k on a suitable mapping torus, where D is a certain 3d Diracoperator. Then because the phases are universal, they can be read off from the Majorana fermion. Formore details, see [25], and for this particular example, also [24]. Another, more concrete way to seethe pattern of phases is to use the identities S = ( ST ) = 1, and the fact that the top-left partitionfunction is invariant under T – though with this approach the quantisation of k is less obvious. Although k appears as the exponent of a 16th root of unity, the shift k → k + 8 is a gaugetransformation, so k is valued mod 8 not mod 16. We could have made k manifestly mod-8 valued bymaking a different gauge choice. But the one we have chosen is more convenient in the long run. – 19 –o such integer k can be found, we do not have a consistent symmetry. Otherwise wedemand that the partition functions are sensible, as before. But because we may haveneeded to adjust their phases to gauge-fix the diagram, it only makes sense to demandthat they have a sensible expansion into characters up to an overall phase.There is one final consequence of the anomaly, and that is the possible appearanceof an unpaired Majorana mode in the frustrated sectors. This means that for thebackgrounds APAP APP PAP PAP with a symmetry defect that wraps vertically, we should allow a possible overall factorof √ Solving the Constraints We turn now to the task of solving the above constraints. To do this, we will recast asubset of the constraints as a set of matrix equations, solve them, and then check thatthe resulting solutions satisfy the remaining constraints.We begin by deriving the set of matrix equations. The partition function on APAP is encoded by the matrix A ij = N ij s ij where we recall that N ij is a known matrix of 0s and 1s encoding the partition functionon APAP , and the s ij are a set of unknown signs – unknown except for that of the identityoperator, which we set to be s = +1. Meanwhile, the corresponding matrix for the APAP partition function must take the form[ √ e iθ B ij where B is an unknown matrix of nonnegative integers, the phase θ is arbitrary, andthe √ APAP and APAP are related by an S -transformation. This fact is expressed by S A S = [ √ e iθ B Actually, since S is real, the phase e iθ must equal ± 1, so we obtain S A S = ± [ √ B (4.2)– 20 – second equation arises by considering a T transformation of APAP . Such a transfor-mation preserves the background, but contributes an anomalous phase of e πik/ . Thisfact is expressed by T − B T = e πik/ B (4.3)Equations (4.2) and (4.3) will be all we need to determine the symmetries. We shallanalyse them separately depending on whether the √ If (4.2) contains no √ 2, then we can easily show that the only solutions are the trivialsymmetry and fermion parity. To do this, we will make use of various Galois-theoreticresults that were used extensively to solve the bosonic version of this problem [7]. Asthese results will play an important role both in this section and the next, we beginwith a brief review of these ideas.The entries of the modular S -matrix belong to the cyclotomic field Q ( ζ n ) where n = 2 m ( m + 1), and ζ n = e πi/n is an n th root of unity. This field is acted on by Galoistransformations σ h , labelled by elements h ∈ Z ∗ n , via σ h ( ζ n ) = ζ hn The action of σ h on the modular S -matrix is [32] σ h ( S ij ) = (cid:15) h ( i ) S σ h ( i ) j where (cid:15) h ( i ) is a sign given by (cid:15) h ( r, s ) = η h (cid:15) m ( hr ) (cid:15) m +1 ( hs )while σ h is a permutation that will not be of interest to us. Here we have also introduced η h = σ h ( √ n ) / √ n , a computable but irrelevant sign, as well as another sign (cid:15) m ( x ) = sign sin (cid:16) πxm (cid:17) known as an su (2) affine parity, defined for all x (cid:54) = 0 mod m .We can use these facts to derive a useful constraint