Classifying three-character RCFTs with Wronskian index equalling 0 or 2

In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers [n,l]. n is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and l, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study [3,0] and [3,2] MLDEs in order to classify the corresponding CFTs. We reduce the problem to a “finite” problem: to classify CFTs with central charge 0 < c ≤ 96, we need to perform 6, 720 computations for the former and 20, 160 for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity. In the [3,0] case, for 0 < c ≤ 96, we obtain many character-like solutions: two infinite classes and a discrete set of 303. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify [3,2] CFTs for 0 < c ≤ 96: each CFT in this case is obtained by adjoining a constant character to a [2,0] CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.


Introduction
Two dimensional conformal field theory (CFT) is a subject of primary importance relevant to many areas in physics and mathematics; [3][4][5][6] comprise a partial list of references.In physics it is important for string theory both in perturbation theory, initially and later for various aspects of non-perturbative string theory also.Two dimensional CFTs describe two dimensional critical systems at RG fixed points.They have contributed to many areas of mathematics including representation theory, infinite dimensional algebras, theory of modular forms, etc.An important subclass of two dimensional conformal field theories is that of rational conformal field theories (RCFTs).In these theories the central charge c and the conformal dimensions of the primary fields, the hs, are all rational numbers [9].An infinite number of RCFTs are known to exist.For example the infinite series of Virasoro minimal models [1] (see also [3]), and the infinite set of Wess-Zumino-Witten conformal field theories (WZW CFTs) [2].Even though an infinite number of RCFTs are known, there is no known classification of them.
One classification scheme for RCFTs is based on the structure of their torus partition functions.For a RCFT, the torus partition function can be expressed as a sum of holomorphically factorised terms.The holomorphic factors are the characters of the RCFT and the number of characters, denoted by n, is an important detail of a RCFT for this classification scheme.Another important detail of a RCFT relevant for this classification scheme arises as follows.The n characters are the linearly independent solutions of a single n-th order ordinary differential equation on the moduli space of a torus, a modular linear differential equation (MLDE) [9,10,12].One then considers the Wronskian of these solutions which is a modular function of weight n(n − 1).The Wronskian index of the RCFT, denoted by l, is a number that takes non-negative integer values and is associated with the structure of zeroes of the Wronskian and can be expressed purely in terms of the RCFT data (the central charge, the number of characters and the conformal dimensions of the primary fields).In this classification scheme, a RCFT is identified in terms of these two numbers [n, l], the number of characters and the Wronskian index.
Remarkably, and this is the primary reason this classification scheme exists, the form of the MLDE gets fixed by this same pair [n, l].Actually the coefficient functions of the MLDE are fixed by [n, l], apart from some undetermined parameters.Thus two RCFTs with the same [n, l] values are both solutions to the same MLDE but for a different set of parameters in the MLDE.One thus sets up a MLDE for a given number of characters and a given Wronskian index and studies all character-like solutions (that is solutions with nonnegative integer coefficients in their q-expansion) and obtains all RCFTs with that number of characters and that Wronskian index and thus classifies them.This is the MLDE approach to the classification of RCFTs, first carried out in [11], where a [2, 0] MLDE was studied and all [2, 0] RCFTs were classified.The program has been implemented for [2,2] first in [13] and then in [14][15][16]; then partially for three-character RCFTs in [12,20,23,35,45,46].
In this paper, we take on the task of classifying three-character CFTs.We do restrict ourselves to the cases of l = 0 and l = 2, i.e. [3,0] and [3,2] CFTs.The study of three--2 -character CFTs are at the least one level harder technically than two-character CFTs.One aspect of this can be seen from the form of the MLDEs involved.In a MLDE (2.8), the coefficient functions φ r which are modular w.r.t the full modular group are fixed by the Wronskian indiex l; the functional form is completely fixed leaving only a few arbitrary parameters.For two character MLDEs studied so far in the literature, viz.[2,0], [2,2] and [2,4], the MLDE has only one arbitrary parameter (the [2,4] MLDE does have two parameters to begin with but one of them gets fixed by other considerations [17]).On the other hand both the [3,0] and [3,2] MLDEs have two arbitrary parameters.Solving the MLDE means also finding the value of the parameters for which solutions exist.Hence obtaining two-character RCFTs involves scanning the set of possibilities for one parameter which is the infinite set of all rational numbers.While solving for three-character RCFTs (for l ≤ 2), one would need to scan over the infinite set of pairs of rational numbers.
