Modular Differential Equations with Movable Poles and Admissible RCFT Characters

Studies of modular linear differential equations (MLDE) for the classification of rational CFT characters have been limited to the case where the coefficient functions (in monic form) have no poles, or poles at special points of moduli space. Here we initiate an exploration of the vast territory of MLDEs with two characters and any number of poles at arbitrary points of moduli space. We show how to parametrise the most general equation precisely and count its parameters. Eliminating logarithmic singularities at all the poles provides constraint equations for the accessory parameters. By taking suitable limits, we find recursion relations between solutions for different numbers of poles. The cases of one and two movable poles are examined in detail and compared with predictions based on quasi-characters to find complete agreement. We also comment on the limit of coincident poles. Finally we show that there exist genuine CFT corresponding to many of the newly-studied cases. We emphasise that the modular data is an output, rather than an input, of our approach.

Originally the method was applied to cases with two or three independent characters satisfying what is now called the "non-zero Wronskian condition" 1 , which is the vanishing of a non-negative integer ℓ proportional to the number of zeroes of a certain determinant (we explain this in more detail in the following Section). ℓ is known as the Wronskian index. The work of [1] completely classified admissible solutions to the two-character MLDE with ℓ = 0.
Here "admissible" means the solutions have non-negative integral Fourier coefficients, and also that the identity character is normalised to start with unity, reflecting uniqueness of the vacuum state. For two characters, it turned out that the admissible solutions all correspond to CFTs though there are a couple of subtleties that we will not go into here.
The three-character case, again with ℓ = 0, was investigated for the first time in [2] where several interesting results were found, but not a complete classification of admissible solutions. These results were extended many years later in [25,26] and then more completely in [10,[13][14][15] resulting in a complete set of admissible characters of which a large fraction could be identified as CFTs. In [22], additional information was used to tabulate the complete set of CFTs with three characters and ℓ = 0.
Studying MLDE and admissible solutions beyond ℓ = 0 is more difficult and there are very few papers in this direction. For the two-character case, solutions with ℓ = 2 were considered in [25,27,28], while solutions with ℓ = 4 were analysed from the MLDE perspective in [9,27,29]. Reference [14] studied three characters for ℓ = 2 and [13] classified solutions with three, four and five characters and ℓ = 0. To our knowledge, no analysis of the ℓ ≥ 6 case has been carried out even for two characters. Indeed there seems to be a consensus that the MLDE approach is intractable for ℓ ≥ 6, and to our knowledge no attempt has been made to formulate and solve the MLDE in such cases 2 . This will be the main focus of the present work.
When ℓ < 6 the poles in the MLDE, if any, must be located at the special points τ = ρ or τ = i in moduli space. On the other hand for ℓ ≥ 6, the MLDE can have poles at generic points in moduli space. Hence we refer to the ℓ ≥ 6 case as having "movable poles". To be clear, this only means that their locations are free parameters in the equation, but of course for any particular admissible solution the poles will take fixed values. Also it is important to emphasise that the solutions of the MLDE have no poles and are regular everywhere, leading to completely regular candidate partition functions. The poles are present only in the coefficient functions of the MLDE itself.
There exist other approaches beyond MLDE that have provided insights into admissible characters with ℓ ≥ 6. These approaches avoid explicitly solving, or even formulating, an ℓ ≥ 6 MLDE. For example, [3] employed a novel construction of Hecke operators on vectorvalued modular forms. On the other hand, [6,30] proposed the method of "quasi-characters" about which we will say more below.
The primary motivation of the present work is to study modular differential equations and their admissible solutions for arbitrary ℓ ≥ 6. We will restrict our attention to the case of two characters (second-order equations), though some results will be more general. Despite the complications due to the presence of both movable poles and "accessory parameters", we will be able to make progress using the following strategy. We first of all parametrise such generic MLDEs in a useful way, which in fact can easily be extended beyond the case of two characters. Next we impose single-valuedness of the solutions around all the poles of the MLDE, leading to a set of equations relating the accessory parameters to the locations of the poles. These equations define a hypersurface in the space of poles and accessory parameters.
Looking at the asymptotic region of this hypersurface relates the MLDE for a given ℓ to that for ℓ − 6, corresponding to one of the poles migrating to infinity. This allows us to determine the possible critical exponents for all ℓ ≥ 6 in terms of those for ℓ = 0, 2, 4, which are already 2 However there were some remarkably prescient observations in this direction in the concluding section of [27].
known. This in turn makes it easier to solve the MLDE explicitly, as we show explicitly in the cases of ℓ = 6, 8. Once there are two or more movable poles this becomes more difficult so we follow a slightly different strategy in the ℓ = 12 case.
In [6] a complete classification scheme for admissible characters for the case of two characters and arbitrary ℓ was provided in terms of "quasi-characters", using inspiration from mathematical works [31,32]. The present work, based on MLDE, provides an alternate route to the same result and the two can therefore be compared. We do so at various stages and find complete agreement. This encourages us to hope that the present approach can lead to new results for three or more characters where a full classification based on quasi-characters is not available (though partial results can be found in [8]).
Before going on, let us mention two important points that will provide some context for our work. First, there exists an elegant approach to the classification of vector-valued modular forms due to Bantay and Gannon [33,34]. This approach relies on the classification of modular data, which we do not assume in our work, so it may be considered a complementary point of view. Recently this approach was applied to the classification of n ≤ 4 character solutions in [23].
The second point, already alluded to earlier, is that classifying admissible characters is necessary but far from sufficient to classify CFT. In particular we know of infinite families of admissible characters that cannot correspond to any CFT. One of the most explicit tools to find genuine CFT within families of admissible characters is the coset construction [35][36][37].
A version of this where the numerator is a meromorphic CFT [38,39] was applied to the explicit construction of new CFT with small numbers of characters, and their classification, in [18,22,23,25,40]. In the present work we do not address the problem of classifying actual CFTs within the space of admissible characters, rather our focus is purely on admissible MLDE solutions. Nevertheless, towards the end we will provide explicit examples of genuine CFT corresponding to the characters we construct, which makes it clear that the sub-space of CFT within the space of admissible characters is well-populated even for ℓ ≥ 6.

MLDEs in τ -space
We now move on to the construction of MLDE of nth order and ℓ > 0 and the study of their admissible solutions. We label such MLDE by (n, ℓ).