on B ij . Starting from (4.2),applying σ h , and comparing the result back to (4.2) gives B ij = (cid:15) h ( i ) (cid:15) h ( j ) B σ h ( i ) σ h ( j ) – 21 –rucially, all the entries of B ij are nonnegative. This means that if ever the product ofsigns (cid:15) h ( i ) (cid:15) h ( j ) equals − h , then both sides must be zero. We learn that B ij obeys the parity rule B ij (cid:54) = 0 only if (cid:15) h ( i ) = (cid:15) h ( j ) ∀ h (4.4)The set of all pairs ( i, j ) obeying this condition was determined in [7], and by Result 4of that paper, they have conformal dimensions satisfying h i − h j ∈ Z ∪ Z ∪ Z Now we recall (4.3), whose ( i, j )th component reads e πi ( h j − h i ) B ij = e πik/ B ij The phase on the left hand side can never be a nontrivial power of an eight root ofunity. We learn that k = 0, or in other words, that the symmetry is non-anomalous.The existence of a non-anomalous symmetry is a powerful statement. It implies thatif we perform a GSO projection, then the symmetry survives in the resulting bosonictheory [33–36]. Let us call the original symmetry α , and denote by ι ( α ) its imagein the bosonic theory. Then ι ( α ) commutes with ι (( − F ). In the bosonic minimalmodels, the only such symmetry is ι (( − F ) itself, hence ι ( α ) = ι (( − F ), which inturn forces α = ( − F . We therefore learn that the original symmetry must have beeneither trivial or fermion parity, as claimed.We note that it would have also been possible to derive this conclusion directly fromequations (4.2) and (4.3), using technical arguments entirely parallel to those in [7].However the above argument, similar to those used in [37], is more transparent. In the more novel case that (4.2) contains a √ 2, we show that solutions exist only for m = 3 , m = 11 , 12 in the exceptionals.As a first step we show that m = 3 , √ ∈ Q ( ζ n )This is true if and only if n is a multiple of 8, which in turn requires m = 0 , B ij . We find that it is modified bythe presence of the √ 2, and now takes the form B ij (cid:54) = 0 only if (cid:15) h ( i ) = f ( h ) (cid:15) h ( j ) ∀ h (4.5)Here f ( h ) = σ h ( √ / √ √ 2, and takes the explicit form f ( h ) = (cid:40) +1 h = 1 , − h = 3 , i, j ), we write it in terms of affine parities as (cid:15) m ( hr i ) (cid:15) m +1 ( hs i ) = f ( h ) (cid:15) m ( hr j ) (cid:15) m +1 ( hs j ) ∀ h ∈ Z ∗ n Let us assume that m = 0 mod 4, since the case m = 3 mod 4 can be treated almostidentically. Then the above condition is equivalent to (cid:15) m ( hr i ) = f ( h ) (cid:15) m ( hr j ) ∀ h ∈ Z ∗ m (cid:15) m +1 ( hs i ) = (cid:15) m +1 ( hs j ) ∀ h ∈ Z ∗ m +1) The solutions to the second equation were determined in [7], where Result 3 states that s i = s j or s i = m + 1 − s j . Meanwhile, the solutions to the first equation are determinedby the following conjecture: Conjecture Suppose n is a multiple of 4, 1 ≤ x, y ≤ n − 1, and (cid:15) n ( hx ) = f ( h ) (cid:15) n ( hy )for all h ∈ Z ∗ n . Then the only solutions are • n = 4 and ( x, y ) = (1 , • n = 8 and ( x, y ) = (1 , • n = 12 and ( x, y ) = (1 , , (2 , , (4 , • n = 24 and ( x, y ) = (1 , , (5 , n, x, y ) → ( n, y, x ) , ( n, n − x, y ) , ( n, x, n − y ) , ( kn, kx, ky )It is computationally trivial to verify this conjecture up to n = 1000, and we expect,but have not proved, that it holds for all n .We now know all solutions to (4.5). Without loss of generality, s i = s j . (This usesthe equivalence ( r, s ) ∼ ( m − r, m + 1 − s ) of the Kac Table.) The values of r i and r j are given by – 23 – if m = 4 a , ( r i , r j ) = ( a, a ), • if m = 8 a , ( r i , r j ) = ( a, a ), • if m = 12 a , ( r i , r j ) = ( a, a ) , (2 a, a ) , (4 a, a ), • if m = 24 a , ( r i , r j ) = ( a, a ) , (5 a, a ),and those related to them under ( r i , r j ) → ( r j , r i ) , ( m − r i , r j ) , ( r i , m − r j ).Next, we use this result to compute