Towards a Classification of Two-Character Rational Conformal Field Theories

We provide a simple and general construction of infinite families of consistent, modular-covariant pairs of characters satisfying the basic requirements to describe two-character RCFT. These correspond to solutions of generic second-order modular linear differential equations. To find these solutions, we first construct"quasi-characters"from the Kaneko-Zagier equation and subsequent works by Kaneko and collaborators, together with coset dual generalisations that we provide in this paper. We relate our construction to the Hecke images recently discussed by Harvey and Wu.


Contents
3 Two characters: complete classification for ℓ < 6 1 Introduction The traditional classification of Rational Conformal Field Theories (RCFT) in 2d is based on either (chiral) symmetry algebras or lattices. In the former approach, after selecting a particular chiral algebra or set of algebras, one uses the structure of null vectors to write down a minimal series of CFT's that furnish realisations (not necessarily unitary) of the given algebra [1,2] (for more examples see [3] and references therein). Among possible chiral algebras, Kac-Moody algebras are special because one can consider affine models containing all integrable primaries at a given level, namely WZW models, and then take cosets of them. This coset construction generates a vast supply of RCFT's including the various minimal series already obtained by null-vector methods. The approach based on lattices includes the famous c = 1 classification [4,5] and the constructions based on even, self-dual lattices as reviewed for example in [6]. These two approaches, the chiral algebra/WZW coset approach and the lattice approach, are not complete -we know RCFT's that belong in neither of these classes [7,8]. Moreover, there are simple coset models which do not come from affine WZW theories [9]. Hence it can be useful to organise RCFT's not by these criteria but by simplicity, defined as the number of independent scaling exponents or primary fields, roughly the same as the number of characters.
The c = 24 models classified in [10] are examples of one-character CFT's where there is an extremely large chiral algebra, including many spin-2 generators. The only primary field of this extended algebra is the identity. Several two-character RCFT's have likewise long been known [11,12]. A complete classification of the simplest models could be more useful than the traditional approach for various purposes, including applications in condensed-matter systems as well as for studies from a mathematical viewpoint. An additional application of simple rational CFT's has emerged in the context of four-dimensional supersymmetric gauge theories [13][14][15].
An approach to the classification of RCFT with a small numbers of characters was proposed long ago in [11]. It is based on the observation that the characters of a rational CFT are vector-valued modular forms of weight zero and satisfy a Modular Linear Differential Equation (MLDE) (this fact was independently noted in [16]). Thus, to classify them we may start from a general class of MLDE and identify values of the parameters for which they determine "admissible" sets of candidate characters -those whose q-expansion has non-negative integer coefficients 1 . After finding admissible characters there are well-defined procedures, using the modular transformation matrix S and its relation to the fusion rules via the Verlinde formula [17], to determine whether they truly describe a consistent RCFT, and if so to solve the resulting theories. These procedures were implemented in considerable detail in [12,18] and some recent work in this direction can be found in [19]. There is also now a considerable mathematical literature on using MLDE to find vector-valued modular forms of CFT type, of which some relevant works are [20][21][22][23][24][25][26][27][28] and additional ones will be described in what follows.
Modular linear differential equations are characterised by an integer n, the number of characters, which is also the order of the differential equation, as well as an integer ℓ ≥ 0, ℓ = 1 which specifies the number of zeroes in moduli space of the Wronskian determinant of the solutions (this concept will be explained in more detail below). In [11] all admissible characters with n = 2, ℓ = 0 were classified and form a finite set. All but one have been identified with definite RCFT's, while one has a degenerate vacuum and was initially rejected for this reason. It is now understood to be an Intermediate Vertex Operator Algebra (IVOA) [29] based on the notion [30] of Intermediate Lie Algebra. Related mathematical work on the 2nd order ℓ = 0 equation may be found in [31][32][33][34][35][36].
Subsequently the case n = 2, ℓ = 2 was classified in [12,37] 2 . Again there is a finite set of admissible characters. A remarkable property of these characters, shown in [9], is that they are in one to one correspondence with the ℓ = 0 theories and satisfy a "generalised coset" relationship with them. This in particular proves that the ℓ = 2 characters correspond to consistent RCFT's. Like its counterpart at ℓ = 0, one of them has a degenerate vacuum and in this respect resembles an IVOA. We will refer to characters having a degenerate vacuum state as being of "IVOA type" although there is no implication that an intermediate Lie algebra as in [29,30] is involved.
The status for larger values of ℓ is as follows. It was argued in [12] and explicitly verified in [37] that there are no admissible characters with ℓ = 3. In fact [12] argued that with two characters, there can be no admissible solutions for any odd ℓ. For ℓ = 4, [12] observed that admissible solutions can exist, and searched for them, but then concluded that there are no new ones other than tensor products. More recently, [37] failed to find any models with ℓ = 4 but they did not make an exhaustive search. However, admissible characters with ℓ = 4 do in fact exist. Most of these correspond to tensor product RCFT's, while precisely three of them turn out to be irreducible in this sense and we will describe them below. Two of them first appeared in [39], whose authors restricted their study to potentially unitary CFT's. Out of the three that we find, the first two agree with [39] while the third is in the Lee-Yang class, hence the corresponding CFT -if any -would be non-unitary.
To summarise, for two-character theories the parameter ℓ labelling the differential equation has to be odd, and for ℓ < 6 the classification leads to a finite set of characters for ℓ = 0, 2 that have been identified as RCFT's, while for ℓ = 4 there is also a finite set, most of which are tensor products but a few are new admissible characters. They are yet to be identified as RCFT's.
The classification for ℓ ≥ 6 is more complicated and has remained largely unexplored until recently. It is known that taking tensor products increases the value of ℓ, but of course we would like to discover "irreducible" theories that are not tensor products. It has long been known [18] that irreducible RCFT's with ℓ arbitrarily large do exist. A simple example noted in [37] lies on the orbifold line of a free boson theory with c = 1 at radius R = √ 2p. This theory has p + 4 characters and ℓ = 3(p − 1) for arbitrary integer p ≥ 1. However, this series starts at five characters. The first significant insight into this question for two-character theories, from the point of view of classification, was recently provided by Harvey and Wu in [40]. These authors defined a novel type of Hecke operator that acts on vector-valued modular forms to provide new ones. In this process the number of characters is preserved, and under certain conditions the positivity of coefficients in the q-expansion is also preserved, so that the Hecke images are admissible characters. These Hecke images have increasing ℓ and it is easy to find admissible characters with arbitrarily large values of ℓ (as well as the central charge c).
In this paper we re-examine the classification problem for ℓ ≥ 6. We find two distinct methods, both based on a notion of "quasi-characters" which we introduce, that generate infinite families of new admissible characters with arbitrarily large values of the parameter ℓ and the central charge c.
The organisation of this paper is as follows. In Section 2 we start by reviewing onecharacter (often called "meromorphic") CFT's and place them in the context of the modular linear differential equation (MLDE) approach. While the results are not new, some of the details are not widely appreciated and will be useful to provide a context for the ensuing discussion of two-character theories. Thereafter in Section 3 we discuss in some detail the status of two-character RCFT with ℓ < 6. In addition to the review material there will be some new results that contradict previous claims. We will also highlight several important subtleties.
In Section 4 we summarise several ways in which characters solving a 2nd-order MLDE can "fail" to be admissible. In this category are pairs of characters where one is modular invariant on its own, pairs that give rise to negative but nonetheless integral fusion rules, characters where one achieves integrality of Fourier coefficients only if the vacuum state is chosen to be degenerate, and pairs with exponents that differ by an integer, which correspond to logarithmic CFT. But the most important category for us, neglected in many previous works, are those where all the "degeneracies" are integers but some are negative. We call these "quasi-characters" and will find that they are useful building blocks for admissible characters. We go on to show that a certain parametrisation introduced by Kaneko and collaborators [31][32][33] gives rise to infinite families of such quasi-characters with ℓ = 0. We generalise these considerations to find a fresh series of quasi-characters with ℓ = 2. The two sets turn out to obey the coset/bilinear relation of [9] just as the genuine admissible characters do.
Finally in Section 5 we present our main results: that infinite sets of admissible characters with arbitrarily large values of ℓ (and c) can be constructed using quasi-characters. We describe two ways to do this. One of these ways involves multiplication of a quasicharacter with a suitable function of the j-invariant, and the other involves taking linear combintations of different quasi-characters. A key outcome is that for ℓ ≥ 6 one can find infinitely many admissible characters with the same central charge, though it is also easy to construct examples with large ℓ as well as c. We then compare the results of our constructions to the Hecke image approach of [40]. We conclude by listing a number of open questions raised by our work.

Basic features
We start our discussion with one-character theories because they carry some important lessons. Such theories have a partition function of the form: where χ(τ ) is holomorphic. Defining q = e 2πiτ , the character behaves near τ → i∞ as where c is the central charge. It has long been known that for this partition function to be invariant under SL(2, Z), c must be a multiple of 8. For c = 8 there is a unique character χ E 8,1 that is modular invariant upto a phase. The resulting CFT is the WZW model for the algebra E 8 at level k = 1. For c = 16 there is again a unique character, where j(τ ) is the Klein j-invariant, which happens to be equal to χ E 8,1 3 , and N is, at this stage, any real number. Note that modular invariance alone puts no condition on N . Thus at c = 24 there appear to be potentially infinitely many characters, depending on the added constant N . Now we need to invoke a physical requirement. For the resulting partition function to describe a conformal quantum field theory, the coefficients in the expansion of χ(τ ) in powers of q must describe degeneracies of states. Hence they must be non-negative integers. This is already true of all the coefficients in the expansion of j, but can potentially be violated by our choice of the constant N . Indeed the q-expansion of the above character is: We see that the second coefficient is integer only if N is integer, and is non-negative if N ≥ −744. This still leaves us with an infinite set of admissible characters that might describe a CFT, one for every integer N ≥ −744. The next question is whether all of these describe actual CFT's. This is precisely the problem that was addressed in 1991 by Schellekens [10], who argued that there are just 71 consistent single-character CFT's with c = 24. Some features of his result are worth noting here: (i) there are less than 71 different characters that correspond to consistent CFT's, because in many cases a given character describes more than one CFT. This is analogous to the situation at c = 16 where there is a unique character but two different CFT's, (ii) the minimal value of N , namely −744, is realised as a rather special CFT. In this case the expansion of the character is missing the term of order q relative to the leading term. This means the CFT, if there is one, does not have any Kac-Moody symmetry (because among all possible chiral symmetry algebras of spin-n, only spin-1 generators create a secondary at the first level above the identity state). And indeed, not only is there a CFT at this value but it is a celebrated one: the Monster CFT [41,42].
Beyond c = 24, as observed for example in [11], the single character of a one-character theory must be a function of the Klein j-invariant of the form: Notice that the c = 8 and c = 16 characters arise by choosing δ = 1 3 , 2 3 respectively with β = 0 and P wτ (j) = 1. Clearly there is an infinitely number of potential characters with c > 24, and by examining the behaviour as q → 0 we find the central charge of the corresponding CFT (if there is one) would be But as we already noted in the case of c = 24, only a subset of these will be admissible as characters, namely those whose Fourier coefficients are all non-negative integers (in particular this subset has w i = 0). By carefully choosing the polynomial one can find the admissible characters. The next step would be to repeat the procedure of Schellekens and find out how many of these correspond to consistent CFT's. Unfortunately for c ≥ 48 this seems a very hard problem and remains open at present.