Bases of modular forms
We will choose a convenient basis of holomorphic modular forms of SL(2,Z). These can have any non-negative even weight w > 2. A generic modular form of this weight is denoted M w (τ ).
These form a multiplicative ring generated by the Eisenstein series E 4 (τ ), E 6 (τ ), which we normalise so that their q-expansion starts with 1. We also use the cusp form: The Klein j-invariant, which will play a key role later on, is given by: Torus moduli space has cusps at τ = ρ ≡ e 2πi 3 and τ = i. We have E 4 (ρ) = E 6 (i) = 0. In the first case it is a fractional zero of order 1 3 and in the second, of order 1 2 . Thus E 3 4 and E 2 6 , of weight 12, both have a single full zero. The most general modular form with a single full zero is a linear combination of these two. Alternatively, and more usefully for us, it can be parametrised up to an overall constant as E 3 4 − p ∆ for some (in principle complex) number p. In this form the leading coefficient in a q-expansion is unity independent of p, which follows from the cusp-form nature of ∆. Additionally, it vanishes at the point τ p in the τ -plane where p = E 3 4 ∆ (τ p ) = j(τ p ). Thus p has a clear geometric meaning as the location in the j-plane of the zero of the corresponding form. Generically it is a complex number.
We will need a convenient parametrisation for modular forms of arbitrary weight. To construct a suitable basis we proceed by dividing all possible M w into three classes: Here M w is generated by one of the following: 1, E 4 , E 6 ,  where the covariant derivative in τ , denoted D, is defined by: when acting on a form of weight w. µ 2s are arbitrary parameters and the ϕ 2s are meromorphic modular functions of weight 2s whose poles are governed by the zeroes of the Wronskian, and whose overall normalisations are specified so that their leading term is unity. Explicitly we have: where: τ χ 0 (τ ) D s+1 τ χ 1 (τ ) · · · D s+1 τ χ n−1 (τ ) . . . . . . . . . . . . D n τ χ 0 (τ ) D n τ χ 1 (τ ) · · · D n τ χ n−1 (τ ) (2.6) and D is defined in Eq. (2.4).
It is easy to see from the definition that W n−1 = DW n . From Eq. (2.5) we find the useful relation: The Wronskian index ℓ is defined to be an integer ℓ such that ℓ 6 is the number of zeroes of W n . This number does not have to be an integer because of the possibility of fractional zeroes at the cusps of moduli space, where a zero at τ = ρ counts as 2 3 of a full zero, and at τ = i counts as 1 2 of a full zero. Thus for general RCFT with n characters, ℓ can be any non-negative integer other than 1. Note that if the total number of zeroes is fractional then they must necessarily occur at the cusps, while if the total number is integral (i.e. ℓ is a multiple of 6) then they can occur anywhere in the fundamental region. More generally the fractional part of ℓ 6 describes the zeroes fixed at the cusps, while the integral part describes zeroes that are allowed to be at generic points of moduli space (including possibly the cusps).
This motivates us to define ℓ ρ , ℓ i , ℓ τ to be the contribution to ℓ from the zeroes at ρ, i and generic points respectively. Here ℓ ρ is even, ℓ i is a multiple of 3 and ℓ τ is a multiple of 6 3 and these quantities satisfy: The goal is to classify all possible MLDEs of the form Eq. (2.3) and then find suitable solutions to them. These take the form: We call them admissible when a i,k are integers ≥ 0 for all i, k, which means they potentially correspond to degeneracies of states. Additionally a 0,0 = 1, reflecting non-degeneracy of the vacuum state. In what follows we will establish several properties of the equations for generic (n, ℓ), including a count of the parameters on which they depend. After that we will restrict to n = 2 and consider certain values of ℓ ≥ 6 in some detail and examine families of solutions.
We will start by making a genericity assumption -that for any given ℓ, the largest possible number of zeroes of W n are at generic, distinct points in moduli space, away from each other and from the special points τ = ρ, i. With this assumption, ℓ ρ takes its minimum allowed values of 0, 2, 4 and ℓ i takes its minimum allowed values of 0, 3. Later we will consider what happens when the zeroes merge.
Writing ℓ = 6r + u, 0 ≤ u ≤ 5, the possible cases are: We will also require that the solutions of the MLDE furnish irreducible representations ϱ of the modular group PSL(2,Z). By definition, This matrix will be reducible if it has two coincident entries, i.e. any of the h i is integral or any two h i differ by an integer. Using ϱ(S 2 ) = ϱ (ST ) 3 ) = 1 it can then be shown [10] that S also has two equal eigenvalues and the representation is reducible. It follows that in irreducible representations, none of the h i is integral and no two of them differ by an integer.
Now we turn to the parametrisation of the coefficient functions ϕ 2s (τ ) for s ≥ 2. Writing ℓ = 6r + u, u = 0, 1, · · · , 5 as above, we consider the different u values separately as they have slightly different characteristics. From the definition of ϕ 2s in Eq. (2.5) and the fact that W n has precisely ℓ 6 zeroes, it follows that ϕ 2s can be expressed as a ratio of holomorphic modular forms such that the denominator has weight 2ℓ. This follows from the fact, mentioned earlier, that a full zero (ℓ = 6) is achieved by a general weight 12 modular form E 3 4 − p∆. To achieve the desired modular weight, the numerator of ϕ 2s must be modular of weight 2ℓ + 2s.
From Sub-section 2.1, we find that, under the genericity assumption, these denominators of weight 2ℓ can be parametrised as follows: Thus for all u ̸ = 1 the denominators have exactly r full zeroes, whose locations as a function of τ are determined by the r parameters p I , as well as u fractional zeroes whose locations are fixed and hence they are not associated to any free parameters. For u = 1 we instead have r − 1 full zeroes, two zeroes of order 1 3 at τ = ρ and a zero of order 1 2 at τ = i. Applying Eq. (2.7), we find: By inspection we see that in every case, the expression has a leading term ℓ 6 as q → 0. Since we are normalising every ϕ 2s to start with 1, it follows that µ 2 = ℓ 6 . We now consider the behaviour of solutions around τ → i∞, where the appropriate coordinate is q = e 2πiτ → 0, by inserting the leading behaviour χ i ∼ q α i + O(q α i +1 ) into the MLDE. In a CFT, the exponents α i determine the central charge c and the conformal dimensions h i via: where h 0 = 0, corresponding to the identity primary. Expanding: and inserting this as well as Eq. (2.9) into the MLDE Eq. (2.3), at leading order we find the indicial equation: If the roots of this equation are α i , i = 0, 1, · · · , n − 1 then we see that: where we used µ 2 = ℓ 6 and ϕ 2,0 = 1. The above equation is the valence (or Riemann-Roch) formula.
The lower order terms in Eq. (2.15) are straightforward but tedious to write explicitly, and they similarly allow us to determine the parameters µ 4 , µ 6 , · · · µ 2n in terms of the critical exponents α i , i = 0, 1, · · · , n − 1. We refer to the parameters µ 2s as rigid parameters since they are completely determined by the critical exponents. Conversely if we know the µ i then they determine the critical exponents.

(2, ℓ) MLDE
In this paper we will work with two characters, yet keeping the Wronskian index ℓ arbitrary.
To our knowledge this region of (n, ℓ) space has not previously been investigated barring some insightful observations in [27]. Let us mention that for two characters, the concept of movable poles essentially corresponds to the "non-rigid" case from the perspective of Fuchsian differential equations. These are the cases where the parameters in the equation are uniquely determined by the exponents of the solutions. Technically the rigid cases among the (2, ℓ) family arise for ℓ = 0, 2. However as we will argue below, admissible characters for ℓ = 4 are completely determined in terms of those for ℓ = 0. Thus the non-rigid cases of interest start at ℓ = 6, which is also where movable poles first arise. So for practical purposes we can think of "non-rigid" (2, ℓ) MLDE as being equivalent to "MLDE having movable poles". This justifies our use of "rigid" for the parameters µ 2s of the previous sub-section and "non-rigid" for the rest.
The general (2, ℓ) MLDE is: where ϕ 2 , ϕ 4 are meromorphic modular forms of weight 2 and 4 respectively. Now consider the coefficient function ϕ 4 . Its denominator must have (at most) the zeroes of the Wronskian W n . The numerator is then a general modular form of weight 4 higher.
Also the form must be normalised so that its q-expansion starts with 1. This implies that it takes the form: The b 4,I are "accessory parameters" about which we will have a lot to say in the rest of this paper (they carry the subscript 4 because they arise in a weight-four modular function).
Notice that for the middle case there is no pole at τ = ρ due to cancellation of an E 4 between the numerator and denominator. Also the last case has an extra power of (E 3 4 − b 4,r+1 ∆) in the numerator.
Returning to the MLDE Eq. (2.17), we have already determined that µ 2 = ℓ 6 . Also, the leading term of ϕ 4 in a q-expansion is normalised to unity. Then the indicial equation determines: where α i are the critical exponents around q = 0. Also it is known [27] that with two characters, ℓ is always even. Hence W cannot have an odd number of zeroes at τ = i. Then, recalling that ℓ = 6r + u, one is restricted to even values of u. With all the above information, we write the general (2, ℓ) MLDE as follows: ℓ = 6r : χ(τ ) = 0 ℓ = 6r + 2 : χ(τ ) = 0 ℓ = 6r + 4 : Well-studied special cases are the MMS equation [1] which corresponds to ℓ = 0: and the ℓ = 2 equation studied in [27,28]: As we see, these equations have no movable poles.
Returning now to the general case, although p I are generically complex, they are subject to constraints arising from the fact that c and the degeneracies are rational. As we will show below, the symmetric polynomials in the p I must all be real and rational. This generalises the statement in [27]) that a single pole must be real and rational. Let us mention here that for a real pole, τ p lies in the subspace of moduli space for which j(τ p ) is real, namely {Re(τ ) = 0} ∪ Re(τ ) = 1 2 ∪ |τ | = 1. So far we have only considered the indicial equation about q = 0. However, the characters also need to have appropriate behaviour near the cusps τ = ρ, i in order to be single-valued.
We label the exponents around τ = ρ, i as α (ρ) , α (i) respectively to avoid confusion with the exponents α around τ = i∞. Near τ = ρ we introduce a new coordinate: When τ circles ρ by e 2πi/3 , we return to the same point in moduli space. The above change of variables converts this to a regular circle z → e 2πi z, so z is a good coordinate at the cusp.
In this coordinate, E 4 ∼ z 1 3 and j ∼ z as z → 0. The indices at τ = ρ are found by inserting the trial solution Regularity imposes the requirement that α (ρ) is a non-negative multiple of 1 3 . A similar analysis tells us that τ = i is a multiple of 1 2 . Now expanding out the MLDE Eq. (2.20) we get: As we are working near τ = ρ where E 4 and j vanish while E 6 and ∆ tend to finite values, we can replace µ 2 ϕ 2 by u 6 E 6 E 4 . This is because near τ = ρ, µ 2 ϕ 2 has u 6 poles where ℓ = 6r + u, by our genericity assumption. Meanwhile ϕ 4 given in Eq. (2.18) reduces near τ = ρ to: From the definition of the j-invariant we have: 3 , it follows that: From this we can also deduce the behaviour: Now we change variables in the MLDE using Eq. (2.23) and insert Eq. (2.24). In the case u = 0 we find the indicial equation: The solutions are α (ρ) = 0, 1 3 . Thus both solutions exhibit regular behaviour as functions of τ . While this has been a good consistency check, it does not tell us anything new about the parameters in the MLDE.
Next, for u = 2, the MLDE near τ = ρ becomes: Going now to the z-coordinate and inserting χ(z) ∼ z α (ρ) , the indicial equation is: whose solutions are α (ρ) = 0, 2 3 . Again the solution is consistent but does not provide new information.
The situation is different for the last case, u = 4. Eq. (2.26) tells us we have non-trivial behaviour for both ϕ 2 and ϕ 4 . This indicial equation now becomes: where: From this we learn that α As we have seen, these exponents are non-negative multiples of 1 3 , which leads to the unique solution α . It now follows from Eq. (2.33) that: This was previously noted in [6] for the case ℓ = 4. Here we see that it is true for all ℓ = 6r + 4 as long as there are precisely two poles (of 1 3 -order each) at the cusp τ = ρ and the rest are at generic values away from the cusp. We can think of this result as determining b 4,r+1 in Eq. (2.34) in terms of the other b 4,I . Then the (so far) independent coefficients are b 4,I , I = 1, 2, · · · , r and p I . Thus, despite the appearance of an apparent additional parameter b 4,r+1 , the case ℓ = 6r + 4 is actually similar to the cases ℓ = 6r, 6r + 2 in that all of them have precisely 2r parameters of which r correspond to poles p I of the coefficient functions and the other r are the b 4,I . As indicated above, these are the accessory parameters familiar from Fuchsian differential equations.
In standard treatments of the MLDE, starting with [1], one now solves the equation order by order using the Frobenius method, and imposes admissibility at each successive order, in particular non-negative integrality of the Fourier coefficients. We will do this eventually, but here we pause to rewrite the MLDE treating j(τ ), rather than τ , as the independent parameter. This makes it somewhat easier and more intuitive to write out general MLDEs and impose their single-valuedness around poles of the coefficient functions. Of course the Fourier coefficients in the j variable have no integrality restrictions, so we will need to return to the τ -coordinate in order to check integrality of the coefficients in the q-expansion and thereby determine admissibility of solutions.
3 MLDEs in j-space In this Section we return to the general case of n characters and study MLDEs in a formalism where the independent variable is the Klein invariant j(τ ) rather than τ (this was explored for special cases in [26,27]). As a warm-up exercise, let us consider the one-character case.
First we fix ℓ = 6r. Then the most general allowed character is: where p I are a set of r complex numbers that describe the zeroes of the character (which is the same as the Wronskian in this case). The MLDE satisfied by this character is trivially seen to be: where: Here we have labelled the first non-trivial coefficient as ψ 2 (j) in keeping with the convention used for the MLDE in τ , though here it does not reflect the modular weight, since everything is modular invariant (up to possible phases). Also note that the coefficients µ 2s are now absorbed into the normalisation of the ψ 2s .
The generalisation of the above to the case of ℓ = 6r + u is straightforward: the character acquires an extra multiplicative factor of j 1 3 for each zero at τ = ρ and a factor (j − 1728) 1 2 for a zero at τ = i (we are not requiring admissibility at this stage, which would have ruled out the latter). Then the coefficient function ψ 2 (j) acquires an additive term: for each zero at ρ, i respectively.
Moving on to the n-character case, the MLDE in terms of the independent variable j can be written: The modular invariants ψ 2s can have poles at the special points j = 0, 1728 as well as at generic points j = p I , I = 1, 2, · · · r. The solutions can be expanded as follows around the special points: Similarly around each of the generic poles j = p I , we parametrise the solutions as: One should keep in mind that the a i,k with superscripts (ρ), (i), (I) have no particular integrality property.
The relevant Wronskians are defined similarly to Eq. (2.6) 4 : and: As noted in [27], W r (j) necessarily have poles, unlike W r (τ ). The poles are introduced by the powers of dj dτ that relate two Wronskians. For example the relations between the Wronskians W n (j) and W 0 (j) with that of the Wronskians W n (τ ) and W 0 (τ ) are as follows (similar but more complicated relations can be found for all the W r ): Using Eq. (2.27), we see that: Thus dj dτ has two zeroes of order 1 3 at τ = ρ and one of order 1 2 at τ = i, so W (j) acquires n(n−1) 3 poles at ρ and n(n−1) 4 poles at i. It also has the zeroes of W n (τ ). So we could define a new Wronskian index ℓ j such that ℓ j 6 gives the total number of zeroes of W (j): We see that for n = 1, ℓ j = ℓ, while for n = 2, ℓ j = −7 + ℓ as one can read off from Page 434 of [27]. Actually this relation is slightly misleading as it does not contain all the information: the extra poles contained in the first term on the RHS are necessarily at the points τ = ρ, i and are not free to move. So it is better to break up ℓ j into contributions from zeroes/poles at ρ, i and generic positions (as we did before for ℓ): Now, positive values of these quantities denote zeroes while negative values denote poles.
Recall that in the τ -space case, the terms are individually ≥ 0 and ℓ ρ , ℓ i and ℓ τ are multiples of 2,3,6 respectively. Then, taking account of the new poles introduced by the change of variables, we get: Positivity of ℓ p , ℓ i , ℓ τ then induces obvious lower bounds on ℓ j ρ , ℓ j i , ℓ j τ . Notice that it is possible for ℓ j ρ , ℓ j i to vanish due to cancellations between poles induced by the change of variables to j and zeroes of the original Wronskian at the special points ρ, i.
From the above considerations, we can readily fix the first coefficient function ψ 2 (j) in Eq. (3.5), which is given by: The result is:

(2, ℓ) MLDE in j-space
We now again specialise to the case of two characters, keeping the Wronskian index arbitrary.
The first step is to determine the remaining coefficient function ψ 4 (j) in the MLDE for this case. From the definition we have: Now, Inserting the behaviour χ i ∼ j α (ρ) i near j ∼ 0 (τ = ρ) we find: The reason to write the first correction to W 0 is that the leading term can vanish, if α 1 vanishes. However since the two exponents must be distinct, the leading term of W 2 cannot vanish. Thus we have ψ 4 (j) ∼ j −2 unless one of the exponents vanishes, in which case The exponents α Writing ℓ = 6r + u, we have u = ℓ ρ = 0, 2, 4 respectively for the cases ℓ = 6r, 6r + 2, 6r + 4.
It follows that the exponents are as follows: (these facts have already been derived in terms of the τ coordinate in Sub-section 2.3, but here our goal is to derive everything independently in the j coordinate). Thus in the first two cases the leading term in W 0 indeed vanishes and the subleading term has to be used. We see that the behaviour of ψ 4 (j) in the three cases is ∼ j −1 , ∼ j −1 , ∼ j −2 respectively.
Next we consider the behaviour near j = 1728 (τ = i). Similar arguments tell us that This time we have (see Eq. (A.4)): in every case, so the leading term always vanishes and we have a simple pole in j − 1728.
From the r generic zeroes of W 2 at j = p I , we get a simple pole at each of these points.
Finally, the τ → i∞ behaviour requires that the overall power of j as j → ∞ is −2. Hence ψ 4 (j) must contain, in the numerator, a generic polynomial in j of degree r for ℓ = 6r, 6r + 2 and of degree r + 1 for ℓ = 6r + 4. Thus finally we get: ℓ = 6r : These expressions can easily be confirmed by explicitly changing variables from τ to j in Eqs.
(2.12), (2.18). However, the methods we have used to arrive at them are useful in the general case (higher than second-order) and one does not need to invoke the MLDE in τ to write the equations in j-space.
By considering the indicial equation around j = 0 (τ = ρ), we will again find, in the ℓ = 6r + 4 case, that it is possible to fix b 4,r+1 in terms of the remaining coefficients. As a result, once we impose the indicial equations there are 2r independent coefficients in every case, namely the p I and b 4,I with I = 1, 2, · · · , r. Note that the MLDE is totally symmetric under permutations of the p I and also under permutations of the accessory parameters b 4,I .
The differential equations in the j plane that were discussed above are examples of Fuchsian differential equations (FDE) with regular singular points. However they have some special features. A general FDE with regular singular points is of the form: where the coefficient functions α i (x) have at most poles of order i at the regular singular points. However due to our genericity assumption, the Wronskian has only simple zeroes at generic points p I . Hence in our case, all the coefficient functions have only simple poles (with the exception of poles at τ = ρ, which are double poles whenever the Wronskian index is equal to 4 mod 6). This means that ab initio they span a more restricted set than general Fuchsian differential equations with regular singular points. We will revisit this issue later on when we move away from the genericity assumption by allowing movable poles to coalesce.
Meanwhile, as already noted above, our b 4,I correspond in the language of FDE to what are called "accessory parameters".
3.3 Reduction of ℓ = 6r + 4 to ℓ = 6r Let us note an important general lesson that is exemplified by Eq. (3.23). In the third line, the lower of the two exponents is 1 3 . This means we can take any solution of the ℓ = 6r + 4 MLDE and write it as: where ζ(j) has an expansion about j = 0 in positive powers of j. Then, as is easily verified, ζ(j) solves an MLDE with (n, ℓ) = (2, 6r). This means that, in terms of having a well-defined power-series expansion about all the singular points of the MLDE, every ℓ = 6r + 4 solution factorises into the E 8,1 character j 1 3 times a solution of the (2, 6r) equation. It is not, however, necessarily the case that both factors are admissible. In particular, it is possible for ζ(j) to be a non-admissible character while j 1 3 ζ(j) is admissible. A very striking example, noted in sub-section 5.1 of [6], is that the characters of the c = 33 CFT of [41], which has Wronskian index ℓ = 4, can be written as the product of j 1 3 times a solution with c = 25 and ℓ = 0. However, the c = 25 solution has some negative coefficients in its q-series and therefore does not count as admissible (as we will see below, it is actually a quasi-character). Yet, after multiplying it by j 1 3 it becomes admissible and in fact a genuine CFT. But this CFT, despite the factorisation described above, is by no means a tensor product of two other CFTs. The factorisation of solutions described above for ℓ = 4 is easily seen to persist for all ℓ = 6r + 4. Hence we no longer need to discuss MLDEs for the case ℓ = 6r + 4, even though we have formulated them above. All we need to remember is that admissible solutions in these cases are found by considering all integral (not only admissible) solutions of the ℓ = 6r equation, multiplying each one by j 1 3 and then testing for admissibility. On the other hand, in the first two lines of Eq. (3.23), the lower of the two exponents is 0. This tells us that we cannot extract a positive power from the character and still hope to find a positive power-series expansion in j. Moreover this fact persists for ℓ = 6r, 6r + 2 as long as the genericity assumption is obeyed. So even relaxing admissibility, the characters in these cases do not factorise.