There is another sense in which the problem of classifying three-character CFTs is harder.
There already exists an infinite number of such CFTs.In a recent work [39], with the intention of correctly placing the known classes of CFTs within the present classification scheme, we computed the [n, l] values for an infinite number of CFTs.This allows us to obtain a partial classification of all CFTs with a given [n, l].We call this a partial classification because it partially answers the question "what is the classification for all CFTs with this given [n, l]?" The partial classification for [3,0] CFTs from that paper is given here (3.11).As one can see, there are two infinite classes (second and fourth lines), 7 irreducible CFTs and 8 tensor product CFTs.Thus, even before we start the problem, we are told that there will be an infinite number of solutions, at least.This should be contrasted with what happens for [2,0] CFTs.The partial classification from [39] reveals that there are a finite number of such CFTs viz.8 (and it eventually turns out that the full classification [11] also has only these 8 CFTs and nothing else).Here in this paper, at the end of the day, we do reproduce the partial classification of [39] for [3,0] CFTs and find two other infinite classes of solutions as well as a discrete set of CFTs and we may have a few (7) potentially new CFTs.The subsequent case of [3,2] CFTs is intriguing in it's own way.There is not a single such CFT among the infinite number of CFTs that we studied in [39].Here in this paper, we do manage to classify all [3,2] CFTs up to central charge 96 and we find exactly nine new CFTs.But in a sense, they are not as interesting since it seems that they are related to the nine [2, 0] admissible character solutions and it is not clear if they should be considered as new CFTs.In any case, we have achieved classification.
In this paper, we obtain solutions to the MLDEs by directly solving them; the MLDE is a homogenous linear differential equation and one can solve it with the Frobenius method.
Although we do give a way of thinking about the MLDE as an inhomogenous MLDE (one order lower), which turns out to be useful to understand features of some of the solutions, our main method is the direct Frobenius one.One directly plugs in the q-expansion of the characters: ) 0 = 0 into the MLDE and solve it order by order.At the lowest order, one gets a nth order polynomial equation, the indicial equation, whose roots are the indices α i .Higher orders give expressions for the f (i) n 's which are rational functions of the parameters of the MLDE and the index α i .Then one focusses on the identity character whose index α 0 (for unitary conformal field theories) is related to the central charge (α 0 = − c 24 ) and each of it's Fourier coefficients are (non-negative) integers.There is a lot of purchase that comes out of this integrality.Typically one imposes the fact that m n 's (the coefficients of q n in the identity character) are integers (just for a few n's) which constrains the possibilities for α 0 .One kind of equation that comes out of this analysis is a polynomial equation for α 0 with integral coefficients; the integral-root theorem then tells us the the kind of rational values that α 0 and hence c can take.For example in the [2, 0] and the [2,2] cases [14], the integrality of m 1 implies that for c = p q with p and q coprime, q can be either 1 or 5 (we give more details for [2, 0] in section 2.2 below).This means that 5c is always an integer; hence for a fixed range of central charge, say 0 < c ≤ 1, there are now only 5 possibilities viz, 1  5 , 2 5 . . . 5  5 .Prior to this stage of the analysis, we had an infinite number of possibilities for c, even for a finite range.Hence, the problem from here on, we say, has become a "finite" problem.
For the [3,0] and the [3,2] cases, we have to impose integrality of both m 1 and m 2 to get a polynomial equation for α 0 with integral coefficients.It then turns out that, for the [3, 0] case, 70c is an integer, the problem becomes "finite" and for 0 < c ≤ 1 there are 70 possibilities For the [3,2] case, the corresponding number is 96×210 = 20160 1 .This strategy of making the problem "finite" is a very important first step in our procedure to classify threecharacter CFTs.Further steps are described in the subsequent sections; they involve, for each of these finite possibilities, finding simultaneous solutions to a pair of Diophantine equations, computing higher order Fourier coefficients and checking for positivity.
Disclaimer : In this paper, when we say, we solved a certain MLDE and obtained a CFT, what we really would have done is, we solved that MLDE and obtained a character-like solution a.k.a admissible character solution i.e. a solution whose Fourier coefficients are non- 1 It turns out that for the [2, 0] case there is a further constraint for the central charge viz.c < 10, which makes even the range of the central charge finite.This makes the number of possibilities actually finite: 49.