Modular linear differential equations
Let us now phrase the above results in the language of differential equations. Though essentially trivial, this will give a nice introduction to the integer ℓ mentioned above, and set the stage for the the study of 2-character theories for arbitrary ℓ. We start by noting that for any of the 1-character theories, the single character satisfies the trivial equation: The purpose of writing this equation is that it provides a way to "experimentally re-discover" the characters. To do this, we rewrite the above equation as: and allow φ 2 to be an arbitrary function of τ . Since the character is modular invariant (upto a phase) it has weight 0 under SL(2, Z). It is easy to see that differentiating in τ increases the modular weight by 2. Hence for the equation to be invariant, the function φ 2 (τ ) must transform with weight 2 under SL(2, Z) modular transformations: One may think that φ 2 has to be holomorphic, but in fact it is generically meromorphic. Referring to Eq. (2.6), we see that φ 2 has a pole whenever χ has a zero. And despite φ 2 having such a pole, the solution of the differential equation is perfectly holomorphic (away from τ → i∞, as always). We now classify the differential equation by the number of zeroes of χ. This is generically the same as the number of poles of φ 2 = − Dχ χ , though sometimes the zeroes of χ could be partly cancelled by zeroes of Dχ. Because of the nature of the torus moduli space where τ takes values, the number of poles of any meromorphic function is quantised as ℓ 6 where ℓ = 0, 2, 3, 4, · · · . The reason why fractional powers arise is that the space has orbifold points at τ = ρ ≡ e 2πi 3 of order 3, and τ = i of order 2. As a result, one can have poles of order 1 3 and 1 2 respectively at these points, corresponding to ℓ = 2, 3 respectively. One gets ℓ = 4 by having coincident poles at ρ, and ℓ = 5 by having a pole each at i and ρ. Finally with ℓ = 6 one can have a pole anywhere in moduli space (we call this a "bulk pole"). Bulk poles can be present in principle for all higher values of ℓ except ℓ = 7.
It will be useful to note that all modular forms under SL(2, Z) are polynomials in the Eisenstein series E 4 and E 6 of weight 4 and 6 respectively. If one wants meromorphic forms, they can be constructed by taking ratios. Due to the location of the zeroes of E 4 and E 6 , when they appear as factors in the denominator they contribute ℓ = 2 and ℓ = 3 respectively.
Since the moduli space is compact, the total number of zeroes and poles of the character must be equal. This is a special case of the Riemann-Roch theorem. The pole is only at infinity, where we assume the behaviour χ ∼ q α . We have the identification α = − c 24 . Meanwhile the number of zeroes is just ℓ 6 . It follows that: or equivalently c = 4ℓ (2.9) which from Eq. (2.5) tells us that ℓ 6 = w ρ + w i + w τ . Now let us try to build Eq. (2.7) starting with the simplest case, ℓ = 0. There are no holomorphic modular forms of weight 2, so we must have φ = 0. The unique solution is χ = 1 (we are free to choose the normalisation). This is the trivial character. The next case is ℓ = 2. In this case we must take φ 2 = µ E 6 E 4 , where for the moment, µ is an arbitrary real coefficient. Since the q-expansion for the Eisenstein series is known, the differential equation can be solved order by order in q and gives rise to a character that is modular invariant (upto a phase) for every value of the free parameter µ. The resulting character will have a power-series expansion of the form: Note that we have chosen the coefficient of the leading term to be 1. This is because the corresponding state is the vacuum state of the theory, which is expected to be unique. Since ℓ = 2, from Eq. (2.9) we have α = − 1 3 , so the central charge of the associated CFT (if it exists) is c = 8. Solving recursively to the first two orders, we find: µ = −α, a 1 (µ) = 744µ (2.11) which in turn tells us that µ = 1 3 and a 1 = 248. Subsequent orders will determine a 2 (µ), a 3 (µ) etc. The remarkable fact is that all of these numbers are non-negative integers. Moreover, the value of a 1 determines the dimension of the Kac-Moody algebra of the theory, in this case 248. Thus we have rediscovered the E 8,1 character.
One can repeat this exercise for ℓ = 3 by writing the equation: This time, Riemann-Roch tells us that α = − 1 2 which means c = 12. Recursively solving the equation gives: In this case we see that at the first level above the ground state, the "degeneracy" is a negative integer. This fails the test to be an admissible character and therefore there is no one-character CFT with c = 12. Nevertheless, it turns out that this case satisfies a weaker consistency condition: the degeneracies, although sometimes negative, are nonetheless all integers. In what follows we will find that such "quasi-characters" can be useful as building blocks, though by themselves they do not describe any CFT.
It is straightforward to continue in this way. At ℓ = 4 one finds a unique character having c = 16 and a 1 = 496, which is the dimension of E 8 ⊕ E 8 as well as Spin(32)/Z 2 . The case of ℓ = 5 has φ 2 = µ 1 Again we find that the character has integral coefficients, but some are negative. At ℓ = 6 we have φ 2 = µ 1 1728 is the modular discriminant (in our conventions). As expected, this has a "bulk pole" where E 3 4 + µ 2 ∆ = 0. This case has α = −1, corresponding to c = 24. Inserting this and solving the equation recursively, we find: Continuing to solve for the character to one higher order, we find that a 2 = 196884, independent of the value of µ 2 , and all subsequent terms are non-negative integers and independent of µ 2 . From the above equation it is clear that to have an admissible character we must take µ 2 = N , an integer ≥ −744. Indeed, the character in this case is simply χ ℓ=6 = j(τ ) + N . As stated above, there are 71 consistent CFT's having characters in this set, while all the remaining (infinitely many) characters of this form seem to be unrelated to any CFT. The reader may have noticed that we could have simply integrated the differential equation Eq. (2.7) instead of solving it by a series expansion in q. Indeed there are far simpler ways to extract the above results on admissible characters, and Eq. (2.4) summarises the complete set of results for the one-character case, which have long been known in the mathematical literature -so there was no need to resort to differential equations in the first place. But the reason we have gone through this exercise is to highlight a number of features that will for the most part generalise to the two-character case, where a general solution analogous to Eq. (2.4) does not exist. The relevant features of the one-character case are: • there are admissible, holomorphic characters for arbitrarily large values of ℓ, • when ℓ lies in the range 0 ≤ ℓ < 6, there is a finite set of admissible characters for ℓ = 0, 2, 4 and none for ℓ = 1, 3, 5, • every admissible character in this range corresponds to a consistent CFT, • for ℓ ≥ 6 one encounters situations where the characters depend on arbitrary integers and are admissible over infinite ranges of these integers. However, most of them do not correspond to genuine CFT's.

The general theory and the map to hypergeometric equations
Analogous to Eq. (2.7) for the one-character case, we can write the general two-character differential equation. A new feature relative to the one-character case is that we need to use covariant derivatives, defined by: where k is a positive even integer and E 2 (τ ) is the second Eisenstein series, which is a holomorphic connection on moduli space. It transforms inhomogeneously under modular transformations, such that if f (k) (τ ) transforms with weight k under SL(2, Z) then: transforms with weight k + 2n. We denote D (0) simply by D. The special case D n (0) is the only one relevant for CFT characters, since those have zero weight 3 . Hence it will be simply written D n . With these definitions, the general MLDE for two characters is: where φ 2 , φ 4 are meromorphic and modular of weight 2,4. The characters that solve the equation take the form: Notice that we have treated the two characters somewhat asymmetrically. The first one, which we associate to the identity character, has the exponent α 0 = − c 24 . It is normalised so that its series expansion starts with 1, since this is the vacuum degeneracy and one normally expects a non-degenerate vacuum state. The series coefficients m i then have to be non-negative integers in order to be interpreted as degeneracies. The second character corresponds to the non-trivial primary field, and has the exponent α 1 = − c 24 + h where h is the (chiral) conformal dimension of the primary 4 . The second character is normalised by multiplying with a positive integer D which is the degeneracy of the primary state. The expansion coefficients a i do not need to be integers. It is sufficient if they are rational numbers with a denominator that divides D, so that the products D a i are integers. If one knows the characters then in principle one can compute the modular transformation matrix S defined as follows: If this matrix turns out to be unitary then the partition function: will be modular invariant. However, it is also possible that the non-identity character has a multiplicity M , corresponding to the fact that a single character corresponds to more than one primary. Such will be the case when there is some discrete symmetry such as complex conjugation that causes the primaries to appear in pairs or larger multiples. In this case, the modular-invariant partition function will be: It is important to distinguish the degeneracy D from the multiplicity M . The former tells us only that a particular primary is degenerate, the latter says that there are M different primaries, each with degeneracy D, whose character is the same. In the presence of a multiplicity, the matrix S ij will not be unitary but rather will satisfy: If the characters χ 0 , χ 1 were known, the coefficient functions φ 2 , φ 4 in the differential equation would be determined in terms of the Wronskians: as: The differential equation is labelled by the number ℓ 6 of zeroes of the Wronskian W (in the one-character case of Section 2 the Wronskian was just the character χ) which in turn labels the maximum allowed poles of the modular forms φ 2 , φ 4 . We will examine the solutions to this requirement below. But before we turn to that question, let us observe some general features. Suppose we consider the subfamily of differential equations where the poles are located only at ρ, i (as explained above, this is the general case for ℓ < 6 while it is a special case for ℓ ≥ 6). Then, as studied in [12,18], the differential equation Eq. (3.3) can be mapped to a hypergeometric equation and its solutions explicitly found in terms of hypergeometric functions. [12] went further and used the Klein j-invariant itself, known as a "hauptmodul" of SL(2, Z), as the variable for the differential equation. In terms of this, one can actually bypass the differential equation and write the characters as functions of j. Suppose the characters behave near the points τ = ρ, i, i∞ where j → 0, j → 1728, j → ∞ as: where r ρ , s ρ ≥ 0 and ∈ Z/3, while r i , s i ≥ 0 and ∈ Z/2. Then, the Riemann-Roch theorem tells us that: Equating α 0 , α 1 to − c 24 , − c 24 + h respectively, where h is the chiral conformal dimension associated to the character χ 1 , the above relation becomes: Defining: it was shown in [12] that: the characters can be expressed in terms of j as: Armed with these results, we first summarise all cases with ℓ < 6.
3.2 The case of ℓ = 1, 3, 5 The first equation in Eq. (3.15) already tells us that ℓ = 1 is not possible, given that the RHS is either 0 or else has a minimum value of 1 3 . Requiring that there are two independent solutions when expanded about the singular points ρ, i, imposes t ρ , t i / ∈ Z (if this were not true, then we would find a logarithmic behaviour at either of the special points). This immediately implies, using Eq. (3.14), that w i ∈ Z and ℓ ∈ 2Z. Thus, when ℓ is an odd integer, the space of solutions to the MLDE is one dimensional. This rules out two-character CFT with odd ℓ. Note that this argument doesn't change when there are additional zeroes elsewhere in the moduli space, the only condition being imposed is that the critical exponents at the singular points do not differ by an integer.