Determining accessory parameters: the (2, 6) case
Having dealt with the indicial equations about j = 0, 1728, the next step is to study the indicial equations around j = p i . This will determine all the accessory parameters b 4,i in ψ 4 (j). Let us start with a particular case, the (2, 6) MLDE which from Eq. (3.25) has the form: A priori it has 2 parameters, b 4,1 and p 1 . In this case we have ℓ ρ = ℓ i = 0 and ℓ 1 = 6.
Let us examine the leading behaviour of the characters about j = p 1 . Since p 1 is not a special point in moduli space (by the genericity assumption), the critical exponents around it must be integers. We substitute the expansion Eq.
so the exponents are α 1,1 for this solution, and continuing in this way we are guaranteed to determine the subsequent coefficients. If we insert the other value α It is important not to identify these two solutions with the two characters χ0(τ ) and χ1(τ ) that form the two independent CFT characters with integral expansions in q. The reason is that here we are expanding around a point inside moduli space instead of the point τ → i∞. Hence each pair is in general a linear combination of the other pair.
Now the recursion relation that should have determined the next coefficient a (1) 0,2 does not contain that variable. This is a consequence of the integral difference in indices that we noted above. Instead, it gives us a constraint on b 4,1 : Thus the above constraint is the condition that the second solution does not have a logarithmic piece. If we do not implement the constraint, one of the characters becomes multi-valued around a zero of the Wronskian and has to be rejected on physical grounds 6 .
Substituting Eq. (3.31) into Eq. (3.32) we get: After multiplying by all the denominators, this becomes a quadratic curve in Using α 0 = − c 24 and α 1 = c 24 − 5 6 (the latter follows from Eq. (2.16)), we see that this is positive for all c ̸ = 10, and the quadratic is a hyperbola. At c = 10 the curve degenerates to a parabola. From Eq. (3.33), notice that when b 4,1 = p 1 then we have p 1 = 0 or p 1 = 1728, in other words the pole has to be at one of the cusps τ = ρ, τ = i of moduli space. The curve Eq. (3.33) determines the accessory parameter in terms of the pole p 1 .
It is useful to consider the asymptotic region of the curve Eq. (3.33) as p 1 → ∞. In this limit, the Wronskian no longer has a zero in the finite region of moduli space, hence now we should find solutions with ℓ = 0 and this is indeed what happens as we will see later in several examples. Defining: we see that at large p 1 , Eq. (3.33) becomes: with the solutions: Since ℓ = 6, we have from Eq. (2.16) that α 0 + α 1 = − 5 6 . This allows us to simplify the above equation to: Such objects are called "weak VVMF" in [26].
Next we consider generic values of ℓ and show that the accessory parameters are determined similarly. As we will see, this allows the complete determination of α 0 , α 1 for all ℓ in terms of those for ℓ < 6 which are already known.

Determining accessory parameters: the general case
In the most general case with two characters and arbitrary ℓ, as long as the singularities are well-separated the phenomenon is very similar. For each singularity p I we get one constraint on the combined set of b 4,I and p I by trying to calculate the second-order coefficient a (I) 0,2 . This provides us with a set of simultaneous equations for the b 4,I . In the cases ℓ = 6r, 6r + 2 this is sufficient to determine all the b 4,i in terms of the p I . We now consider each family in more detail.
In this case, we have r full zeroes that are well-separated from each other and from the special points j = 0, 1728. Substituting Eq. (3.8) into the ℓ = 6r case of Eq. (3.25) and equating the coefficient of (j − p I ) α (I) i −1 to zero, we get the indicial equation: and hence (α At the next order in the expansion, setting to zero the coefficient of (j − p I ) α (I) i we get: which is manifestly a generalisation of Eq. (3.30). Choosing α (1) 0 = 0 in the above, we get: At the next order, we set to zero the coefficient of (j − p I ) α (1) i +1 . First let us examine the term that contains a (I) 0,2 : Since this is proportional to the indicial equation, the above expression is identically zero and hence the dependence on a (I) 0,2 drops out. In its place, we find a set of constraint equations on the parameters of the MLDE (assuming the p I are distinct from the accessory parameters b 4,J ): Now substituting the value of a Thus we get a set of r coupled equations for the accessory parameters b 4,J which can be solved in principle to determine them as functions of the p I .
We can think of these equations as defining a sub-manifold or algebraic variety in the 2r-dimensional parameter space of the p I and b 4,I . We will call them "accessory equations".
If we multiply out all the denominators, these become a set of r polynomials of degree 2r.
The special case in the last section, Eq. pr fixed. Then the equations for I = 1, 2, · · · , r − 1 become: while the equation for I = r becomes: The second equation determines x r in terms of the product of exponents α 0 α 1 : We can simplify this using the valence formula Eq. (2.16) which tells us that α 0 + α 1 = 1−ℓ 6 . Then: Meanwhile, the first set of equations is precisely the one for the MLDE with r replaced by r − 1, i.e. Wronskian index ℓ replaced by ℓ − 6, with the replacement: Applying the procedure recursively, this equation determines α 0 , α 1 (up to exchange of characters) for all ℓ = 6r given their values for ℓ = 0, which are known from [1].
For the case of ℓ = 6r + 2, we again have r distinct full zeros at p I , and also a zero of order 1 3 at τ = ρ (j = 0). The genericity assumption also says that p I ̸ = 0, 1728. This case is very similar to the ℓ = 6r case and we again find (α At the next order we have: At the next order we find the constraint:  Since we have argued above that the ℓ = 6r + 4 case can always be reduced to ℓ = 6r by extracting a factor j 1 3 , we do not need to find the accessory equations separately for that case. Hence at this stage our analysis of accessory equations is complete. To summarise, what we have learned from the asymptotic analysis is that Eq. (3.49) is true for all ℓ = 6r + u where u = 0, 2. Although there are two choices in this equation, it is clear that they are related by an exchange of characters. We can invert Eq. (3.49) and iterate it u times (where ℓ = 6r + u) to get: Here we have chosen α 0 = − c 24 and α 1 = − c 24 + h. So the above equations tell us that: Thus, using only the MLDE for generic ℓ, we have demonstrated that the central charge and conformal dimension of a solution for any ℓ = 6r, 6r + 2 are related as above to those of an MLDE solution with ℓ = 0, 2 (with a corresponding result for ℓ = 6r + 4 following from factorisation of the solutions in that case). As we show below, this perfectly agrees with the analysis from quasi-characters [6].

Admissible range of central charges for (2, ℓ) solutions
In this sub-section, we study the admissible range of central charges for (2, ℓ) solutions, based on the asymptotic analysis and knowledge of the admissibility range for (2, 0) and (2,2) solutions. Then we will present the results for ℓ = 6, 8, 12, 14, which will be used in upcoming sections.
We first note that Eq.
respectively. Imposing unitarity via h (ℓ) > 0, we also get the lower bounds: One of the four possibilities in Eq. (3.56) can be ruled out, namely To see this, let us suppose it is allowed. Then the unitarity bound Eq. (3.57) for c (ℓ) gives . However, the unitarity bound directly implies that c (ℓ−6) > 2(ℓ − 7). Thus we have a contradiction and the above possibility is ruled out.
It follows that at each step we can only have the following three possibilities: We can now recursively work out the ranges for any given ℓ = 6r, 6r + 2 starting from the known ranges for ℓ = 0, 2 [1,25]: and applying all the possibilities above subject to the constraint Eq. (3.57). We find 7 : range for (2, 0) and (2, 2) solutions, we note that for any ℓ = 6r, or 6r + 2 we have c (ℓ) = n + c (ℓ=0) , m + c (ℓ=2) , where n, m are non-negative integers (as ℓ is a non-negative integer).
This fact was first noted in [43] where the result was derived using representation theory of PSL(2, Z). Here we have derived it using just the MLDE approach. Similar results about the modular data for n-character admissible solutions with n = 3, 4, 5 have been obtained in [13].
It is worth exploring if those results can also be derived within the MLDE approach.

Detailed solution for the case of one movable pole
With the understanding of this system that we have described above, it is relatively straightforward to directly find the most general admissible solutions of the MLDE in the case ℓ = 6, where there is one movable pole p 1 and one accessory parameter. We first present the solution and then examine its relation to the quasi-character approach.
We will study the (2, 6) case in detail, with some formulae reserved for Appendix C.2, while a similar analysis for the (2,8) case can be found in Appendix D. For future use, a review of the Frobenius solution for the (2, 0) and (2, 2) MLDEs is presented in Appendix C.1. This contains formulae that will be needed below.