There is a similar constraint in the [2,2] case [14] that makes the range of the central charge itself finite and again results in a finite number of possibilities for c.For three-character CFTs at least for the [3, 0] case, we cannot expect such a finiteness for the range of c simply because even the known CFTs in this class have an infinite range of central charge (3.11).
-4 -negative.Strictly speaking,, we should say we solved the MLDE and obtained an admissible character or character-like solution.In some cases, the character-like solution does constitute the characters of a known CFT and hence our words are justified; in other cases, our words are not justified, especially in situations where the solution is a hitherto unknown one there is more work to be done, before we declare it a CFT.This paper is organised as follows.In section 2 we give technical details pertaining to MLDEs and the calculations needed to achieve classification of RCFTs.Section 3 contains the classification of three-character RCFTs.We first study the [3,0] case in 3.1: the classification of up to central charge 96 results in a large number of solutions, which are divided into five mutually exclusive and exhaustive categories, which we describe successively in 3.1.1,3.1.2,3.1.3,3.1.4and 3.1.5.We then study the [3,2] case in 3.2.In section 4, we collect some observations on the various new solutions of section 3.1, that seem to suggest that there are novel coset dual pairs of CFTs among them.We conclude in section 5 where we consolidate our results and give future directions.The appendix A tabulates all the character-like solutions of both [3,0] and [3,2] MLDEs.

Modular Linear Differential Equations
The characters of a RCFT are a set of n functions, χ i (τ ), on the moduli space of the torus such that the torus partition function of the RCFT can be written as: The characters have a q-expansion: where q = e 2πiτ and f n are the Fourier coefficients of the ith character.Here α i are the exponents, where h i are the conformal dimensions of the primaries and c is the central charge.Note that, a single character can correspond to multiple primaries.The modular invariance of the torus partition function, implies that the characters transform as vector-valued modular forms: where V ik are unitary matrices in the n-dimensional representation of SL(2, Z).
The characters of a RCFT are solutions to an ordinary differential equation.The appropriate notion of derivative is that of the Serre-Ramanujan derivative.We denote the upper half plane by H, The Serre-Ramanujan derivative operator (see, for example, section 2.8 of [7] and exercise 5.1.8 of [8]) is defined as follows: where E 2 (τ ) is the weight 2 Eisenstein series.The operator D k is a linear operator which maps weight k modular objects to weight k + 2 modular objects and satisfies the Leibniz rule.This is the differential operator with which one sets up modular linear differential equations.
With this we can write the most general n th order MLDE [11]: where the solutions χ i are the characters of a n-character RCFT.In the above, the coefficient functions φ r (τ ) are modular w.r.t the full modular group SL(2, Z) (even if the characters χ i (τ ) themselves need not be, being modular w.r.t some congruence subgroup of SL(2, Z)) and are given by rational functions of the Eisenstein series E 4 (τ ) and E 6 (τ ).Just as for differential equations with ordinary derivatives, the co-efficient functions can be expressed in terms of the Wronskians of the solutions.Define, the Wronskian, W , of (2.9) is given by W ≡ W n .The coefficient functions of (2.8) and the Wronskians (2.9) are related by:

The Valence Formula and the Wronskian Index
For a modular function f of weight k, ν τ 0 (f ) denotes the order of pole/zero of f at τ = τ 0 .
Let us now look at the valence formula (see, for example, section 3.1 of [7]) which states that, where ω is one of the cube root of unity, F is the Fundamental Domain and the above the summation means that the sum excludes τ ∈ H which have Re(τ ) = 1 2 or, which have both |τ | = 1 and Re(τ ) > 0. Now, at τ → ι∞, χ i ∼ q α i and hence, W ∼ q n−1 i=0 α i .Hence, W has a − n−1 i=0 α i order pole at ι∞.The last three terms on the left hand side of (2.11) together can be written as l 6 where l is a non-negative integer.Hence, the valence formula now reads (2.12) Now using (2.3), we have l, which is some information about the zeroes of the Wronskian (at the cusp points and the interior of the fundamental domain), is the Wronskian index of the RCFT.This Wronskian index l, which is expressed purely in terms of the RCFT data (as seen in (2.13)), proves to be important for the classification of RCFTs.