the possible values of the phase e πi ( h i − h j ) amongall solutions. Using the identity 2( h i − h j ) = ( m + 1) r i − r j m mod 1, we find the followingcontributions to the set of values taken by e πi ( h i − h j ) : • m = 4 a → ζ ± a • m = 8 a → ( − a • m = 12 a → ζ ± a , ζ ± a • m = 24 a → e πi ( h j − h i ) B ij = e πik/ B ij In view of the possible phases that can arise on the left hand side when B ij (cid:54) = 0, it iseasy to see that the anomaly k must satisfy m = 4 mod 8 = ⇒ k = odd m = 0 mod 8 = ⇒ k = evenAll of the above analysis also goes through unchanged if instead m = 3 mod 4. Com-bining the results from the two cases, we conclude that m = 3 , ⇒ k = odd m = 0 , ⇒ k = evenAn even value of k is incompatible with our assumption of a √ k were even, then we could stack with an even number 8 − k of copies of the Majorana fermion, obtaining a system with no anomaly yet still withan unpaired Majorana zero in the frustrated sector. This gives a contradiction, since– 24 –he presence of an unpaired zero mode is an exclusive feature of anomalous theories.The only way to avoid this contradiction is if m = 3 , m = 0 , B ij failing to be a nonnegative-integer matrix.The only exceptions are m = 7 , 8, where there is a single solution for B ij , but this thengoes on to fail (4.3).To make further progress with the remaining cases, we deal with the infinite seriesand the exceptional models in turn. For the infinite series when m = 3 , N ij takes the form N = (cid:77) i ∈ KT / (cid:48) ξ i =1 i i (cid:48) (cid:18) (cid:19) i i (cid:48) i = ( r, s ) ∈ KT there is anassociated sign ξ i = ( − ( m +1) r + ms +1 One can show that ξ i = ξ i (cid:48) , and that ξ i = +1 = ⇒ i (cid:54) = i (cid:48) . (Here ( r, s ) (cid:48) = ( m − r, s ) isthe involution (3.4) introduced in Section 3.) These facts are necessary to ensure theabove block decomposition makes sense.First we use the consistency conditions to constrain the form of A ij , the matrixof unknown signs corresponding to APAP , using arguments analogous to [7]. Because S N S = N , which follows from S -invariance of APAP , N ij obeys the parity rule (4.4).Since A ij = N ij s ij , so too does A ij . By acting on (4.2) with a Galois transformation,comparing it back to (4.2) and invoking the previous fact, we find that A ij obeys a modified permutation rule A σ h ( i ) σ h ( j ) = f ( h ) A ij This states that the many unknown signs in A ij are in fact related. Rather thanusing the full power of this equation, though, we shall only use it for the special case– 25 – = m ( m + 1) − 1. In this case σ h ( i ) = i (cid:48) and f ( h ) = − 1, so we obtain A i (cid:48) j (cid:48) = − A ij This states that the unknown signs within each 2 × A ij are related, and that A ij takes the form A = (cid:77) i ∈ KT / (cid:48) ξ i =1 i i (cid:48) (cid:18) (cid:19) i (cid:15) i η i i (cid:48) − η i − (cid:15) i (4.6)where (cid:15) i and η i are a set of unknown signs associated to the elements of KT / (cid:48) , and thesign corresponding to the identity operator is (cid:15) = +1.We are now in a position to rule out all but a finite number of models as having nosymmetries. Recall that B ij obeys the modified parity rule (4.5). By the conjecture,this implies that unless m ∈ { , , , } , we have B i = B i = 0By (4.2) this tells us that A annihilates the vector S i from both sides. Using (4.6) andthe fact S i = S i (cid:48) > 0, we obtain the contradictory equations (cid:15) i = ± η i showing immediately there are no solutions.Finally we return to the special cases m ∈ { , , , } that were exempt from theabove no-go analysis. Listing the solutions for these cases is a purely finite problem,which can easily be done manually. We find the following results: • m = 3 The m = 3 model has two symmetries. We specify them by writing outthe partition function diagram (4.1) explicitly. For the first one, we have( χ + χ )( χ − χ ) √ χ + χ ) χ √ χ − χ ) χ ( χ − χ )( χ + χ ) √ χ ( χ + χ ) √ χ ( χ − χ )with anomaly k = 1. The other symmetry is given by flipping the diagramupside down, an operation which corresponds to composing with ( − F , and hasthe opposite anomaly k = − 1. These symmetries are of course simply left andright chiral fermion parity of the Majorana fermion.