The case of ℓ = 0
The first attempt at classifying MLDE was carried out in [11] for the case of two characters and ℓ = 0. The map to the hypergeometric equation was not used there and was in fact proposed later, independently in [18] and [12]. From the above discussion it is clear that for ℓ = 0, the forms appearing in Eq. (3.3) are φ 2 = 0 and φ 4 = µE 4 where µ is a free parameter. Hence the equation is: We will refer to this as the MMS equation 5 . As shown there, the parameter µ in the equation is related to the central charge as µ = − c(c+4) 576 . Using the map to hypergeometric equations described in the previous subsection, we find that the general solutions as: This can then be used to generate the q-expansion. We now summarise the results. For generic rational values of µ, the q-expansion of the solutions of Eq. (3.19) has rational "degeneracies" but the rational numbers have growing denominators as we go up in the order of expansion. Such solutions are eliminated. However, for some special values of µ, the expansion coefficients become integer or rational with a bounded denominator. Some of these cases involve negative integer coefficients and were also eliminated 6 . At this stage, precisely 10 solutions survived -each having the property that (possibly after choosing an overall integral normalisation) every coefficient in the q-expansion, upto a large order, was a non-negative integer. The 10 solutions in [11] are listed in Table 1. This is admittedly an empirical method, wherein integrality is verified to a high power of q and then conjectured to hold to all orders. Its success lies in the fact that at least in the 5 Note a change of normalisation in the definition of the parameter µ relative to that in [11], namely µMMS = −4µ here . 6 These are the "quasi-characters" that will play an important role later in the present work.

No.
µ c h m 1 D M Theory ℓ = 0 case, each of the resulting characters can be given a definite physical interpretation in the language of CFT. This interpretation then amounts to a "physics proof" of the desired integrality to all orders. There are also other ways to prove integrality without having to identify with a concrete CFT, these will be discussed below. Of these 10 pairs of characters, the most straightforward are the seven cases numbered 2-8. These are Wess-Zumino-Witten (WZW) models at level k = 1 for the corresponding Lie algebra. It is remarkable that the MLDE approach, at these specific values of µ, gives us complete information about the characters of these models, which can be independently obtained from the famous Kac-Weyl formula (see for example [43]). These straightforward cases will help us highlight a few important features. In every case, the number m 1 , defined in Eq. (3.4), corresponds to the dimension of a simple Lie algebra. Here, χ 0 is the identity character. Recall that each character has its leading exponent, thus for two character theories one has α 0 , α 1 . As mentioned earlier, the natural choice is to take α 0 to be the smaller root of the equation. In these cases this choice is reaffirmed by the consistent identification of the characters with known, unitary RCFT's. We see that for the non-identity character, the leading term corresponds to a positive integer degeneracy D (see Eq. (3.4)) which, in every case, turns out to be the dimension of the fundamental representation of the associated Lie algebra. In the approach of [11] this fact was not assumed, but rather it emerged upon seeking the minimum degeneracy factor that ensures integrality of all the coefficients in the q-expansion of the non-identity character.
Notice that in some cases the multiplicity M defined in Section 3.1 is different from 1. For A 2 and E 6 we have M = 2. This arises from the fact that the fundamental representation is complex. Also for D 4 we have M = 3, associated to the triality between the vector, spinor and conjugate spinor representations, which ensures that all have the same character. These multiplicities are not required to make the q-expansion integral, so one might ask how they were discovered in the MMS approach. In fact this was done in subsequent works, [12,18], where the associated characters were computed exactly in terms of hypergeometric functions. Using these, the modular transformation matrix S defined in Eq. (3.5) was computed. Four of them turned out to be unitary while the remaining three did not. Instead they satisfied Eq. (3.8) with M = 2 in two of the cases, and M = 3 in the final case.
Finally, the fusion rules for these theories are computed using the Verlinde formula [17]: It is a rather miraculous feature of the Verlinde formula that the left-hand-side turns out to be a non-negative integer in all consistent CFT's, while the right-hand side is made from a matrix whose entries are not even rational in general. This was verified for each of the above seven theories in [18].
We must now consider the remaining three cases, numbers 1, 9 and 10 in the table. Each of these teaches us a new lesson and leads to valuable insights for the general case. Let us first address entry No.10. Notice that in the table we have chosen the multiplicity of the non-trivial primary to be 0. What does this mean? It turns out that the WZW model for E 8 at level 1 has a single character. This is related to the fact that in E 8 , the fundamental and adjoint representations are the same. In fact we already encountered this theory in our discussion of one-character theories in a previous Section. It is obvious that any modular-invariant set of p characters will not only solve a differential equation of degree p, but also differential equations of every degree p ′ > p. In the latter case, we will find p characters that transform among themselves under SL(2, Z) as well as p ′ − p "spurious" characters. What we are seeing in entry No.10 is such a recurrence of the E 8 character, as part of a 2-character set where the second character is spurious.
Assembling the entries 2-8 and 10 in the table, we find the level-1 WZW models for a series of Lie algebras A 1 , A 2 , G 2 , D 4 , F 4 , E 6 , E 7 , E 8 . The MLDE approach unites them despite their having different Kac-Moody algebras. Subsequent Deligne [44] proposed that the corresponding Lie algebras form a series with special representation-theoretic properties (similar observations have been made by Cvitanović [45]).
Let us now turn to the remaining cases, entries Nos. 1 and 9 of the table. If the first one were a CFT it would seem to contradict some familiar wisdom: all unitary theories with c < 1 are minimal models [1], but there is no unitary minimal model with c = 2 5 . So the theory, if it exists, must be non-unitary. This by itself is acceptable since we are not seeking to classify only unitary theories. However, computation in [18] of the fusion rules using Eq. (3.21) revealed a more serious problem: some of the fusion rule coefficients were found to be negative integers. Concretely, the modular transformation matrix for this theory is: Inserting this in Eq. (3.21) leads to: This is inconsistent as it stands, since the fusion rules of a consistent RCFT (even nonunitary) are supposed to be non-negative integers. The remedy, as noted in [18], is to exchange the roles of the two characters. The exponents α i for this theory are − 1 60 and 11 60 . While solving the MLDE, it was assumed that the former is to be identified with − c 24 , leading to the values in the table. Suppose however that we identify the latter exponent with − c 24 . In this case, we find that c = − 22 5 , h = − 1 5 . These are the correct values for the famous (non-unitary) Lee-Yang minimal model! The matrix S now has its rows and columns permuted and becomes: The result is merely a change of sign of some of the entries, but it has the desired effect.
The fusion rule coefficients are now: We will refer to the original ordering of characters (α 0 < α 1 ) as the "unitary presentation", and the reverse ordering (α 1 < α 0 ) as the "non-unitary presentation". Note that a given theory will be consistent only in one of these two presentations. If it is consistent in the unitary presentation, it still needs to be verified whether it is unitary or non-unitary (the only thing that is assured in the unitary presentation is that the conformal dimension h is positive, this is a necessary but not sufficient condition for the theory to be unitary).
However if a theory is consistent in the non-unitary presentation, it is necessarily nonunitary -since in this case h is negative. (Meanwhile, somewhat confusingly, the matrix S is unitary in both presentations.) To summarise what we have learned about entry No. 1 of the table, this is inconsistent in the unitary presentation but potentially consistent (the fusion rules are non-negative) in the non-unitary presentation. Moreover in the latter presentation the corresponding RCFT is known to exist and is a well-known minimal model. Note that one could consider exchanging characters in the other cases (Nos. 2-8), then one would find a system with some negative fusion rules. Our consistent policy, which will continue in the following sections, is to classify only theories with a positive central charge -these may of course be either unitary or non-unitary. Then, only if the fusion rules turn out to be negative, we consider the non-unitary presentation and check if the fusion rules have now become positive.
We now move on to entry No. 9 of the table. The matrix S of this theory is precisely the same as that of entry No. 1, thus in the unitary presentation it is given by Eq. (3.22) while in the non-unitary presentation it is the one in Eq. (3.24). It follows that the fusion rules are inconsistent in the unitary presentation and consistent in the non-unitary presentation. This tempts us to simply exchange the characters, as we did in case No. 1. Unfortunately we encounter a new problem having to do with the degeneracy factor D. In entry No. 1, the degeneracy of the non-identity character (in the unitary presentation) turned out to be 1. This was fortuitous, and ensured that upon exchanging characters to go to the non-unitary presentation the vacuum of the theory continues to be non-degenerate. Unfortunately in entry No. 9, one finds D = 57. Thus in the non-unitary presentation the identity character, and hence the vacuum of the theory, is degenerate. For this reason, entry No. 9 is inconsistent as a CFT both ways: in one presentation the fusion rules are negative while in the other, the vacuum is degenerate. Hence it was rejected in [18]. Nonetheless it is interesting that this case appears to lie "between" the E 7 and E 8 WZW models -but there is no WZW model occupying such a position and even within finite-dimensional Lie algebras, no mathematical structure of this sort was known at the time.
However the situation changed in 2004 when it was proposed [30] to augment the Deligne series by the inclusion of a structure called E 7 1 2 that lies "between" E 7 and E 8 . This was argued to fill a "hole" in Deligne's classification. The numerology of E 7 1 2 matches beautifully with entry No. 9 in the MMS series, for example the number of states above the identity character (in the unitary presentation) is 190, corresponding to the "dimension" of E 7 1 2 , suggesting that what was found as entry No. 9 in [11] is a kind of generalised CFT based on a generalised level-1 WZW model based on E 7 1 2 . More recently it has been proposed [29] that these characters be considered as those of an "Intermediate Vertex Operator Algebra" (IVOA). This has some, but not all, of the features of a Vertex Operator Algebra (VOA) which is the mathematically precise formulation of conformal field theory. So far it is not understood whether IVOA can be thought of as a generalised kind of RCFT in a physically useful way. If so, one would still need to decide whether it should be considered in the unitary presentation (giving up positivity of the fusion rules, but retaining a non-degenerate vacuum) or in the non-unitary presentation (retaining positive fusion rules but allowing a degenerate vacuum). For the present purpose we will leave this issue open, but will typically consider this theory and its analogues in the unitary presentation.
There is one more exotic case, not in the table, that is worth mentioning. In [11] it was noticed that the first-level degeneracy m 1 of the identity character diverges at c = 10. In recent years it has been understood that such theories have logarithmic characters [46,47], where one of the characters (which we take to be the identity character) has a power-series expansion of the form: In the context of the mathematical literature on MLDE this was noted in [35], building on previous results of [32,33]. It was shown in [35] that for such logarithmic solutions to the ℓ = 0 equation, the coefficient functions have non-negative integer coefficients in their q-expansion for the values c = 10, 22, 34, 46, 58, 106, 226. The authors described this kind of system as a "non-rational Vertex Operator Algebra". Logarithmic characters arise whenever the conformal dimension of the primary field becomes an integer. At ℓ = 0, it is easily seen from the Riemann-Roch equation that this happens whenever c = 2(6m + 5) = 10, 22, 34, · · · which includes all the cases found in [35].
A particularly simple example arises at c = −2, where the two characters are: and it is easily verified that they are exchanged by modular transformations. Logarithmic CFT's are interesting in their own right, but also because they bound certain "regions" in parameter space where regular CFT's are found as we will see below.
The reason we have spend considerable space discussing exotic types of characters: non-unitary, IVOA type and logarithmic, is that they occur repeatedly in investigations of MLDE. The first two will play a particularly important role in our discussion of quasicharacters below.