[2, 0] MLDE
In this subsection, we will illustrate the theory with the example of [2, 0] that first appeared in the classic paper [11].What we do in the following is slightly different from [11], but it has the advantage that it can be generalised to the three-character story in section 3. Also thus we will recall the classification of [2, 0] CFTs which show up in multiple ways in the classification of both [3,0] and [3,2] CFTs that we take up in section 3.
Any 2-character MLDE has the form: with φ 1 (τ ) modular of weight 2 and φ 0 (τ ) modular of weight 4. From (2.10), φ 1 (τ ) = For vanishing Wronskian index l = 0, W has no zeroes and φ 1 (τ ), φ 0 (τ ) are SL(2, Z) modular forms (i.e.holomorphic) of weights 2 and 4 respectively.Hence they are respectively 0 and E 4 .Thus we see how the [n, l] values determine the MLDE in the [2,0] case to be where µ 1,0 is the one single parameter of this MLDE; we use a notation µ a,b to be the constant parameter that multiplies the monomial E 4 (τ ) a E 6 (τ ) b .Now to obtain [2, 0] admissible character solutions and CFTs, we plug in (2.2) into (2.15); at the leading order we obtain the indicial equation We evaluate the above equation for the identity character α 0 = − c 24 and use it to express the parameter in terms of α 0 µ 1,0 = −144 α 2 0 + 24α 0 . (2.17) Having thus converted the parameters into expressions pertaining to the identity character, we now get to the next to leading order of (2.15), which when evaluated for the identity character and further using (2.17), we obtain Here, m 1 is the Fourier coefficient (of q 1 ) of the identity character and is a non-negative integer.Define and (2.19) becomes a quadratic equation for N with integer coefficients and by the integral root theorem we can conclude that N is an integer.−120α 0 is an integer means that 5c is an integer.This is the stage at which the problem becomes "finite": for the finite range 0 < c ≤ 1, there are only five possible values viz.
10−c and since the left hand side is positive as well as the numerator of the right hand side, the denominator needs to be positive which implies c < 10.Hence we find that the allowed values of the central charge are the 49 values 1 5 , 2 5 , . . . 48 5 , 49 5 or 1 ≤ N ≤ 49.Now we compute higher Fourier coefficients.The Frobenius method gives the Fourier coefficients of a character as rational functions of the index and the parameter, which on using (2.17) and the fact that l = 0, become rational functions of the central charge.We then just compute for each of the 49 cases and eliminate those cases for which the co-efficients are negative.It turns out that there are 10 admissible character solutions, given in table 1.
Of these, the first eight admissible character solutions correspond to CFTs.The first entry needs an explanation.What one obtains is two indices − 1 60 and 11 60 .The way the -9 -discussion has been set up, we equate the smaller index to be − c 24 and the bigger one to be h − c 24 .This gives us a unitary CFT interpretation since we obtain positive c and h ( 2 5 and 1 5 respectively).But in reality M(5, 2) is a non-unitary CFT; this is also consistent with the obtained indices.If make the opposite identification, i.e. equate the bigger index 11  60 to be − c 24 and the smaller index − 1 60 to be h − c 24 we would arrive at the correct numbers for M(5, 2) viz.c = − 22  5 and h = − 1 5 .From only the MLDE solution, one cannot decide if it is a unitary CFT or not.Here in table 1, it turns out that only the first entry is a non-unitary CFT and the other CFT entries (two to eight) are all unitary CFTs.In section 3 we have many solutions that do not seem to be associated with any known CFTs; in all such cases, we tabulate the data as if they are unitary CFTs (just as in table 1), but one should keep open the possibility that they could well be non-unitary CFTs (as it happens for the first entry of table 1).
The ninth entry corresponding to N = 38 is not associated with any CFT but it is a genuine admissible character and seems to be associated with some other mathematical structure (see [47]).
The last admissible character solution (for N = 40) is strictly not a two-character CFT.