Solving the MLDE with one movable pole
In this sub-section we will adapt the various elements of the theory of MLDEs developed so far into an organised method to solve them. This method involves incorporation of the accessory equation into the solution from the outset. It allows us, as we will see, to solve the MLDE in the present case completely and thereby derive features of the solution that were suggested by quasi-character theory [6]. The (2, 6) MLDE in the τ -plane is: In this form, we have three parameters, the rigid parameter α 0 α 1 and two non-rigid parameters, the movable pole p 1 and the accessory parameter b 4,1 . We use the first three orders of the Frobenius solution as applied to the identity character. At leading order, we have the indicial equation which determines the rigid parameter in terms of the central charge, and at the second and third order we have the following : Here m 1 and m (6) 2 are the Fourier coefficients of the identity character. The superscript (6) indicates that this is of the (2, 6) solution. The explicit forms of f 1 (c, p 1 , b 4,1 ) and f 2 (c, p 1 , b 4,1 ) are given in Eq. (C.11) and Eq. (C.12).
In the next step, one solves for the three parameters of the MLDE in terms of objects associated to the identity character, namely the central charge c and the Fourier coefficients m (6) 1 and m (6) 2 . This has already been done for α 0 α 1 in Eq. (4.2). For the remaining parameters we obtain: 1 , m 2 ) (4.6) The explicit expressions for the right hand sides can be found in Eq. (C.13) and Eq. (C.14).
We note that both p 1 and b 4,1 are rational functions of m 1 and m 2 with coefficients being rational functions of c. In particular, we see that the movable pole in the (2, 6) solution is rational, as already noted in [27]. Later we will discuss the general version of this statement. The next step is to invoke the accessory equations Eq.
where A 2 (c) and B 2 (c) are given in Eq. (C.15). Consulting Eq. (C.5) we immediately find a relation between the coefficient B 2 (c) above, and the degeneracy for the (2, 0) MLDE solution at a central charge c − 24: Some additional calculation shows that: Thus Eq. (4.7) is the same as: At the next stage, we insert Eq. (4.10) in Eq. (4.5) and Eq. (4.6) to obtain: 1 ) (4.12) The explicit expressions for the right hand sides are given in Eq. Eq. (4.12) are its parametric equations with m 1 serving as a parameter on the curve. We carry on solving the MLDE to higher order. At the next order, after using Eq. (4.10) we obtain the following: where A 3 (c) and B 3 (c) are given in Eq. (C.18). It is again remarkable that m (6) 3 has a linear dependence on m (6) 1 . In the same way as was done above, one shows that m (6) 3 can be written in terms of the (2, 0) solution as follows: which is very similar to the form of m 2 in Eq. (4.10). This motivates us to propose the relation: We have performed a computer check of this phenomenon to order 8. We expect it to hold for all k ≥ 2 and hope to provide a proof in future work.
Notice that we can extend Eq. (4.15) to include k = 1. When we plug in k = 1 in Eq. (4.15) we get m 1 (c)) which is an identity after noting m We should emphasize that Eq. k s and each of Eq. (4.15), for k ≥ 2, leads to a Diophantine equation, after defining N = 5c. The first two are N 4 + (2m In particular this shows directly that N = 5c is an integer. Now, inserting the admissible In this form, we have three parameters, the rigid parameter α 0 α 1 and two non-rigid parameters, the movable pole p 1 and the accessory parameter b 4,1 . We use the first three orders of the Frobenius solution as applied to the identity character. At leading order, we have the indicial equation which determines the rigid parameter in terms of the central charge, and at the second and third order we have the following : Here m In the next step, one solves for the three parameters of the MLDE in terms of objects associated to the identity character, namely the central charge c and the Fourier coefficients m and m (8) 2 . This has already been done for α 0 α 1 in Eq. (4.21). For the remaining parameters we obtain: 1 , m 2 ) (4.25) The explicit expressions for the right hand sides can be found in Eq. (C.21) and Eq. (C.22).
We note that both p 1 and b 4,1 are rational functions of m  and c. Similar to the (2, 6) computation, remarkably, we get the following linear equation in At the next stage, we insert Eq. (4.26) in Eq. (4.24) and Eq. (4.25) to obtain: 1 ) (4.28) The explicit expressions for the right hand sides are given in Eq. (C.23) and Eq. (C.24). These equations now have a geometrical interpretation, similar to the (2, 6) case.
We carry on solving the MLDE to higher order and obtain : We have performed a computer check of this phenomenon to order 8. We expect it to hold for all k ≥ 2 and hope to provide a proof in future work.
We can extend Eq. (4.29) to include k = 1. When we plug in k = 1 in Eq. (4.29) we get m 1 (c)) which is an identity after noting m In particular this shows directly that N = 5c is an integer. Now, inserting the admissible 1 ≥ 0.

Brief review of quasi-characters
A construction of admissible characters for all two-character CFT was presented in [6] 8 . This proposal did not use MLDEs with movable poles (i.e. ℓ ≥ 6) that we are using here, rather it only made use of solutions to the MMS equation, which has ℓ = 0, and a similar equation with ℓ = 2. Now we are in a position to compare our results, obtained from the ℓ = 6 MLDE, with this approach. For this we first briefly review the quasi-character approach and its application to the (2, 6) case (for a detailed exposition with references, see [6]). Then we will compare the results of the present paper with it.
Ref. [6] started from the observation that although the (2, 0) MLDE -the MMS equation -has only finitely many admissible solutions, it has infinitely many more solutions having 8 There is an earlier construction of VVMF due to Bantay and Gannon [33,34], however, that requires advance knowledge of the possible modular data while here we do not make this assumption. Also here, as part of admissibility, we always impose the requirement that the leading term of the identity character is unity. Now we return to the case ℓ = 6. Here one must add precisely two ℓ = 0 quasi-characters differing in central charge by 24. We take one of these to be any of the MMS solutions, denoted χ A i (where A stands for "admissible"), whose central charge lies in the MMS list Eq. (4.35), and the other to be the quasi-character χ Q i with central charge 24 higher. We 9 Although these solutions do not describe CFT, they can still be assigned a value of c by writing their leading critical exponent α0 as − c 24 . 10 These all correspond to CFTs, except for the first and last cases that are "Intermediate Vertex Operator Algebras" [44].
denote the latter central charge by c and the former by c − 24. Thus we form the sum: This sum has the following properties: (i) it has central charge c and satisfies Eq. (2. 16) with Wronskian index 6, (ii) the negative degeneracies of the quasi-character in the sum are potentially cancelled by the positive terms in the admissible character, depending on the value of N 1 . Thus the sum is admissible for N 1 greater than some lower bound, which varies from case to case.
In view of completeness of the above approach, one therefore predicts that all (2,6) admissible characters (and hence all (2, 6) CFT) have central charges:

Comparison of quasi-character and MLDE results
In this sub-section we confront the explicit admissible MLDE solutions described above with the quasi-character approach. The former approach has one free parameter, which we can take to be p 1 describing the location of the zero of the Wronskian, or the Fourier coefficient m 1 representing the degeneracy of the first excited state in the identity module. The two are related by Eq. (4.11). The latter approach has a free parameter N 1 , that also determines the first excited state degeneracy m 1 . Thus p 1 , the location of the movable pole, must be a function of the integer N 1 . We see that admissibility quantises the location of the movable pole and also that the quasi-character parameter N 1 is the natural integer in terms of which this quantisation can be expressed. We now exhibit these relations in all the cases. We will see that all solutions lie on one of the two branches of the hyperbola in Eq. In this case we have: This is also the solution obtained by the quasi-character method In this case we have: This is also the solution obtained by the quasi-character method: In this case we have: This is also the solution obtained by the quasi-character method: This is a curious case, already remarked upon in Section 5.2 of [6]. What happens here is that χ LY n=11 has an integral q-expansion only if the first term of the identity character is normalised to 7, rather than 1. This is the normalisation chosen above. The first excited state "degeneracy" of this quasi-character is −1742 while all others are positive. Since the identity character of the sum will be considered admissible only when its leading term is 1, we must divide the sum by 7 as shown above. As a result the degeneracy of the first excited . This can be any integer, as long as we choose N 1 to be 1742 plus a multiple of 7. With this choice, the sum in Eq. (4.53) has all integral coefficients even after dividing by 7, which is a miracle of sorts since it means all the infinitely many coefficients become multiples of 7 even though neither of the terms in the sum has this property.
Admissible Solutions (v) c = 28, m 1 ≥ 0, m 2 = 97930 + 28m 1 , m 3 = 21891520 + 134m 1 (4.54) In this case we have: This is also the solution obtained by the quasi-character method This is also the solution obtained by the quasi-character method with m 1 = 1 2 (N 1 − 511). Here the quasi-character, when normalised to be integral, starts with 2. For m 1 to be integral, we must choose N 1 to be an odd integer. This is also the solution obtained by the quasi-character method Admissible Solutions (viii) c = 31, m 1 ≥ 0, m 2 = 110980 + 133m 1 , m 3 = 44696513 + 1673m 1 (4.66) This is also the solution obtained by the quasi-character method This is also the solution obtained by the quasi-character method with m 1 = 1 3 (N 1 − 790) and N 1 has to be chosen to be 790 plus a multiple of 3.

Analysis of the accessory equation
Let us now analyse the accessory equation Eq. (3.33) in the case of ℓ = 6 in some more detail.
This equation can be re-written as a quadratic: As noted below Eq. (3.33), this is a hyperbola for all values of α 0 α 1 except when α 0 α 1 = 25 144 , corresponding to c = 10, when it degenerates to a parabola. Remaining away from c = 10, we now analyse the hyperbola in some detail. We will see, among other things, that all (2,6) solutions with N 1 > 0 lie on one branch of the hyperbola, with the other branch corresponding to negative values of N 1 .
To illustrate this, we pick an example. Consider the (2, 6) solution with c = 25. In this case, we have: N 1 = m 1 + 245 (see previous section). For this, the hyperbola is given below.  . Note that, the origin lies on the lower branch. This can be seen from the fact that when N 1 → 0, we have b 4,1 p 1 → 29 5 (whose slope is equal to the purple asymptote), which intersects the lower branch at the origin. This means that the point N 1 → 0 lies on the bottom end of the lower branch. Also, note that when N 1 → ∞, we have b 4,1 p 1 → 1 25 , whose slope is equal to the pink asymptote. This means that the point N 1 → ∞ lies on the left end of the lower branch.
The lower branch of the hyperbola corresponds to characters with N 1 > 0 and the upper branch corresponds to characters with N 1 < 0. To see this note the following argument.
In the next plot, we have all the hyperbolas corresponding to each of the (2, 6) solutions with central charges in the admissibility range: 24 < c (ℓ=6) < 32.

Discussion of the case of two movable poles
We now turn to the case of ℓ = 12. This is important because there are two independent poles p 1 , p 2 and correspondingly two accessory parameters. A number of novel features will emerge in this setting that were not visible for ℓ < 12.