The case of ℓ = 2
This case was considered in [9,12,37,38]. The relevant differential equation is now: The coefficient 1 3 in front of the second term was determined from the indicial equation about τ = i∞, and the parameter µ is related to the central charge via µ = − c(c−4) 576 . The method of solution is as before, either via q-series or via hypergeometric functions. We write down the solution in the latter form: One now proceeds by expanding this in a q-series. The resulting admissible characters are shown in Table 2.
We have not included a "Theory" column in this table, not because the information is unavailable but because it is too long to summarise in the available space. As shown in [9] in the context of the novel coset construction described there, the seven theories numbered 3-9 in the table correspond to very special non-diagonal invariants for certain direct-sum Kac-Moody algebras. All of them are cosets of meromorphic theories of [10] by one of the seven "standard" WZW models in the original MMS series, namely those appearing in entries 2-8 of Table 1. A little inspection reveals that entries 3-9 of Table 2 pair up (in reverse order) with entries 2-8 of Table 1 in such a way that where (c, h), (c,h) are the central charge and conformal dimension in Tables 1 and 2 respectively. This property is a basic feature of the coset construction of [9], where it is also shown that coset duals of theories with ℓ = 0 have ℓ = 2 and vice-versa. The coset property is defined in terms of the Sugawara construction in [9] but can also be expressed through the bilinear relation: No.  where N is a non-negative integer that varies from case to case. As happened with ℓ = 0, again we seem to have found seven clearly consistent sets of characters, which all correspond to RCFT's via the coset construction of [9]. Note in particular that the fusion rules of the cosets are identical to those of the original theories.
What about the remaining three entries of Table 2? The pattern is very much the same as in Table 1. Again, we have three "outliers" which we now describe in turn. Let us start with No. 10. In the unitary presentation, this has central chargec = 24 − 2 5 and primary dimensionh = 9 5 . If we compare it with entry No. 1 of Table 1, we see that c +c = 24, h +h = 2. This again suggests the coset construction of [9], and indeed case No. 10 of Table 2 is the (generalised) coset dual of case No. 1 of Table 1. The reason we call this "generalised" is that these are not theories with conventional Kac-Moody algebras. Instead, the coset relation is defined via the bilinear relation Eq. (3.31), which can be verified even in this case, between the characters of the two theories. An important point to note here is that this coset structure holds when both theories are considered in the unitary presentation. However, in this presentation entry No. 1 of Table 1 has negative fusion rules. Hence the same is true of the dual, entry No. 10 of 2.
If instead we switch to the non-unitary presentation on both sides, the first theory becomes consistent, as seen above (with non-negative fusion rules) and equal to a minimal model. However its dual No. 10 of Table 2 does not. This is because, as seen from the table, the non-identity character has a degeneracy of 32509. Thus after interchange of characters this would be set of characters of Intermediate Vertex Operator Algebra type. Notice that in the non-unitary presentation the two theories would satisfy a different condition from the standard coset condition Eq. (3.30), namely c +c = −24, h +h = −2. For now we are only dealing with admissible characters (non-negative integer coefficients in the q-series), but if one steps outside this class then it is possible to find duals for non-unitary presentations [8].
Next, consider entry No. 2 of Table 2. This entry, with c = 82 5 , clearly pairs up with entry No. 9 of Table 1 in the same way as in the above cases, so it can again be considered the (generalised) coset dual of the IVOA associated to E 7 1 2 , with the coset dual relationship being understood in the sense of Eq. (3.31). Here both theories are of "IVOA type", since their degeneracy factors are 57 and 902 respectively.
Entry No. 1 of Table 2 has an identity character that is modular invariant (upto a phase) by itself. Indeed this is just the character of the famous E 8 × E 8 and Spin(32)/Z 2 WZW models. This is clearly the coset dual of entry No. 10 of Table 1, this follows from the fact that χ 3 Here both theories are completely consistent unitary CFT's and they exhibit a trivial version of the coset relationship.
Finally we note that logarithmic characters exist for ℓ = 2 as well. The Riemann-Roch theorem tells us this happens at specific integer values of the central charge c = 2(6n+1) = 2, 14, 26, · · · . These could be deserving of further study, however that will be outside the scope of the present work.

The case of ℓ = 4
Here we shall complete the classification of all two character theories with l = 4 started in [12,37,38]. In this case the MLDE is: Again the coefficient of the second term is determined by the indicial equation (in fact it is always equal to ℓ 6 ). The coefficient of the last term is determined by considering the indicial equation around τ = ρ. That leaves one free parameter, µ, related to the central charge by µ = − c(c− 12) 576 . We definitely expect to find some admissible characters, namely products of the form χ E 8 χ i where χ i have ℓ = 0. These products all have ℓ = 4 as one can see from the formula for the ℓ-value of the tensor product of two theories with (p, p ′ ) characters and ℓ-values (ℓ, ℓ ′ ) respectively [37]:l The question is whether there are any additional admissible characters that are not of tensor-product form. The value l = 4 corresponds to the Wronskian having a double zero at the orbifold point ρ = e 2πi 3 , and no zeros elsewhere in the moduli space. The characters written as functions of j(τ ) take the following form (upto normalisation): We have chosen c as the free parameter above, and from the Riemann-Roch formula we know that h = c 12 − 1 2 . Now, performing a q-expansion of the above characters, we can derive the following expression which gives: Thus, we see that 5c is an integer, such that the discriminant of the above quadratic equation is a perfect square. This restricts c to 34 values. Next we calculate m i for i ≥ 2 in terms of c. For i = 2 we have: Requiring that m i for i ≥ 2 be a non-negative integer further cuts down the list. This leaves us with 15 possibilities, shown in Table 3.  Table 3. Potentially consistent CFT's with ℓ = 4.
As indicated, entries No. 3-12 in the table are the expected tensor products. Hence we may focus our attention on the remaining five cases. Entry No. 2 is a re-appearance of the one-character theory for E 8,1 . Entry No. 1 has h = 0 and is a logarithmic CFT, falling in the series c = 12m + 6 which corresponds to logarithmic characters at ℓ = 4.
This leaves, as the interesting cases, entries No. 13-15 which have been labelled "A". All three were missed in [12], while No. 13 and 14 (but not 15) appeared implicitly in [38], and No. 14 and 15 (but not 13) have appeared in the mathematics literature [39]. No. 13 with c = 162 5 is in the same fusion-rule category as the Lee-Yang theory and hence, in the given unitary presentation, has negative fusion rules. Upon reversing the roles of the characters, one finds a non-unitary theory with a 310124-fold degenerate identity character, thus in our language it is a candidate theory of IVOA type. We believe that Nos. 13-15 form the complete set of admissible ℓ = 4 characters that are not tensor products of other admissible characters.