– 26 – m = 4 The m = 4 model has a similar structure, with two symmetries relatedby ( − F and opposite anomalies. We shall therefore only give details of the firstone. The partition function diagram (4.1) in this case takes the form( 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 ) √ χ ( χ + χ )+ χ ( χ + χ ) ] √ χ ( χ − χ )+ χ ( χ − χ ) ]and the anomaly is k = − • m = 11 , 12 The models with m = 11 , 12 turn out to have no symmetries. The above analysis can also be carried out for the two exceptional models at m = 11 , m = 3 , − F , with opposite anomalies. Below we list the symmetriesfor the two models. This time we will only list the partition functions on APAP and APAP , as all others are related to these by the same pattern as in previous cases. • m = 11 For the m = 11 exceptional, we have Z (cid:104) τ ; APAP (cid:105) = (cid:88) r =1 , odd ( χ r, + χ r, + χ r, + χ r, )( χ r, + χ r, − χ r, − χ r, ) Z (cid:104) τ ; APAP (cid:105) = √ (cid:88) r =1 , odd ( χ r, + χ r, + χ r, + χ r, )( χ r, + χ r, )with anomaly k = − • m = 12 The m = 12 exceptional is similar, but with the roles of r and s reversed Z (cid:104) τ ; APAP (cid:105) = (cid:88) s =1 , odd ( χ ,s + χ ,s + χ ,s + χ ,s )( χ ,s + χ ,s − χ ,s − χ ,s ) Z (cid:104) τ ; APAP (cid:105) = √ (cid:88) s =1 , odd ( χ ,s + χ ,s + χ ,s + χ ,s )( χ ,s + χ ,s )and anomaly k = 1. – 27 –efore going on, we pause to note that the earlier results for the infinite series can alsobe rewritten to look more like the results above. For m = 3, we have Z (cid:104) τ ; APAP (cid:105) = (cid:88) r =1 , odd ( χ r, + χ r, )( χ r, − χ r, ) Z (cid:104) τ ; APAP (cid:105) = √ (cid:88) r =1 , odd ( χ r, + χ r, ) χ r, while for m = 4 we have Z (cid:104) τ ; APAP (cid:105) = (cid:88) s =1 , odd ( χ ,s + χ ,s )( χ ,s − χ ,s ) Z (cid:104) τ ; APAP (cid:105) = √ (cid:88) s =1 , odd ( χ ,s + χ ,s ) χ ,s We conclude with some comments on our results. First we return to the earlier claimthat the anomalous symmetries, where they exist, exchange the two classes of boundarystates. This follows at a glance from the APAP partition functions of Section 4.2. Indeedthe coefficient of | χ i | in this partition function determines the sign with which thesymmetry acts on the Ishibashi state (cid:107) i (cid:105)(cid:105) . For example, in the m = 11 exceptional, thecharges of Ishibashi states are (cid:107) ( r, (cid:105)(cid:105) (cid:107) ( r, (cid:105)(cid:105) (cid:107) ( r, (cid:105)(cid:105) (cid:107) ( r, (cid:105)(cid:105) +1 − − k being odd. This followsby a stacking argument. Suppose a theory has a boundary state | a (cid:105) and a symme-try U with odd anomaly k . Then if we stack the theory with − k mod 8 copies ofthe Majorana fermion, the resulting theory has a boundary state | a (cid:105) ⊗ | + (cid:105) − k and anon-anomalous symmetry U ⊗ ( − F L . Acting on the state with the symmetry gives anew state U | a (cid:105) ⊗ |−(cid:105) − k , which must lie in the same class since the symmetry is non-anomalous. From Section 1.1, we know that | + (cid:105) − k and |−(cid:105) − k are in different classeswhen k is odd. Therefore so too must | a (cid:105) and U | a (cid:105) , as claimed.