The (2, 12) MLDE and constraints on accessory parameters
The (2, 12) MLDE in the τ plane is given by In the j-coordinate, the same MLDE is given by: The accessory equations for ℓ = 12 can be read off from Eq. (3.45) and after some rationalisation of denominators they reduce to: When one examines the coefficient functions of the MLDEs, both in the τ -space and the j-space, one finds that the poles and the accessory parameters appear in symmetric combinations. Hence, we work with the symmetric parameters: More generally, for ℓ = 6r, we would have P k , B k with k = 1, 2, · · · r, where P k denotes the kth symmetric polynomial in the movable poles and B k denotes the kth symmetric polynomial in the accessory parameters. We will see below that these symmetric parameters always turn out to be rational for admissible character solutions, while the individual poles and accessory parameters need not be. Now, the sum and difference of the two equations in Eq. (5.3) can be written in terms of the symmetric parameters : The accessory equations for the more general case for ℓ = 6r (Eq. (3.44)) can also similarly be recast in terms of the P k 's and the B k 's. Although these look more complicated than Eq. (5.3), we will soon see that the P k and B k are real while the same does not hold for the Now, we will discuss in general terms how one solves the (2, 12) MLDE. We start out as and four others for the Fourier coefficients of the identity character m 1 , m 2 , m 3 , m 4 (to be consistent with our earlier notation these should have a superscript (12) to denote the ℓ = 12 case, but we drop it to simplify the notation). These are four linear equations for the parameters P 1 , P 2 , B 1 , B 2 and we can solve them and obtain the analogues of Eq. and ℓ = 12 cases, we would solve these equations for the 2r variables P k , B k and solve for each of them as a rational function of m 1 , . . . m 2r with coefficients being rational functions of c. Again, we can conclude that the P k and the B k are rational numbers. Thus, any given movable pole is either complex (and occurs with its conjugate) or is real irrational (and occurs with its Galois conjugates) or is rational, and the same for any accessory parameter.
The next step is to bring in the accessory equations Eq. Unfortunately this procedure becomes extremely tedious, so we employ an alternate route below.

(2, 12) admissible characters
We will use quasi-character theory [6] to obtain (2, 12) solutions and make contact with the above analysis. In general terms, quasi-character theory informs us that (2, 6r) admissible character solutions can be found by taking r + 1 summands, each of which is a (2, 0) quasicharacter. The summation will contain r quasi-character parameters, which are non-negative and subject to further restrictions. The precise details and systematics of this procedure has never been worked out for r ≥ 2, and will be addressed in [45]. Here we will content ourselves with working out one example in full detail. According to quasi-character theory, we can pick three (2, 0) quasi-characters in the A 1 class and sum them to obtain a (2, 12) solution: c = 25, h = 1 4 in the following way : where the n = 4 and n = −4 terms are quasi-characters (integral but not positive) while the n = 0 term is an admissible character (for the A 1,1 WZW model). The leading behaviour in the q-series expansion of the identity character corresponds to c = 25 and that of the non-identity character gives h = 1 4 . These numbers ensure that ℓ = 12. But it is not yet clear that these are admissible characters. For that we examine the q-series. For the identity character, we have 24 1 + (−245 + N 1 )q + (142640 + 3 N 1 + 26752 N 2 )q 2 +(18615395 + 4 N 1 + 1734016 N 2 )q 3 + (837384535 + 7 N 1 + 46091264 N 2 )q 4 + . . . (5.7) and for the non-identity character, we have Requiring admissibility of the above q-series up to order q 5 , we find the following restrictions on the quasi-character parameters N 1 and N 2 .
(we have denoted the integer by m 1 anticipating that it will be the degeneracy of the first excited state in the identity character).
Let us consider the relations given in Eq. (5.9). The first expression relates the quasicharacter parameter N 1 to m 1 which is the dimension of the Kac-Moody algebra (if any) of the final theory. On the other hand, the second relation serves as a restriction on N 2 for any fixed N 1 . This restriction will be modified at every order in q. So, to ascertain admissibility of χ in Eq. (5.6), we need to look at the asymptotic growth of the coefficients in the q-series of this character. This is done by considering the Rademacher expansion (see [46] and appendix A of [6]). This assures us that the asymptotic growth of the negative-type quasi-character, χ A 1 n=−4;1 is sub-leading as compared to that of the positive-type quasi-characters χ A 1 n=4;1 and χ A 1 n=0;1 . Thus, Eq. (5.6) will be an admissible character for all N 1 satisfying Eq. (5.9), i.e. N 1 ≥ 245, and finitely many N 2 satisfying some upper bound (not necessarily the one in Eq. (5.9).
Quasi-character theory claims that the two non-negative integral q-series above with c = 25, h = 1 4 are in fact (2, 12) admissible characters. We will check this claim by showing that they solve the (2, 12) MLDE. In particular, first we will compute the point in parameter space where the solution Eq. (5.6) lives, in other words, we will determine the poles and accessory parameters as functions of the quasi-character parameters N 1 and N 2 . Then we will show that this point satisfies the accessory equations Eq. For this, we substitute Eq. (5.6) into the MLDE Eq. (5.1). The two-derivative terms simplify on using the fact that the summands in Eq. (5.6) are solutions to the (2, 0) MLDE: The q-series of the MLDE, at the leading order for both the identity and non-identity character, determines the rigid-parameter in Eq. (5.6) to be the expected α 0 α 1 = 475 576 . To obtain the poles and accessory parameters we consider the first and second order terms in the q-series for the identity and non-identity characters. This gives us four equations for the four variables (the two poles and the two accessory parameters). For the identity character this results in: For the non-identity character we find instead: Solving Eq. (5.11), Eq. (5.19), Eq. (5.13), Eq. (5.20), we obtain solutions for the symmetric parameters in terms of the quasi-character parameters: We see right away that the symmetric parameters are all rational. We then just substitute the q-series expansions Eq. (5.7) and Eq. (5.8) and verify. We have done so for high-enough order to convince us that the quasi-character sum Eq. (5.6) is indeed a (2,12) admissible character for all values of N 1 , N 2 satisfying Eq. (5.9). Now we can solve for the poles and accessory parameters in terms of the symmetric parameters, to find: When A < 0, the movable poles are complex conjugates. Also when A = 0 (which happens for N 2 = 2) both poles coincide and we will discuss this in the next section. Following the discussion of the previous seubsection, we can add three (2, 0) quasi-characters in the A 1 class to obtain a (2, 12) solution: c = 31, h = 3 4 in the following way : where the n = 5 and n = −3 terms are quasi-characters while the n = 1 term is an admissible character (for the E 7,1 WZW model). Now let us consider the leading behaviour in the qseries expansions. The identity character corresponds to c = 31 and that of the non-identity character gives h = 3 4 . Using the Riemann-Roch relation Eq. (2.16), we see that these numbers correspond to ℓ = 12. However, at this stage, it is not yet clear that the q-series expansions are those of admissible characters. For that we examine the q-series. For the identity character, we have 24 7 + (N 1 − 1829)q + (533603 + 133 N 1 + 39 N 2 )q 2 +(309815674 + 1673 N 1 + 1547 N 2 )q 3 + . . . (5.19) and for the non-identity character, we have Requiring admissibility of the above q-series up to order q 5 , we find the following relations: (we have denoted the integer by m 1 anticipating that it will be the degeneracy of the first excited state in the identity character).
Proceeding as before, let us note that, the first of the relations in Eq. (5.21) relates the quasi-character parameter N 1 to the dimension m 1 of the Kac-Moody algebra (if any) of the final theory. However the second relation should be viewed as a restriction on N 2 for any fixed N 1 . This restriction will be modified at higher orders in q. As in the previous sub-section, the Rademacher expansion (see [46] and appendix A of [6]) assures us that the asymptotic growth of the negative-type quasi-character, χ A 1 n=−3;1 is sub-leading as compared to that of the positive-type quasi-characters χ A 1 n=5;1 and χ A 1 n=1;1 . Thus, Eq. (5.18) will be an admissible character for all N 1 , that is, N 1 ≥ 1829 satisfying Eq. (5.21) and finitely many N 2 satisfying some upper bound. Now proceeding as before, we can express the symmetric parameters of the MLDE, namely, P 1 , P 2 , B 1 , B 2 in terms of the quasi-character parameters only. However, in this case these expressions are quite lengthy and hence we do not report them here.