Fusion rules
As discussed above, the modular transformation matrix defined in Eq. (3.5) and the fusion rule coefficients are related by Verlinde's formula Eq. (3.21). In all the above cases, the characters have been determined in terms of hypergeometric functions and this enables us to compute S ij explicitly, from which N ijk may be obtained. A complete classification of possible fusion-rule classes for theories with low numbers of characters was found in [48,49]. Here we compile the results for all known two-character theories with ℓ < 6. For the ℓ = 0 series, some of the fusion rules were found in [17] and more complete results were presented in [18]. Remarkably, as noted in [9] and revisited in [40], the pattern of fusion rules for ℓ = 2 theories follows that of the ℓ = 0 case. We find altogether five different cases that apply to the theories in Tables 1 and 2: (i) For entries No. 1,4,6, 9 of Table 1, entries No. 2,5,7,10 of Table 2 (recall that the latter are cosets paired with the former) and entry No. 17 of Table 3, one finds that there is one primary φ (other than the identity) satisfying the fusion rule: (we follow the convention that any coefficient that is not explicitly written vanishes). These fusion rules are said to be of "Lee-Yang type". They correspond to case A  Table  3, we have one primary φ satisfying the rule: These fusion rules are said to be of A 1,1 type. They correspond to case A  Table  3, there are two non-trivial primaries which we label 1 and 2. They are complex conjugates of each other and have the same character. In this case, as analysed in detail in [18], the 2×2 S-matrix obtained by considering the modular transformations of the two independent characters is not unitary. Hence, to get the fusion rules, one has to enlarge it manually to a 3 × 3 matrix, which does turn out to be unitary, and which can be used in Eq. (3.21) to extract the fusion rules. We find: These fusion rules are of B (1) 2 type in the notation of [48]. (iv) For entry No. 5 of Table 1 and its coset dual, entry No. 6 of Table 2, there are three non-trivial primaries. In the ℓ = 0 case these are straightforwardly identified with the vector, spinor and conjugate spinor representations of D 4,1 . However in the coset dual ℓ = 2 case this identification is not so straightforward, since the theory is not a WZW model but rather a non-diagonal invariant of either (D 4,1 ) 5 or (A 5,1 ) 4 . The fusion rules are: These fusion rules are of type A 3 in the classification of [48]. We observe that there is no new ℓ = 4 theory with these fusion rules.
(v) For entry No. 10 of Table 1, entry No. 1 of Table 2 and entries No. 2,12 of Table  3 there are no non-trivial primaries. In the first case this is the E 8,1 theory and in the second, the E 8,1 × E 8,1 or Spin(32)/Z 2 theory. Both cases re-appear at ℓ = 4. The fusion rules are trivially: These can be called E 8,1 type fusion rules.
In what follows, instead of the notation of [48], we will label the fusion-rule classes (i)-(iv) above as Lee-Yang, A 1 , A 2 and D 4 respectively. Notice that we have realised all the fusion rules in [48,49] that can possibly correspond to two-character theories. Any pairs of characters with ℓ ≥ 6 that one may find, will also fall into one of these fusion classes.

How characters fail
From the examples above, we see that the classification of characters via MLDE occasionally throws up pairs of characters that are admissible in the limited sense of having non-negative integer coefficients, possibly after choosing a suitable overall normalisation, but still fail one or other criterion. We have seen multiple ways in which they can fail, which we summarise here: • At least one of the fusion rule coefficients is negative. However, after reversing the characters one finds a non-unitary theory that is completely consistent.
• At least one of the fusion rule coefficients is negative. After reversing the characters one finds a non-unitary theory whose fusion rules are non-negative, but the identity character is degenerate. We referred to such theories as being "of IVOA type".
• One of the characters is modular invariant by itself (possibly upto a phase). This is not a failure as such, but rather a re-discovery of a known character together with a "spurious" character that has to be discarded.
• The conformal dimension h of the non-identity character equals an integer. This is a logarithmic theory.
It turns out that the most interesting type of "failure" is, however, different from of the above. There are sets of characters that we rejected from the outset because one or more of the coefficients in the q-series, despite being integer, is negative. While admissible characters are finite in number for fixed ℓ (as displayed in Tables 1, 2, 3), there are non-admissible characters whose coefficients are all integers but some are negative, and remarkably there turn out to be infinite numbers of such characters (a finite subset of these were obtained in [38]). We will shortly find a use for them in classifying admissible characters with ℓ ≥ 6. Accordingly, we propose the following: Definition: "Quasi-characters" are a set of solutions to an MLDE having all integer (but not necessarily positive) coefficients in the q-expansion, after selecting a suitable normalisation.
Clearly quasi-characters include admissible characters (those with all non-negative coefficients) as a special case. But, in the context of second-order MLDE, they are far more numerous. For example, while the original work [11] found that there are at most 10 admissible solutions to the ℓ = 0 equation, there are infinitely many more solutions that correspond to quasi-characters. As we will now see, the complete set for ℓ = 0 can be extracted from [32,33] and we will also provide a different complete set for ℓ = 2.
Within the classification of quasi-characters, we empirically find two distinct types. For one class, which we denote Type I, a finite number of coefficients in the identity character are negative after normalising the ground state to be positive. Meanwhile the other character has all positive coefficients. A different class, which we denote Type II, has a finite number of negative coefficients in the identity character when the ground state is also normalised to be negative. Thus, if we parametrise the identity character as: 24 (a 0 + a 1 q + · · · a n q n + · · · ) (4.1) then for the Type I class we have that a 0 > 0, some of a 1 , a 2 , · · · a n are ≤ 0, while all of the a m ≥ 0, m > n. For the Type II class, we find instead that a 0 , a 1 , a 2 , · · · a n ≤ 0 while for m > n, a m ≥ 0. Again, the second character is entirely non-negative. We did not find any examples with a random or alternating collection of plus and minus signs all the way to infinity (which would have fallen outside our classification above). We do not presently have an explanation for this phenomenon. We also found that negative signs occur in at most one out of any given pair of characters, while the other has a completely non-negative q-series.
Let us pause to note that in the context of first-order MLDE, it is easy to generate Type I quasi-characters, the simplest being j(q) + N where N is any negative integer < −744. One also finds Type II quasi-characters, the simplest one being (j(q) − 1728) 1 2 , whose expansion is: As we have done above, in what follows too we will normalise such that only a finite number of terms is negative (possibly including the first term).
For second-order MLDE, objects equivalent to our quasi-characters have been discussed in mathematical works by Kaneko and collaborators, although not precisely in these words nor in the context to which we will be applying them. In the following subsections, we review these mathematical works, which all deal with the ℓ = 0 second-order MLDE, and generalise them to the ℓ = 2 MLDE. In the following Section we describe how to use quasi-characters to construct admissible characters with arbitrary ℓ ≥ 6.

Kaneko-Zagier parametrisation
In 1998, Kaneko and Zagier [31] studied a variant of the MMS differential equation in the context of supersingular elliptic curves and their invariants. Their equation was for weight-k (rather than weight-0) modular forms, and took the form: This is easily transformed to MMS form by the substitution: This results in: which, as we see, is the MMS equation Eq. (3.19) with Since we had previously noted that µ = − c(c+2) 576 , it follows that c = 2k. Kaneko and Koike [32] considered the above equation for the following sets of values: k = 3n + 1 2 , k = 2n + 1, n = 2 mod 3, and k = 4n + 2. Thereafter, Kaneko [33], considered k = 6n+1 5 , n = 4 mod 5. These sets will turn out to be of crucial importance below. From the Riemann-Roch relation − c 12 + h = 1 6 , one finds that h = k+1 6 . As seen in the previous Section, ℓ = 0 characters lie in the range −2 ≤ c ≤ 10 where the endpoints are logarithmic. Correspondingly we have −1 ≤ k ≤ 5. The rational values of k that give admissible characters (including those of IVOA type) are known [11] to be 1 5 , 1 2 , 1, 7 5 , 2, 13 5 , 3, 7 2 , 19 5 . We did not include k = 4 (c = 8) because that is a one-character theory. We see that these cases precisely fall into the four categories, as follows: In the range −1 < k < 5 this is a complete set of k values for the four categories. Moreover each case labels a distinct fusion-rule class, and we have labelled the equation accordingly.
For values of k outside this range, the above series do not give admissible RCFT characters, but remarkably they turn out to provide quasi-characters. Note that the explicit formulae for characters in Eq. (3.20) continue to be valid for quasi=characters -one simply has to substitute c = 2k in those formulae to get the quasi-characters at ℓ = 0 for any k.
To show that these are all quasi-characters, we will use the results of [32,33]. This can be verified by computing explicit solutions as q-series, several examples of which can be found in Appendix A. We will find that both characters have integer coefficients in their q-expansion, but only the identity character (defined as the one with a smaller exponent α) has negative integer coefficients. In a number of examples we find that the number of negative coefficients is finite and of order k, which we conjecture to be a generic feature. Before presenting these results, we consider the ℓ = 2 equation.
We will refer to this as the "dual Kaneko-Zagier equation". Scaling the solutions as in Eq. (4.4) to obtain weight-0 (quasi-)characters, one gets the equation: which is a special subfamily of Eq. (3.28) obtained by setting µ = − k(k−2) 144 . We see again that the associated central charge is c = 2k.
This time the Riemann-Roch theorem says that h = k−1 6 . From Table 2, admissible characters are bounded in a range of central charge 14 ≤ c ≤ 26, again with logarithmic theories at the end-points. This range corresponds to 7 ≤ k ≤ 13. In fact the precise values of k that lead to RCFT are 41 5 , 17 2 , 9, 47 5 , 10, 53 5 , 11, 23 2 , 59 5 where again we have removed the single-character case k = 8 (c = 16). These values fall into the following four series: As in the ℓ = 0 case, the k values listed above are a complete set for the given four series in a particular range, namely 7 < k < 13 (logarithmic end-points excluded). These are the admissible characters. If now we allow k to take values outside this range, then we find that the corresponding solutions to Eq. (4.9) are quasi-characters in every case. Examples of these too are listed in Appendix A. And again, the explicit solutions in Eq. (3.29) continue to be valid with the substitution c = 2k.
In fact we can say more: the coset pairing that was previously observed [9] between ℓ = 0 and ℓ = 2 characters continues to hold for these quasi-characters. It pairs an ℓ = 0 theory with a given value of k with an ℓ = 2 theory with the valuek = 12 − k. We will discuss this relation in a subsequent subsection. But first we will show, both for ℓ = 0 and ℓ = 2, that the quasi-characters can be expressed as polynomials in the characters of a particular "base" theory in that class. This will in particular amount to a proof of integrality of their coefficients.