( Ê8 ) 1 is actually a one-character CFT; it is a [1,2] CFT.The single character of this CFT turns out to be the identity character of the above N = 40 solution.The other character of the solution is an "unstable" character.An unstable character is a character with positive rational coefficients, but there is no positive integer available, which when multiplied makes all the Fourier coefficients integers.The positive integer needed keeps increasing as we increase the order to which we compute.This is in contrast to usual characters, which in this context may be termed "stable" characters, where usually after a certain order of computation one finds a positive integer that when multiplied makes all Fourier coefficients integers not only up to that order but also to all subsequent larger orders of computation.This is the first instance when a CFT with a lower number of characters shows up as a solution to a MLDE with a bigger number of characters along with unstable characters.In our study of solutions to [3, 0] MLDEs, we will find in 3.1.2many two and one character CFTs that appear as solutions accompanied with one or two unstable characters.
The analysis that is presented here in this section (2.2) to derive the famous MMS classification table 1 is not the same as the one that appears in the classic paper [11].They arrive at a finite number of central charges (22) by requiring that the discriminant of (2.21) be a perfect square; they then compute the higher Fourier coefficients for these cases and impose positivity to obtain the classification in table 1.The analysis presented here has the advantage that a slight generalisation works for the three-character MLDEs that we study later in this paper.

Inhomogeneous MLDEs
The MLDEs that appear in the classification scheme being studied here (2.8) and that originated with [11] are homogenous differential equations.In this subsection, we develop inhomogeneous modular linear differential equations.We will show that for every [n, l] MLDE there is an associated inhomogeneous MLDE of order n − 1 which is equivalent to it.That is, the solutions to the homogenous MLDE also solve the associated inhomogeneous MLDE.This point of view for MLDE solutions helps explain some of the solutions that we obtain in section 3.1.3while solving [3,0] homogenous MLDEs.
The starting point is Abel's identity [37] variously referred to as Abel's formula and Abel's differential equation identity.Abel's identity allows the computation of the Wronskian of the solutions of a differential equation without having to solve the differential equation.The identity gives a first order differential equation for the Wronskian; the coefficient functions for this first order differential equation are given by the coefficient functions of the differential equation under study.This theory, given by Abel for differential equations on the complex plane, generalises to the setting of modular differential equations with Serre-Ramanujan derivatives.
First, from (2.9) we obtain DW = W n−1 , which when combined with (2.10) gives the first order differential equation for the Wronskian: [n, 2] obtained as follows.Consider the formula for the Wronskian; we will illustrate this for [3, 0]: and rewrite it as In general, we obtain an inhomogeneous MLDE of order n − 1, the inhomogeneous term given by the Wronskian from (2.22).The Wronskian is a known function at this stage and appears in (2.25), if in addition, we consider χ 1 and χ 2 as given, we can read (2.25) as a differential equation for χ 3 .We then solve this first by obtaining a particular solution.This can be done again with a Frobenius-like procedure.Let us denote the particular solution by A 1 P (q).The general solution to (2.25) is then given by a linear combination of the particular solution and the homogenous solutions.The homogenous solutions of (2.25) are nothing but χ 2 and χ 3 .
Hence the general solution is Let us denote the Fourier expansion of the particular solution and the Fourier expansions of the other characters by Then we have If the values of α 2 and α 3 for which character-like solutions exist are such that − 1 2 + 2α 2 + α 3 and − 1 2 + α 2 + 2α 3 are not non-negative integers, then to get character-like solution for χ 1 , we have to set A 2 and A 3 to be zero.This is what happens in most examples: the first character is nothing but the particular solution of the inhomogeneous MLDE.But one can imagine the following situation.
(i) If − 1 2 + 2α 2 + α 3 is a non-negative integer, then A 2 isn't required to vanish.A 2 can take any positive integral value and we would have a good character solution for χ 1 .(ii) If − 1 2 + α 2 + 2α 3 is a non-negative integer, then A 3 isn't required to vanish.A 3 can take any positive integral value and we would have a good character solution for χ 1 .
We thus have an infinite number of admissible character solutions, parametrised by A 3 , in (2.30).All members of this infinite family have the same indices and hence the same c, h 1 , h 2 and also they have the same Wronskian.But they are different solutions (they differ in the first character).
In our study of character-like solutions to [3, 0] MLDEs we will encounter two infinite families of CFTs (in section 3.1.3)each family having the same c, h 1 , h 2 values, one following (2.29) and another (2.30).