Beyond the genericity assumption: merging of poles
In this section we consider what happens when poles in the original MLDE merge. This first happens when p 1 → 0 or 1728 in the (2, 6) case. Once we reach the value ℓ = 12, we can also have two movable poles p 1 , p 2 merging. We will analyse some special cases below and then draw general conclusions at the end.
6.1 (2, 6) solutions as p 1 → 0 We want to investigate what happens to the (2, 6) solutions if we set p 1 = 0, corresponding to the point τ = ρ. This is our first example of a case violating the genericity assumption.
Let us take this limit directly in Eq. (3.28). It becomes: Now we insert the expansion: We find the indicial equation: Now we come to a key point. Since the lower of the two exponents has shifted from 0 (when p 1 was a generic point) to 1 3 (after p 1 goes to 0) we can make the change of variable: where the function ζ(j) has a sensible expansion in power of j. Indeed, we find that ζ satisfies the (2, 2) MLDE, the middle equation of Eq. (3.25) with r = 0. Recall that the parameters in that equation are α 0 , α 1 , the exponents around τ → ∞ (not to be confused with the α (ρ) i above!). We find the relation: (α 0 α 1 ) ℓ=6 = (α 0 α 1 ) ℓ=2 + 1 6 (6.7) In fact from Eq. (6.6) we already know the exponents of the solution ζ(j) must be: and using the valence formula on both sides, it is easy to check that Eq. (6.7) agrees with this.
In this case we can come to the same conclusion by solving the accessory equation Eq. (3.33) as p 1 → 0. One solution is b 4,1 = 0 and the other is: Inserting this into the MLDE in terms of τ : ζ(τ ) = 0 (6.10) and performing the change of dependent variable: we get the (2, 2) MLDE: In this equation we see the relation (α 0 α 1 ) ℓ=2 = (α 0 α 1 ) ℓ=6 − 1 6 . Thus we learn that, in the cases where the lower of the critical exponents is nonzero, sending the movable pole to the point p 1 = 0 causes the solution to factorise into a product of solutions of an MLDE with lower value of ℓ (in this case a pair of characters with ℓ = 2) times a single meromorphic character j 1 3 which also has ℓ = 2. A priori this may not seem like a "merger" of poles since there was no pole at p 1 = 0 to begin with. However it does count as a merger because a single pole at τ = ρ is three times the minimum allowed pole at that point.
As we will see later, the reason we could simply take the limit in the accessory equation We know from [28] that admissible solutions for the (2, 2) case lie in the range 16 < c (ℓ=2) < 24.
Thus, admissibility of (2, 6) solutions rules out the first case in Eq. (6.13). Then from the second equation above, we have 24 < c (ℓ=6) < 32. We already know the allowed central charges for admissible (2, 6) solutions from Sec. 3.6. Thus we see that tensor-product (2,6) CFTs follow the exact same range and not a subset of it. We will encounter examples later where the product theories occupy a smaller range of central charges.
We will now look at some more examples that present different features, and then turn to the general case.
Let us now insert the following expansion in the MLDE Eq. (6.14): We find the indicial equation to be, The indicial equation tells us that, One could have already deduced this fact from Eq. (A.4), since after taking p 1 → 1728 we have ℓ i = 6. Now the α (i) s must be distinct non-negative multiples of 1 2 . Hence, the possible solutions to the above equation are (0, 3 2 ) or ( 1 2 , 1). The exponents (0, 3 2 ) correspond to either α 0 α 1 = 0 or b 4,1 = 1728. The former can be ruled out since in this case the solution space is 1-dimensional, as argued in the previous sub-section. However b 4,1 = 0 is a possibility and in this case indeed we have the exponents (0, 3 2 ). Solutions, with these exponents, if they exist, would be non-factorisable. Once can see this by following similar arguments of regularity of solution around τ = i as described in the previous sub-section.
The other alternative is that the exponents are ( 1 2 , 1). Inserting this in Eq. (6.16) leads to the constraint: As noted in the previous sub-section, since the lower of the two exponents has shifted from 0 (when p 1 was a generic point) to 1 2 (after p 1 goes to 1728) we can make the following change of variable: where the function ζ(j) is regular around τ = i. Furthermore, we find that ζ satisfies the (2, 0) MLDE and this yields the following relation, (α 0 α 1 ) ℓ=6 = (α 0 α 1 ) ℓ=0 + 1 6 (6.20) In fact from Eq. (6.19) we already know the exponents of the solution ζ(j) must be: In this case also, we can come to the same conclusions as above by solving the accessory equation Eq. (3.33) as p 1 → 1728. This is because, as before, taking p 1 → 1728 in Eq.
6.3 (2, 12) solutions as p 1 → p 2 Next we consider the case where two movable poles coalesce but remain away the points 0 and 1728. This possibility arises for the first time in the (2, 12) MLDE Eq. (5.2). So we put p 2 = p 1 in this equation, to get: Now we expand the characters about j = p 1 as in Eq. (3.8): Due to the double pole, the last term in Eq. (6.23) contributes to the indicial equation, which becomes: Again we cannot read off the exponents directly, but from the above equation we have: Since p 1 is a regular point in moduli space, α 1 must be distinct non-negative integers, so the only possibilities are (0, 3) and (1,2). The former leads to either α 0 α 1 = 0 or p 1 = b 4,1 This in fact already follows from Eq. (A.6), since after taking p 2 → p 1 we have ℓ τ = 12.
Since the α (I) must be distinct non-negative integers, the possible solutions to the above equation are (0, 3) or (1, 2).
With α 0 α 1 = 0, as noted before, we get the solution space to be 1-dimensional. Thus, we can only have the exponents (0, 3) if p 1 = p 2 = b 4,1 or p 1 = p 2 = b 4,2 . At such points, one factor cancels from the numerator and denominator of the (2, 12) MLDE. Thus we get the equation: In this case there is a constraint equation at third order, which is left as an exercise for the reader.
Now we return to the other possible set of exponents, namely (α 1 ) = (1, 2). From Eq. (6.25) we then immediately find the condition: Because the indices now differ by 1 (rather than 2 in the generic case), there is a potential logarithmic singularity in the character χ 0 (j) manifested by a constraint arising at first order beyond the indicial equation (as against second order in the generic case). The mechanism has been discussed before -at this order the coefficient a 0,1 will not appear and instead we will get a constraint.
From the MLDE Eq. (6.23), this constraint is found to be: which is written in terms of the symmetric polynomial basis (see Eq. (5.4)).
In this sub-section, we have shown that when two movable poles p 1 , p 2 coincide with each other (but not with an accessory parameter), the solutions of the MLDE factorise into a product of a meromorphic character and a pair of characters ζ i (j) satisfying an MLDE with ℓ = 0. As we already discussed in sub-section 3.2, this factorisation can means one of two things for an admissible character χ i (j): either ζ i (j) is itself an admissible character, or ζ i (j) is not admissible but becomes admissible upon multiplying by (j − p 1 ).

(6.38)
It is easily verified that the above two equations Eq. (6.37), Eq. (6.38) are in the form: thus they are factorised as in Eq. (6.33).

Analogous considerations for (2, 8) and (2, 14) cases
In this sub-section, we shall first take p 1 → 0 and p 1 → 1728 in the (2, 8) case, and then take p 2 → p 1 in the (2, 14) case. We shall see that, as a result of this procedure, the equations and expressions that come out will be very similar to the (2, 6) and (2,12) The first of these cases corresponds to, as before, b 4,1 = 0 and non-factorised characters.
Regarding the second case, we notice that the difference between the two critical exponents is an integer. This solution is ruled out, as already observed in [27], because the monodromy about τ = ρ would become reducible. The third case allows us to write χ(j) = j 2 3 ζ(j) with ζ(j) having ℓ = 0, and in this case, α 0 α 1 b 4,1 = 1152. One can show that ζ(j) satisfies a (2, 0) MLDE and this in turn gives the following relation, Eq. (6.41) is true for factorised (2,8) solutions of the form: χ (ℓ=8) = j Using the indicial equation Eq. (6.28) and Eq. (6.45), we can solve for B 1 and B 2 , as before, in terms of α 0 α 1 and the movable pole p 1 .
Note that, since the lower exponent, at j = p 1 , is 1 we can have the following substitution: which we can plug in the (2, 14) MLDE (with p 2 → p 1 ) to get, On inserting the values of B 1 and B 2 obtained above, Eq. (6.47) simplifies to: which means ζ(j) solves the (2, 2) MLDE if we identify: The above equation holds for factorised (2,14) solutions of the form: Now let us make a comment on tensor-product (2, 14) CFTs of the above factorised form.
may check whether at least some illustrative CFTs can be found for each of the classes we have considered. We do this here.
There are two cases for which CFTs are already known: the (2, 6) and (2,8) MLDEs with the movable pole away from the boundary of moduli space. For the (2,6) case, this has been done in [30] making use of a relation derived in Eq.(3.6) of [25] that relates three Wronskian indices: L for a meromorphic theory which is the numerator in a coset relation, ℓ for the denominator theory andl for the coset theory 12 . We write the equation for the case of two characters:l Here n ≥ 1 is an integer labelling the sum of dimensions of the non-trivial primary for the denominator and coset theories. One finds n = 2 whenever the coset is of the type where a simple factor of a Kac-Moody algebra is deleted by a corresponding denominator, and n = 1 for non-trivial embeddings of Kac-Moody algebras in the numerator. Now, [30] considers L = 8 and ℓ = 0, corresponding to cosets of a c = 32 meromorphic theory, where the embedding is of the "deletion" type with n = 2. This results inl = 6. Nearly 150 CFTs of this form are listed in Appendix A of [30]. These belong to a class with complete Kac-Moody algebras, which means the stress tensor is pure Sugawara with no additional contribution.
For (2,8), a number of CFTs can be found in [18].  Table 1 of that paper. It may be mentioned that the method used in that work only reproduces theories in the range c < 25. However for the (2,8) case, Eq. (3.60) also allows the range c ∈ (40,48). This can potentially be realised with L = 12 and n = 3 in Eq. (7.1), but it is not clear if n = 3 is allowed for two-character meromorphic cosets, so we leave this question for the future.
Next we move on to the case of (2, 12) with generic poles. Eq. (7.1) tells us that this can be realised by a non-trivial embedding of Kac-Moody algebras (with n = 1) in a c = 32 meromorphic CFT (for which L = 8). Such embeddings have not been completely classified, even for the complete KM algebra case, but one can start with the meromorphic CFTs corresponding to the 132 even, unimodular Kervaire lattices [47] and take a coset by nontrivially embedding A 1,1 into any of the simple factors of the numerator. This will result in the desired (2,12) CFT and one expects that most of them will satisfy MLDEs with generic 12 The equation as written in [25] involves N ≡ c 24 where c is the central charge of the meromorphic theory, but it is easily verified that 6N = c 4 = L.
(non-coincident) poles. As an example, take the Kervaire lattice with root system A 2 9,1 E 2 7,1 and quotient by A 1,1 , embedding it in E 7,1 . The quotient theory has central charge 31 and algebra A 2 9,1 D 6,1 E 7,1 with m 1 = 397. Similarly, the coset of a c = 48 meromorphic theory by A 1,1 with a trivial embedding of the KM algebra (n = 2) will give ℓ = 14. We are not aware of a classification of even, unimodular lattices of dimension 48 even under the restriction of having complete root systems, but one possibility is A 48 1,1 and the trivial embedding deletes one of the 48 factors leaving an ℓ = 14 CFT with Kac-Moody algebra A 47 1,1 . There will surely be many more examples, most of which should have generic poles. Now we turn to a CFT example for (2, 6) with p 1 → 0, a limit studied above that effectively corresponds to coincident poles. As we saw in Sub-section 6.1, in this limit there are two possible sets of exponents at 0, namely (0, 4 3 ) and ( 1 3 , 1). As explained there, these correspond respectively to characters of non-factorisable and factorisable type.
Within the factorisable or ( 1 3 , 1) type, we can easily find a set of tensor-product (2, 6) CFTs with characters of the form in Eq. (6.6), namely χ i (j) = j 1 3 ζ i (j) where ζ(j) are the characters of a (2, 2) CFT. The latter have central charges c ∈ (16,24) and are enumerated in [25]. Since multiplication by j Turning now to the non-factorisable case, where the indices are (0, 4 3 ) and the CFT is irreducible. As shown in sub-section 6.1, this case arises when p 1 and b 4,1 vanish together.
As an example, in Eqs (4.43, 4.44)  3 ) be associated with a coset CFT. We do not know of a deep reason why this should be the case.
We move on to (2,8) with p 1 → 0. Here the possible indices are (0, 5 3 ) and ( 2 3 , 1). We again see that factorised solutions corresponding to the latter case are trivially possible and they lie in the sub-range c ∈ (16,24). To populate the other sub-range in Eq. (3.60), namely c ∈ (40, 48), we now have a possibility: consider quasi-characters for ℓ = 0 in the range 24 < c < 32, for example the c = 25 quasi-character in the A 1 series. While this is not admissible, multiplying it by j 2 3 makes it admissible and it has c = 41. In this case, the result is a tensor product of E 8,1 times the exotic c = 33 theory of [9]. But more general non-tensor-product theories could well exist.
Finally we look at the case of (2, 12). In the coincident limit p 2 → p 1 , one can look for factorisable as well as non-factorisable characters. In the factorisable case one simply has where ζ i is an MMS character hence lies in the range c ∈ (0, 8). The result is in the range c ∈ (24, 32) and will have indices (1,2). Searching for the non-factorisable case with indices (0, 3) is more trivial and we leave it for the future.