Lee-Yang series
This series is defined by choosing the parameter k in Eq. (4.5) as follows: k = 6n + 1 5 , n ∈ N, n = 4 mod 5 (4.11) Recall that the central charge and conformal dimension associated to these characters is: The first case is n = 0, which yields c = 2 5 , h = 1 5 . These are the Lee-Yang characters in the unitary presentation. As we see from the above formulae, all the remaining cases also have a central charge and dimension that is a multiple of 1 5 . Using [48] one sees immediately that the fusion rules associated to them must be of Lee-Yang type. This can also be verified directly by constructing the modular transformation matrix, This explains why we referred to this series as the "Lee-Yang class" (with ℓ = 0). The values n = 0, 1, 2, 3 give genuine ℓ = 0 characters that can be found in Table 1. For all remaining values of n we find quasi-characters.
As shown in [33], the solutions for this series are given recursively in terms of the characters of the Lee-Yang theory: The recursion relation can be written in a (moderately) simple form using a suitable power of the quotient of the two characters, namely: Now, for all k satisfying Eq. (4.11), the solutions are given by χ 5k 0 P n (t) and χ 5k 1 P n (−t −1 ), where P n (t) are polynomials given as follows: Notice that since n = 4 mod 5, P 4 , P 9 , P 14 · · · do not exist. For n ≥ 10, P n (t) are given recursively in terms of P n−5 (t) and P n−10 (t) as follows Now it is easy to prove integrality of the coefficients. Clearly the modular function t has all integer coefficients, and at each step in the recursion only a finite common denominator occurs (which in general grows with n). This ensures that a finite normalisation will render the character integral. The positivity behaviour of the q-coefficients is interesting: the solutions show different behaviours for the two cases n ∈ {0, 1, 2, 3} mod 10 and n ∈ {5, 6, 7, 8} mod 10. In both cases the q-coefficients of the first character (suitably normalised) are all positive integers except for the first ⌊ n 10 ⌋ odd powers of q in the first case and the first ⌊ n 10 ⌋ + 1 even powers of q -including the ground state -in the second case. In both these cases, the second character has all positive integer q-coefficients. Thus, these satisfy our definition of quasi-characters, and moreover the two case precisely fit the definition of what we referred to in the previous sub-section as Type I and Type II respectively. Also, except for the first few values of n, the vacuum character always has degeneracies, requiring an overall integer normalisation to make the coefficients of the q-series integral.
To derive this result, [33] recasts the original differential equation Eq. (4.5) as a differential equation in t, which (in monic form) has rational coefficients. Writing where χ LY 0 was defined in Eq. (4.13), and using Eq. (4.14), the following equation is found: It is then verified that the polynomials P 0 · · · P 8 satisfy this equation (start by choosing arbitrary coefficients in each polynomial, and determine them from the equation). Finally, the recursion relation is verified using Eq. (4.17).

A 1 series and its dual
This series is defined by choosing k in the Eq. (4.5) as follows [32]: With this parametrisation, we have The values n = 0, 1 give genuine ℓ = 0 theories, which can be found in Table 1. For all the other values of n, the solutions of the MLDE are quasi-characters and are given recursively in terms of the characters of the A 1,1 theory: The quasi-characters are polynomials of the above two characters. Consider the ratio This is related to the modular lambda function as t(τ ) = λ(2τ ). Then the quasi-characters for all n are given as χ 6n+1 0 P n (t) and χ 6n+1 1 P n (t −1 ), where the polynomials P n (t) are given as follows (this can be derived from Theorem 1(iv) of [32]): We find that this series of quasi-characters is of type I for n = 0, 1 mod 4 and of type II for n = 2, 3 mod 4.

Dual A 1 series
Here we propose a dual A 1 series of quasi-characters which satisfy the l = 2 differential equation. This is defined by choosing k in Eq. (4.9) as This corresponds to the following values of central charges and conformal dimensions These quasi-characters are also expressed as polynomials of the A 1,1 theory, and for n ≥ 1 are given by χ 6n−1 0P n (t) and χ 6n−1 1P n (t −1 ).
and for n ≥ 5,

A 2 series and its dual
The A 2 series occurs at central charges c = 2k = 4n + 2, n = 2 mod 3. For n = 0, we have the A 2,1 theory with characters is the weight 1 Eisenstein series at level 3. The quasi-characters are given as polynomials in the above characters. However, in the A 2 and D 4 classes as well as their duals, the pattern is different from the previous cases in that the second character is not obtained from the first by a simple substitution. Hence we only quote the identity character here. Using t = χ 1 χ 0 3 , we have χ 2n+1 0 P n (t) for the identity character, where: and for n ≥ 6, we have The l = 2 series of A 2 quasi-characters occur at c = 2k = 4n − 2, n = 1 mod 3. The identity character is given as χ 2n−1 0P n (t) where: and for n ≥ 7,

D 4 series and its dual D 4 series
The D 4 series of quasi-characters occur at central charges c = 2k = 12n + 4. For n = 0, we get the D 4,1 affine theory whose characters are is the weight 2 Eisenstein series at level 2. Again, the quasi-characters are polynomials in the above characters. Consider the ratio t = χ 1 χ 0 2 , then the identity quasi-character for any n is given as χ 3n+1 0 P n (t) with: and for n ≥ 2, Finally, the l = 2 series of D 4 quasi-characters occur at central charges c = 2k = 12n − 4. The identity character is given as χ 3n−1 0P n (t), with: and for n ≥ 3,P

General approach to recursion relations
Here we briefly describe a more general approach to recursion relations that applies for any k. Consider the solutions of the original Kaneko-Zagier equation Eq. (4.3). Recall that these have modular weight k and are denoted f (k) . These solutions have the following important recursive property [32]: given a solution f (k) of the equation with parameter k, the function f (k−6) defined as: is a solution of the same equation with parameter k − 6. Here, is the Rankin-Cohen bracket of two modular forms of weights k and l. Also, given two solutions f (k) and f (k−6) of the equations with parameters k and k − 6 respectively, the solution of the equation with parameter k + 6i + 6 for all i ≥ 0 is given by Thus, for each solution f (k) of the original equation, the solutions of differential equations with parameters k − 6 and k + 6 exist and are given in terms of f (k) by the simple recursive formulae Eqs.(4.34),(4.36). Now from [11] we know the set of all solutions with integer coefficients for k lying in the interval [−1, 5] (to get the weight-k solution one has to multiply the solutions of [11] by η 2k ). Then we can use Eq. (4.34) to calculate the solutions corresponding to the interval [−7, −1]. Finally we can use this set in Eq. (4.36) to find all the solutions for arbitrary values of k. This enables us to list all the quasi-characters for the l = 0 case, starting from the characters. One can think of the range c ∈ [−2, 10] (equivalently k ∈ [−1, 5]) as a "fundamental domain" for ℓ = 0 quasi-characters. We have noted previously that the end-points of this fundamental domain are logarithmic CFT's. The approach of this subsection will reproduce precisely the quasi-characters of the previous sections, though not directly in polynomial form.
We can carry out the same procedure for the dual equation in Eq. (4.8). Although it is quite different from the Kaneko-Zagier equation, we have found that its solutions have a similar recursive property. Given a solution f (k) of Eq. (4.9), the function f (k−6) defined as: is a solution of the same equation with parameter k − 6. Given the solutions f (k) , f (k−6) , we can find the solution f (k+6i+6) recursively, as follows: The proof of the above recursion relations closely follows that of the l = 0 case in [32] with slight modifications. From the definition of the Rankin-Cohen bracket, we know that: Here, we have used the Ramanujan identities D (4) E 4 = − 1 3 E 6 and D (6) E 6 = − 1 2 E 2 4 . Taking another derivative of the bracket, we can show that Dividing the above equation by ∆, and using D (12) ∆ = 0 gives us the dual KZ equation with the free parameter k replaced by k − 6, and shows that is its solution. The other recursion relation, is also proved in a similar manner. Note that the coefficient in front of f (k+6i−6) in Eq. (4.38) differs from the l = 0 case. The fundamental domain for k in this case will be chosen as [7,13] because this gives the admissible characters. Here too, logarithmic characters lie at the boundaries of this domain.

Modular transformations for quasi-characters
An important operation that we will consider in the following Section is the addition of quasi-characters. When suitably done, this can remove minus signs and convert quasicharacters into admissible characters. However, the process of addition must commute with modular transformations. For this to be true, the different quasi-characters being added must all have the same modular S and T matrices. These have been computed for admissible characters in [12] in terms of trigonometric functions, and in [18] in terms of Γ-functions. Because the solutions of the ℓ = 0 differential equation have universal expressions in terms of hypergeometric functions that depend only on k, the modular transformation matrices likewise can be expressed for all cases as a function of k by simply extending these formulae to all k. The result can be written: where: We see that the result is periodic under the shift k → k + 12, but not any smaller shift. This formula has a nice property under the inversion k → −2 − k. This shift preserves the Kaneko-Zagier parametrisation and is known to interchange the characters. One can show that under this shift, the above formula transforms by an exchange of the two rows and the two columns.
As an explicit example, for the Lee-Yang class of quasi-characters one finds: The shift k → −k − 2 discussed above acts on the label n of this case as n → −2 − n and we can see that the expected pattern holds. For example if we consider n = −10, this will have an S given by the second line in the above equation. Under inversion it becomes n = 8 which corresponds to the fourth line. Now if we take the matrix in the fourth line and exchange the role of identity and non-identity character (which is done by exchanging the rows and then exchanging the columns) then we get back the same matrix as in the second row. We would also like to have an analogous result for the ℓ = 2 quasi-characters. In principle this can be deduced by the same methods as above. However there is a simpler route. In the following subsection we will demonstrate a coset relation between quasicharacters with ℓ = 0 and those with ℓ = 2. Since the coset pairing leads to a bilinear relation that is modular invariant, it follows that the modular transformation matrix for any ℓ = 2 quasi-character is the inverse of that for its ℓ = 0 coset dual. This ensures in particular the same periodicity property as above when k → k + 12.

Coset relations among quasi-characters
If we compare the quasi-characters for ℓ = 0 and ℓ = 2, we find that they are related as follows. Consider the former at some value of k and the latter at a value ofk such that k +k = 12. From Riemann-Roch it also follows that the conformal dimensions h,h associated to the two sets satisfy h+h = 2. Using c = 2k, these are the same coset relations obeyed by the admissible characters at ℓ = 0 and ℓ = 2, as shown in [9]. Following the previous discussion, it should not be surprising that this coset (or more properly, bilinear) relation continues to hold for quasi-characters. Combining Eqs. (3.20,3.29) it can be shown that: where we have denoted the ℓ = 2 quasi-characters asχ i , and f k = − 1728(k−4) 2(k−5 . The above equation is basically the same as Eq.(4.10) of [9].
It is pleasing to see that this relation, once thought to hold only for a finite set of admissible characters, holds for an infinite set of quasi-characters. One should not be surprised to see that the additive constant is, in general, a fraction. The reason is that the characters listed in Eqs. (3.20,3.29) are normalised so that their q-series starts with 1. However, with this normalisation, quasi-characters are not always integral, but have bounded denominators. Thus upon multiplication with a suitable integer they become integral, but now must be thought of as being in the IVOA category. It is clear that the same normalisation that makes each quasi-character integral will also make the constant on the RHS of Eq. (4.44) integral.