– 28 –e would also like to return to a subtlety we felt was best left unaddressed inSection 4, but are now in a position to close. A theory with a Z symmetry actuallyhas six more partition functions that we did not consider. These fit into an orbitdiagram, analogous to (4.1), which looks like Z (cid:104) τ ; PAP (cid:105) Z (cid:104) τ ; APP (cid:105) Z (cid:104) τ ; APAP (cid:105) Z (cid:104) τ ; PP (cid:105) Z (cid:104) τ ; PP (cid:105) Z (cid:104) τ ; PP (cid:105) T S = ζ − k T = ζ k S = ζ − k T S = ζ k T = ζ − k S = ζ k In the models with an anomalous symmetry, all these partition functions are zero.This is because when k = ± 1, the diagram violates the relations S = ( ST ) = 1 thatmust be satisfied by any minimal model partition functions, so there are no nonzerosolutions. With these partition functions now in hand, one might ask why we did notdemand the consistency of symmetry-projected traces, such as12 (cid:16) Z (cid:104) τ ; PAP (cid:105) + Z (cid:104) τ ; PP (cid:105)(cid:17) which counts bosonic states on a periodic circle frustrated by a symmetry defect. Theanswer becomes clear if we look at the Majorana fermion. Here, the answer would be1 √ χ + χ / )( τ ) χ / ( τ )which is inconsistent even for a fermionic theory. The issue is of course that with anunpaired Majorana mode, there is no separation into fermionic and bosonic states, andsuch symmetry-projected traces are meaningless. We conclude that there is no needto demand consistency of the symmetry-projected traces, and the constraints we haveimposed on our symmetries are all that there are.Next we attempt to shed some light on the observation that, from our results, thefollowing properties of fermionic minimal models appear to be equivalent:1. equalclass sizes ⇐⇒ 2. existence of ananomalous symmetry ⇐⇒ 3. vanishing of the PPsector partition functionIndeed, all three conditions are satisfied by m = 3 , m = 11 , 12 from the exceptionals. It is natural to ask whether the above superfi-cial equivalences are in fact honest equivalences. Below we will outline some argumentsthat show that for some for them at least, the equivalences are indeed honest.– 29 – ⇒ As we have seen, when an anomalous symmetry is present it exchanges the twoclasses of boundary states. This trivially implies they have equal sizes. ⇒ Here we can argue the contrapositive as follows. SPT classes form an affine space:they can be compared, but there is no preferred choice of one phase as trivial. Ifhowever a model has boundary state classes of different sizes, then we seeminglyhave a way to distinguish one class over the other. But this is not a contradiction,as we should remember that for every fermionic minimal model listed in Section 2,there is another related by stacking with ( − Arf [33–35], and whose boundarystate class sizes are reversed. So it is acceptable for a theory to have differentclass sizes as long as Z and Z ( − Arf are different theories. The condition forthis is that Z does not vanish in the PP sector. ⇒ This observation, that anomalies can force the vanishing of the Ramond sector,has been noted in various places in the literature. See for example [23], where itwas explained using the algebra obeyed by ( − F L , ( − F R and ( − F .For the above implications, we do not know of arguments in the other direction.Finally we comment on our Conjecture in Section 4.2. In earlier work on the bosonicminimal models, the role of this conjecture was played by Result 14 of [7]. This resultwas proved by relating it to a theorem about the classification of simple factors ofJacobians of Fermat Curves [38]. Given the close connection between the fermionic andbosonic minimal models, it is natural to ask if our conjecture has a similar interpretationin terms of Fermat curves. Whether or not it does remains an open question. Acknowledgements The author thanks Shu-Heng Shao for the initial idea for this paper, David Tong forguidance and comments on the draft, Joe Davighi and Nakarin Lohitsiri for manydiscussions about anomalies, and Pietro Benetti Genolini, Avner Karasik and CarlTurner for other enlightening discussions. 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