Discussion and conclusions
Since this has been a lengthy discussion, let us review the chain of arguments that led to our understanding of MLDEs with movable poles. The first step is to suitably parametrise the MLDE. This was done in Section 2.17 in the τ parameter and Section 3 in the j parameter. Next, in Section 4 we returned to the single-pole MLDE as a function of τ and computed the Frobenius solution. Inserting the known allowed values of c then determined both the characters completely up to an arbitrary integer. One cannot reduce further since it is known [3,6] that there is exactly one free integer parameter for each movable pole. We were also able to related our results precisely with the quasi-character construction of [6] which constructs admissible characters for generic Wronskian index without any use of the corresponding MLDE, and precise agreement was found. An analogous discussion in Section 5 considered the case of two movable poles. In Section 6 we considered what happens when one violates the genericity assumption by merging poles. The equation and solutions remain well-defined when one pair of poles is merged, though they become singular if we simultaneous merge more than two poles. Finally in Section 7 we gave just a few examples of CFTs for the various cases we considered, showing that they are populated by genuine CFTs, and leaving a more detailed analysis for the future.
Our analysis makes it clear that whenever there are movable poles, there is an equal number of free integer parameters in the admissible solutions. This fact has previously been noted in [6], but here we have re-obtained it directly from MLDE. This means there is an infinite set of admissible characters for every ℓ ≥ 6. However it can be argued that the number of CFTs for a given ℓ ≥ 6 is finite. For example, a result of [18] implies that every (2, 6) [2, 10, 13-15, 22, 25, 26] are all at ℓ = 0. Things are only slightly better using the alternate approaches of Hecke operators and quasi-characters. Ref. [3] constructed some admissible characters for the (3,6) case as Hecke images of the Ising model characters. Several sets of quasi-characters solving the (3, 0) MLDE were found in [8] and their linear combinations were shown to provide admissible characters with ℓ = 6. A concrete example of a (3, 6) CFT was also provided in this reference. Beyond this, the space of admissible characters and CFT for n ≥ 3 characters arising from MLDE with movable poles, is essentially unexplored. It should certainly be possible to gain some insights into at least the (3, 6) case from the MLDE following the methods used here.
Another space of MLDEs that is largely unexplored is the (n, 0) class -with n characters but no poles. Like the (2, ℓ) case studied in the present paper, (n, 0) also involves a proliferation of parameters for n ≥ 6, however clearly these do not correspond to poles or accessory parameters and one has to find a useful interpretation. Moreover the number of exponents is n, if this is large the analysis may be quite difficult. Nevertheless, as the existing literature shows, a lot can be learned bout RCFT by exploring modular differential equations, and we hope to report on more of the open problems in the future.

A Critical indices at the poles
In this Appendix we review the leading behaviour of the character χ(j), about various points in the upper half plane, following [27].
About τ = ρ we have j → 0 and the leading behaviour of the characters are parametrised as: 1 must be non-negative multiples of 1 3 to ensure regularity of the characters around τ = ρ. Now let us compute the leading behaviour of the Wronskian W (j) about τ = ρ: where we used the fact that E 4 = j 1 3 ∆ 1 3 and both ∆ and E 6 are non-vanishing at τ = ρ. Thus, we get: About τ = i we have j → 1728 and the leading behaviour of characters is parametrised as: 1 being non-negative multiples of 1 2 . This is to ensure regularity of characters around τ = i. Now let us compute the leading behaviour of the Wronskian W (j) about τ = i: From this we get: where we used the fact that E 4 and ∆ are finite at τ = i. Then we have: Next let us study the leading behaviour of the Wronskian about a movable pole say, j = p 1 . We parametrise this by: 1 being non-negative integers to ensure regularity of characters around j = p 1 . Now the leading behaviour of the Wronskian W (j) about j = p 1 is: and we find: .
(A.6) B ℓ is even for 2-character solutions In this appendix we will show that for 2-character solutions, ℓ is always even. This result was first obtained in [27] using monodromy arguments for solutions around τ = i. It was shown that if ℓ is odd then the monodromy is reducible, implying the solution space becomes one-dimensional and hence is not allowed. Here we will approach the problem in a slightly different way but will arrive at the same conclusion.
Comparing the above MLDEs to the ones given in Eq. (3.25), we notice a striking difference, namely the term 1 2(j−1728) is missing in the first-derivative term. We shall see that the absence of this term is crucial to ruling out odd ℓ values.
(C.7) We will now use the above results to understand the case of MLDEs with movable poles, specifically (2, 6) and (2,8). In the next step, we solved for three parameters in terms of objects associated to the identity character viz. the central charge c, the Fourier coefficients m 1 and m 2 . For the non-rigid parameters, we obtained the equations Eq. (4.5) and Eq. (4.6) where f 3 (c, m 1 , m 2 ) and f 4 (c, m 1 , m (6) 2 ) are given by: 1 , m     Next we rewrote the pole and accessory parameters only in terms of c and m 1 ; after substituting Eq. (4.10) in Eq. (4.3) and Eq. (C.11). We obtained Eq. (4.11) and Eq. (4.12) where the f 5 (c, m 1 ) and f 6 (c, m In the next step, we obtained the third Fourier coefficient of the identity character in Eq. (4.13) where A 3 (c) and B 3 (c) are given by: We now give formulae that will be referred to in the main text of the paper, for the (2,8) MLDE. In the first step, one computes the first three orders of the Frobenius solution for the identity character and obtains Eq. In the next step, we solved for three parameters in terms of objects associated to the identity character viz. the central charge c, the Fourier coefficients m 2 ) and f 4 (c, m 1 ) and f 6 (c, m

D Some (2, 8) MLDE solutions and quasi-characters
Here we analyse the (n, ℓ) = (2, 8) MLDE in the same was as was done for (2,6) in Section 4. The MLDE in the j-coordinate is given by: ∂ j + α 0 α 1 (j − b 4,1 ) j(j − 1728)(j − p 1 ) χ(j) = 0 (D.1) Using the series expansion χ i = ∞ k=0 a (1) i,k (j − p 1 ) k+α (1) i , a (1) i,0 ̸ = 0, the indicial equation around j = p 1 is: At first subleading order, for the solution α At second order beyond this, we find (as expected) that the a and using the methods explained in Section 4, we obtain: correspond to admissible characters 13 with ℓ = 2 [25]. As before, linear combinations of these quasi-characters make up admissible characters with increasing values of ℓ, this time in the family ℓ = 6r + 2, and all such characters are generated.
We can now list the admissible (2,8)  with the identification, m = N 1 . This solution appears in [18]. 13 Again these all correspond to CFTs, except for the first and last cases that are Intermediate Vertex Operator Algebras [44]. A new feature here is that a single set of admissible characters corresponds to more than one CFT.
with the identification m = N 1 .