Summary of the properties of quasi-characters
It is useful to summarise the basic properties of quasi-characters that we have discussed above. This is done in Table 4.  Table 4. Quasi-characters with ℓ = 0, ℓ = 2.
An important consequence of our discussion is that the sets of quasi-characters listed in the table, both for ℓ = 0 and ℓ = 2, are complete. This can be argued as follows. The periodicity discussed above is general for any value of k. Thus if the list of quasicharacters were not complete, there would be additional candidate quasi-characters in the "fundamental domain" k ∈ [−1, 5] for ℓ = 0 or k ∈ [7,13] for ℓ = 2. But this cannot be the case, since we have a complete classification in this region. Another way to argue this is that the fusion-rule classification [48,49] for two-character theories admits conformal dimensions which can only be multiples of 1 2 , 1 3 , 1 4 , 1 5 . Now the range of k considered in the various series above exhausts all possible such values.
Finally we note that an infinite set of ℓ = 4 quasi-characters can be obtained by simply multiplying those for ℓ = 0 by j 1 3 . The argument above can be extended to show that these are similarly complete. Note that multiplying ℓ = 2 quasi-characters by the same function cannot lead to a complete set for ℓ = 6 because from this case onwards, addition of quasicharacters consistent with modular invariance becomes possible and gives more general solutions.

Generating characters for ℓ ≥ 6
The main result of this Section (and of the paper) is that despite being inadmissible as RCFT characters, the quasi-characters provide useful building blocks to construct admissible characters with higher values of ℓ. This is done by removing the negativity in the q-coefficients, which can be done using two different methods as we now describe.

Products of quasi-characters with one-character theories
The first observation is that products of quasi characters of Type I with a sufficiently high degree polynomial in χ E 8,1 = j 1 3 can eliminate all of the negative signs in the former (this was correctly anticipated in [38]). In this process, the vector-valued modular transformations of the quasi-characters are preserved upto a phase, since j 1 3 is modular invariant upto a phase all by itself. The resulting characters, if shown to correspond to a consistent RCFT, will correspond to new theories and not tensor products of old ones. The simple reason is that the original quasi-character, being of type I, has a certain number of negative coefficients in its q-expansion and does not qualify to be a CFT -only its product with a power of j 1 3 qualifies. It is easy to verify that if we tensor j r 3 with a pair of quasi-characters having a particular value of ℓ, the result has: In many cases the vacuum of the quasi character is degenerate, hence the resulting set of new characters will be of "IVOA type". The first non-trival example of this kind is provided by the three admissible characters that we found with ℓ = 4 in Subsection 3.5. These correspond to the following values of c, h: (c, h) = ( 162 5 , 11 5 ), (33, 9 4 ), (34, 7 3 ) (5. 2) It is easy to verify that each of these character pairs corresponds to the product of χ E 8,1 = j 1 3 with the ℓ = 0 quasi-characters having: in Eq. (4.5). Notice that the h-values are preserved in the process: the above quasicharacters and the final admissible characters both have h = 11 5 , 9 4 , 7 3 . Also note that the k-values in Eq. (5.3) belong to the series of quasi-characters k = 6n+1 5 , n = 10; k = 3n + 1 2 , n = 4; k = 2n + 1, n = 6 respectively, which fall in the Lee-Yang, A 1 and A 2 classes respectively. From Eqs.(A.1),(A.17) and (A.33) we see that each of these has a single negative coefficient m 1 at first level above the vacuum state in the identity character. Multiplication by j 1 3 ∼ q − 1 3 (1+248q +· · · ) will add 248 to this coefficient, so if m 1 ≥ −248 then the final character after multiplication will have all positive integral coefficients and thereby become admissible. Inspection of the three examples reveals that their values of m 1 are −244, −245 and −247 respectively, satisfying the requirement. Within the space of ℓ = 0 quasi-characters, it is easy to verify that these are the only ones satisfying m 1 ≥ −248 (we do not count the admissible characters with ℓ = 0 since those give rise to tensor-product theories at ℓ = 4), which explains why there are precisely three new admissible characters with ℓ = 4.
Let us generate a few more examples involving type I quasi-characters. In the Lee-Yang class we may consider n = 20, 30, 40 for which the q-expansions are given in Eq. (A.4). We have verified that the following are the "minimal" admissible characters that can be made from these by multiplication by powers of j In general, whenever we start with quasi-characters at ℓ = 0 we find that the products made by tensoring with j r 3 have ℓ = 4r. If we start with dual quasi-characters at ℓ = 2, we will find theories with ℓ = 4r + 2. This exhausts all even values of ℓ.

Sums of quasi-characters
In this section we discuss how linear combinations of quasi-characters can be used to create admissible characters with higher values of ℓ. In this approach, we add different q-series, each corresponding to a quasi-character, such that the sum is an admissible character with only non-negative q-coefficients. Of course, arbitrary sums of characters destroy modular invariance so these sums need to be taken carefully. To preserve modular invariance we can only add two characters which transform in the same way under modular transformations. As we now show, the resulting theory will have the l-value increased in multiples of 6.
Let us right away consider an example. Take the ℓ = 0 quasi-characters in the Lee-Yang family corresponding to n = 0 and 10. From Eq. (4.43) these have the same modular transformations and so the result will also transform in the same way. Introducing an integer N 1 , we consider the family of characters: For th identity character i = 0, the q-expansion of this sum is: =q − 61 60 (1 + (N 1 − 244)q + (N 1 + 169641)q 2 + (N 1 + 19869896)q 3 + · · · ) (5.6) while for i = 1 (the non-identity character) the q-expansion is: q 71 60 (310124 + 27523505q + 1012864984q 2 + · · · ) + N 1 q 11 60 (1 + q 2 + q 3 + · · · ) =q 11 60 (N 1 + 310124q + (N 1 + 27523505)q 2 + (N 1 + 1012864984)q 3 + · · · ) (5.7) Now if we choose N 1 ≥ 244, we have eliminated all negative signs and the resulting characters are admissible. To find out more about the potential CFT that they describe, notice that for the identity character, the leading exponent is that of the quasi-character of higher central charge (n = 10 in this example) while for the non-identity character the situation is reversed: the exponent is that of the quasi-character of lower central charge (n = 0 in this example). It follows that the CFT would have c = 122 5 and h = 6 5 , from which we find that ℓ = 6. Thus by simply adding two quasi-characters with ℓ = 0, we have found an infinite set of admissible characters with ℓ = 6, one for each integer N 1 ≥ 244. While we do not necessarily expect there to be an RCFT for each of these cases, this achieves the goal of generating large classes of admissible characters.
The next example is striking because, for some choices, it allows us to convert a quasicharacter with a degenerate identity field (what we called IVOA type) to an admissible character with a non-degenerate identity. We add quasi-characters, again for the Lee-Yang series, but for the values n = 11 and n = 1. The result, for the identity character, is: 60 7 + (−1742 + N 1 )q + (722729 + 14N 1 )q 2 + (133716590 + 42N 1 )q 3 + (7374239425 + 140N 1 )q 4 + (220691372762 + 350N 1 )q 5 + (4460548657432 + 840N 1 )q 6 + (68133599246580 + 1827N 1 )q 7 + · · · (5.8) Like the previous case, this one again has ℓ = 6. In this case the original identity character for n = 11 was of IVOA type, and in fact 7-fold degenerate as we see from the 7 multiplying the leading term. This means the higher degeneracies were not divisible by 7, if they had been so then we could have normalised the character to have a non-degenerate vacuum state. However after adding characters as above, a miracle takes place when N 1 = 1742, its lowest allowed value. In this case the first level degeneracy above the identity vanishes, but all the higher degeneracies become multiples of 7. Thus we find: 60 (1 + 106731q 2 + 19112822q 3 + 1053497615q 4 + 31527426066q 5 + 637221445816q 6 + 9733371775602q 7 + · · · ) (5.9) We are now in a position to drop the leading 7 and find an admissible character with a non-degenerate vacuum. We see the encouraging fact that quasi-characters of IVOA type (which are the generic type) can lead to regular non-degenerate admissible characters upon being added to each other. By examining numerous examples we have found that this generically seems to happen at least for the minimal allowed value of the integer constant.
In general one can take sums (with integer coefficients) of any number of quasicharacters that all have the same modular transformations and generate large sets of admissible characters. The question is then, what is the ℓ-value of the result. We can provide a general formula for this. Consider a set of ℓ = 0 quasi-characters all lying in the same class (but not necessarily the Lee-Yang class). Label them by the parameter k. As we have seen, the shift k → k + 12 leaves the modular transformations invariant. Thus, we may consider sums of the form: Assuming N pmax = 0, the critical exponents of the resulting characters are: from which it follows that ℓ = 6p max . This agrees with our previous examples where p max was 1 and we found ℓ = 6. Thus we see that it is no problem to generate infinite sets of admissible characters with arbitrarily large values of ℓ just by adding a number of quasi-characters, and choosing the integers N p to ensure that all minus signs are removed. The above was for ℓ = 0 quasi-characters. Had we instead started with ℓ = 2 quasicharacters, we would end up with ℓ = 6p max + 2. Finally, we have seen that all ℓ = 4 quasi-characters are products of j 1 3 with ℓ = 0 quasi-characters. By adding these to each other as above, we can generate admissible characters with ℓ = 6p max + 4. Thus we have shown how to generate infinitely many admissible characters for all even values of ℓ.
In this Section, so far we have only considered Type I quasi-characters with a single negative coefficient. However the procedure works equally well with more negative coefficients. For example one can consider the quasi-character in the A 1 case at ℓ = 0 and n = 12, see Eq. (A. 19). This has negative coefficients in front of q, q 3 and q 5 . On taking an arbitrary linear combination of this with the n = 8 and n = 4 quasi-characters in Eqs.(A.18) and (A.17) respectively, as well as the well-known n = 0 characters (which correspond to the SU 2,1 WZW theory), we find that the identity character goes as: Notice that, because of the addition process, there are now minus signs at all orders from q to q 5 . However for suitable choices of the integers N 1 , N 2 , N 3 one can easily ensure that all these terms become non-negative. The subsequent terms in the above character from O(q 6 ) onwards are all positive linear combinations of the N i , so they will remain positive if we choose all N i non-negative, and it is clear that this allows for infinitely many choices 7 .
In the previous subsection we saw how to get admissible characters by multiplying quasi-characters by j r 3 . One may wonder whether this approach is exhaustive, generating all admissible characters with ℓ ≥ 4. In such a situation, potentially that method would yield identical results to the one explained in this subsection. However it is easy to see that this is not the case. For example, starting from a given quasi-character with ℓ = 2, we can get a single (potentially admissible) character at ℓ = 6 upon multiplying by j 1 3 . However the methods of the present section allow for infinitely many characters, all with the same central charge and conformal dimension, at ℓ = 6. Thus the method of adding quasi-characters is more powerful.
Nonetheless the method of the previous subsection is essential. If we are interested in admissible characters with ℓ = 4 mod 6 (for example ℓ = 10) we can only get them by the addition method if we start with ℓ = 4 quasi-characters. But these, in turn, can be generated by multiplying ℓ = 0 quasi-characters by j 1 3 . Thus it appears that the "seed" quasi-characters that could potentially generate all admissible characters, are those which we have described based on the Kaneko-Zagier parametrisation and its dual (ℓ = 0 and ℓ = 2 respectively) as well as j 1 3 times the ℓ = 0 quasi-characters. Just using these three sets, one can generate infinitely many admissible characters for all even ℓ ≥ 6 by adding quasi-characters.