A Classification of three-character CFTs
In this section, we study three character CFTs.First we study three character CFTs with a vanishing Wronskian index i.e. [3,0] CFTs and subsequently we study three character CFTs with a Wronskian index equalling two i.e. [3,2] CFTs.We have been made aware by the recent concurrent work [41] that for three character MLDEs new admissible character solutions are expected when the Wronskian index is a multiple of 3.
Hence they are respectively 0, E 4 and E 6 .Thus we see how the [n, l] values determine the MLDE in the [3, 0] case to be This is a two parameter MLDE.Let us denote the three characters by χ 0 (q), χ 1 (q), χ 2 (q) where χ 0 (q) is the identity character.The characters have the q-expansion χ i (q n a positive integer.α i is referred to as the index of the character.The three indices are named such that α 0 < α 1 < α 2 .The three rational numbers that define a three-character RCFT viz. the central charge c, the conformal dimensions of the non-identity characters h 1 and h 2 with h 1 < h 2 , are taken to be related to the indices : A linear differential equation, when solved by a Frobenius-like method, determines the ratios . For the identity character, f (0) 0 = 1; in the following we will write f and all the m n 's are non-negative integers.For the other characters, i = 1, 2, f (i) 0 need not be 1 and and hence in our search for character-like solutions we will require that the ratios determined by the differential equation are all positive rational numbers such that upon multiplying by a suitable positive integer all of them become positive integers.This step depends on the order n up to which one solves the differential equation.We will denote this positive integer by D 1 and D 2 for the characters χ 1 and χ 2 respectively.f (1) 0 and f (2) 0 cannot be determined by the method pursued here which is the Frobenius-like method for solving linear differential equations; they can be any multiple of D 1 and D 2 respectively.
At the leading order in the q-expansion, the differential equation (3.2) gives the following indicial equation whose solutions are the indices of the characters: α 0 , α 1 , α 2 .At the next to leading order in the q-expansion, the differential equation (3.2) gives the following equation We now evaluate the two equations (3.2) and (3.3) for the identity character and interpret these equations as two equations for the two parameters which give the solution We have thus determined the two parameters of the MLDE in terms of objects associated with the identity character viz.α 0 and m 1 .We now consider the differential equation (3.2) CFTs also show up as solutions within this scheme.We can illustrate this with the example of the non-unitary Virasoro minimal model M(7, 2) for which we have c = − 68 7 , h1 = − 2 7 , h2 = − 3 7 which corresponds to indices α0 = 17  42 , α1 = 5 42 , α2 = − 1 42 which are not ordered according to α0 < α1 < α2.But if we rename the indices α0 = − 1 42 , α1 = 5 42 , α2 = 17 42 so that they are correctly ordered α0 < α1 < α2 and define central charges and conformal dimensions α0 = − c 24 , α1 = h1 − c 24 , α2 = h2 − c 24 we get c = 4 7 , h1 = 1 7 , h2 = 2 7 which looks like a unitary CFT.And this is how the non-unitary CFT M(7, 2) shows up within our scheme of computation (see the second row of table 11).It turns out [38] that the central charge c has a physical meaning as the central charge that controls renormalisation group flows between non-unitary CFTs (it is called c eff in [38]).Another non-unitary CFT that shows up in our tables is M(5, 2) ⊗2 which has c eff = 4  5 (see third row of table 11).Thus among the many solutions that we obtain in this paper which are completely unknown or not yet fully characterised as CFTs, there could be non-unitary CFTs also.And we would have tabulated their c eff .
-15 -at the next order in the q-expansion, which when evaluated for the identity character, is an equation for m 2 .After using (3.5) we obtain a polynomial equation in the variables α 0 , m 1 and m 2 .We observe that if we define a new variable the polynomial equation is the following one with indeterminates N , m 1 and m 2 .
If we consider (3.7) as a polynomial equation to determine N , which at this stage is a rational number, we find that the coefficients are all integers.Given that the coefficient of N 4 is 1, which is the reason why we choose the definition (3.6) in the first place, we can use the integer-root-theorem to conclude that N has to be an integer.This is a important turning point in our analysis.N which was just rational is now restricted to be an integer.This means that the central charge, which is given by c = N 70 is a rational number whose denominator (not necessarily in the p q form with p, q coprime) is always 70.This also means that the central charge when expressed in it's irreducible form p q with p, q coprime, q can take values 1, 2, 5, 7, 10, 14, 35 The roots of the quadratic equation determine the other two indices α 1 and α 2 and eventually h 1 and h 2 .For rational roots the discriminant has to be the square of a rational number, but since the discriminant is an integer (on account of all the coefficients being integers), it needs -16 -to be a perfect square to be able to result in rational roots.If we denote the discriminant by k 2 with k chosen to be positive, we have We note that (3.9) is an equation where all the indeterminates viz.N , m 1 and k are positive integers and hence a Diophantine equation.The roots are then given by 58800 with α 1 the smaller root and the α 2 the bigger one.