Relation to Hecke images
In [40], Harvey and Wu introduced novel Hecke operators that act on vector-valued modular forms which occur as RCFT characters and give rise to new sets of potential characters. These Hecke images generically have increasing values of ℓ, and under certain conditions they are admissible in the sense we have used in this paper. Let us briefly review their construction as it applies to the two-character case. Consider the q-expansions of the characters of a particular RCFT: In the second expression above, N is the common denominator of the original exponents α i , which are therefore written as n i N , and the summation is now over The integer N is called the conductor of the CFT. Then, the Hecke image T p of the above characters is defined to be the new q-series: Here, p ∈ N is a prime and is relatively prime to N . The new exponents α for a certain matrix ρ ij which is completely determined by p. Finally the q-expansion can be recast in a standard form using coefficients c using Eq. (5.14). The Hecke images (T p χ) i will be admissible according to our definition for certain values of p which ensure that the c (p) i determined by the above procedure are non-negative. For Hecke images of ℓ = 0 theories, the case predominantly considered in [40], the ℓ-value of the resulting characters is found by applying the Riemann-Roch theorem: The main virtue of the Hecke image procedure is that we can easily generate infinite sets of admissible characters with arbitrarily large values of ℓ as well as c.
However this procedure has a limitation. If we want to find ℓ (p) = 4 mod 6, then we must have p = 3 mod 6. But this is never realised, because p has to be co-prime to the conductor N , and in two-character theories the conductor is always a multiple of 3. Therefore in particular one can never get an ℓ = 4 theory as a Hecke image. This point was noted in [40] but incorrectly attributed to the fact that [37] Table 5. Admissible Hecke images with ℓ = 6 theory. As we have seen in Section 3.5, there are indeed admissible characters with ℓ = 4 that escaped the notice of several previous works. Yet, they are not Hecke images. This example shows that Hecke images do not produce a complete set of admissible characters. One of our methods in this paper, of taking sums of quasi-characters, increases ℓ in multiples of 6. Therefore this procedure also cannot give us ℓ = 4 characters. This proves that it, too, does not produce a complete set of admissible characters. In fact the only way to generate ℓ = 4, as we have observed above, is by taking the product of j This leaves open the possibility that taking Hecke images and taking sums of quasicharacters could be equivalent to each other, in the sense that the set of characters obtained by both procedures could be the same. Let us examine this for the special case of ℓ = 6. We start by classifying all primes p such that starting from an ℓ = 0 CFT, we generate ℓ = 6. Starting from the ℓ = 0 Lee-Yang CFT, the only primes p that generate an admissible ℓ = 6 Hecke image are p = 61, 67, 73, 79 8 . If we start with the ℓ = 0 A 1,1 theory, the primes are p = 25, 31. On the A 2,1 theory one can use the prime p = 13. Finally on D 4,1 one can use p = 7. These exhaust all the cases leading to admissible ℓ = 6 characters using Hecke images. The central charge and conformal dimension of these characters are given in Table  5.
We will now show that there is a sum of ℓ = 0 quasi-characters that reproduces each of these cases. In fact, all one has to do is set k = c 2 for each entry in the table, and then consider the sum: We have verified that for some value of N 1 in each case, this set precisely reproduces all the Hecke images in Table 5. Moreover, as explained in [40], one is allowed to consider sums of Hecke images. Likewise, we can consider varying the coefficient N 1 . One finds that the two procedures agree. For example, for the first entry in Table 5 one has: where of course χ LY is the same as χ k= 1 5 . The LHS is the sum of a Hecke image and a character, while the RHS should be seen as the sum of two quasi-characters, the second one being in fact a character. For the remaining entries of the table, one has the general result: for suitably chosen N ′ 1 , where on the right-hand side, one has to adjust p according to the value of k (this can be read off from the table). We conclude that the space of (sums of) Hecke images at ℓ = 6 is precisely equal to the space of sums of quasi-characters. We conjecture that this equivalence, between sums of Hecke images and sums of quasicharacters, is true in general.
The relation between Hecke images and linear sums of quasi-characters, for twocharacter theories, is reminiscent of a well-known phenomenon in the study of Hecke images of modular functions. If we act with a Hecke operator on j(τ ), the result is a sum of jfunctions evaluated at (shifted) multiples of τ of the form nτ and τ +i n . On the other hand, the result of this action can be written as a polynomial in j(τ ) using meromorphy of j. Comparing coefficients on both sides, one finds that linear combinations of the q-coefficients of j are equated to sums of powers of the same coefficients. For the present case, the Hecke image of a particular character χ as defined in [40] provides an analogue of the left-hand side of this relation, since its q-coefficients are linear combinations of the q-coefficients of χ. Meanwhile, recall that our quasi-characters can be written as polynomials of characters (see for example the discussion in Subsection Eq. (4.3)). Thus on the RHS we encounter powers of coefficients of the same character χ. Our conjectured equivalence then becomes a nice analogue of the famous result for modular functions.

Conclusions and Discussion
In this paper we introduced the concept of quasi-characters that have all q-coefficients integer, but finitely many are negative. The complete family of quasi-characters for ℓ = 0 and ℓ = 2 was presented. A complete family of ℓ = 4 quasi-characters is obtained by multiplying those of ℓ = 0 by j 1 3 . Quasi-characters are naturally labelled by a rational number k = 1 2 c where c would be the central charge in the admissible case. We have argued that quasi-characters of Type I (the ground state degeneracy is positive) can be used to generate admissible characters by addition. One takes positive integer linear combinations of these objects for sets of values of k differing by 12, choosing the positive integers such that the result is admissible. In this way ℓ is increased in multiples of 6. Repeating the process for the three "base" sets of quasi-characters (ℓ = 0, 2, 4) gives us the possibility of generating admissible characters for every even ℓ. It is very plausible (though we do not yet have a proof) that this procedure generates all possible admissible pairs of characters.
We have conjectured that the addition of quasi-characters yields exactly the same family of admissible characters provided by taking linear combinations of Hecke images following [40]. Our procedure may be simpler because we have an explicit formula for every quasi-character, namely Eqs. (3.20, 3.29, 3.34) with c = 2k and the values of k being read off from Table 4. These then have to simply be inserted into the summation formula Eq. (5.10). We have noted that this postulated equivalence between Hecke images and linear combinations is an interesting analogue of well-known facts for standard Hecke operators and modular functions.
For one-character RCFT, the problem of classifying admissible characters has long been solved, as it just amounts to constructing polynomials in j with suitably chosen coefficients, and possibly multiplying them with a power of j 1 3 . Now we have a similarly simple algorithm to find admissible characters in the two-character case. The next step would be to carry out a programme on the lines of [10] to identify which of this large set of characters can actually describe consistent conformal field theories.

A Examples of quasi-characters
In this Appendix we list several indicative examples of quasi-characters in all four classes: Lee-Yang, A 1 , A 2 and D 4 , for ℓ = 0 and 2. From these examples we will try to deduce a rule for the number and location of occurrence of minus signs. In every case c = 2k, as explained in the main body of the paper, and k is given by a different formula in terms of an integer n for every class. These examples are restricted up to order q 8 in every case for brevity of presentation only. We have in fact examined them to much higher orders, and in particular have verified that in all the given examples, no minus signs occur beyond the order quoted.
Though we are only quoting terms up to O(q 8 ), we have verified these properties upto far higher orders in every case. These examples were the basis of our observation, in the main text, that the positivity/negativity properties form two distinct classes. Here, n = 0, 1, 2, 3 mod 10 lie in one class and n = 5, 6, 7, 8 mod 10 lie in the other. These correspond to what we have denoted Type I and Type II respectively. We will now see that this feature persists for higher n, with the number of negative coefficients for the identity character increasing with n.
Next we list the identity quasi-character for n = 15, 25, 35: n = 15 : (A.6) n = 25 : n = 35 : In these cases of type II, negative coefficients occur at levels 0,2 for n = 15, at levels 0,2,4 for n = 25, and at levels 0,2,4,6 for n = 35. As we have said previously, the properties for n = 10p + 5 also carry over identically to n = 10p + 6, 10p + 7, 10p + 8. We see that type I and type II solutions in this family have distinctive signatures of how their positive and negative signs occur. The pattern is that for the identity character, the first ⌊ n 10 ⌋ odd-level coefficients are negative in the type I case, while the first ⌊ n 10 ⌋ even-level coefficients are negative in the type II case. Meanwhile the second character has all positive integral coefficients. We conjecture that this pattern persists for all values of n.
As n increases, the number of negative signs increases in χ 0 , as can be seen in the following examples: n = 25 : (A.14) n = 35 : (A.15) n = 45 : As in the Lee-Yang series, the dual Lee-Yang series exhibits a pattern of signs, though in this case the negative signs do not necessarily alternate.