The procedure4 to obtain character like solutions to (3.2) can now be stated.
(i) We solve the two Diophantine equations (3.7) and (3.9) simultaneously.We fix a value of N and then obtain solutions to (3.7).Each such solution is a set of positive integers for N , m 1 and m 2 .We take the m 1 and N values of each such solution and see if we can find positive integer values for k that satisfy (3.9).At the end of this procedure, we will have a set of four positive integers, for N , m 1 , m 2 and k which simultaneously solve (3.7) and (3.9).
For example, for 0 < N ≤ 70, we obtain six simultaneous solutions to (3.7) and (3.9) given in Table 4. (3.10) which further means that we have the c, h 1 and h 2 of the putative CFTs.But we have imposed positivity on the Fourier coefficients of the character only upto order q 2 and only for an irreducible three-character CFT, that is, it is not a tensor-product of theories with fewer characters.We can locate them in the partial classification (3.11)   ⊗2 is not a CFT, but very much an admissible character solution [47].
Of all the solutions in category I, two of them are non-unitary CFTs, the second and first entries of tables 5 and 6 respectively.See footnote 3.

Category II
Even at the outset, it should already be stated that all the solutions in category II, except possibly two, are not to be considered as admissible-character solutions that will give consistent CFTs.This is because while all the three characters satisfy the positivity constraints, there are one or more characters that are "unstable".An unstable character is a character with positive rational coefficients, but there is no positive integer available, which when multiplied makes all the Fourier coefficients integers.The positive integer needed keeps increasing as we increase the order to which we compute.This is in contrast to usual characters, which in this context may be termed "stable" characters, where usually after a certain order of computation one finds a positive integer that when multiplied makes all Fourier oefficients integers not only up to that order but also to all subsequent larger orders of computation.
In the original MMS classification for [2, 0] CFTs [11], given here in table 1, of the ten solutions that pass the positivity criteria, only nine of them are admissible character solutions with only "stable" characters.Eight of them (the first eight in table 1) are genuine CFTs and one of them (the ninth entry of table 1), is related to another mathematical structure [47].
But that paper does include a solution (the last entry of table 1), one of whose characters is an "unstable" character and the other "stable" character is the character of the ( Ê8 ) It is an interesting question to ask: which two-character CFTs along with one "unstable" character are solutions to [3, 0] MLDEs?There is also the related question: which onecharacter CFTs along with two "unstable" characters are solutions to [3,0] MLDEs?The there is a solution to the [3, 0] MLDE (3.2), two of whose linearly independent solutions are the characters of M(5, 2) and the other is a solution with negative Fourier coefficients at low orders (reminiscent of quasi-characters [20]).This occurs for the N = 28, m 1 = 1, m 2 = 1, k = 67032 solution to the Diophantine equations.But we do not report this here in the tables.
The last two solutions left to discuss are curious ones: S. No.4 and S. No.15 of table 7.
In each case all characters are "stable" characters.For S. No.4, two of the characters are the characters of the ( D4 ) 1 CFT and the third character is the non-identity character of the ( D4 ) 1 CFT.Thus this solution has the curious feature that two of the three characters are identical.This means that Wronskian vanishes, which does not contradict the fact that all the two-character CFT is a [2,2] CFT that appeared in [14].GHM-dual CFTs.All of them belong to only one particular case of GHM-duality among the many that are studied in [15].This is the case where both the denominator CFT D as well as the GHM-dual CFT C are [3, 0] CFTs.It is convenient and suggestive to use the notation D for C. The central charges and the scaling dimensions of the two CFTs, in this particular case of GHM-duality, are related as follows: We collect all the solutions in category IV here in table 8.Each of the GHM-dual CFTs are duals of category I CFTs.From our computations, and from (3.12) which comprise necessary conditions, we can only identify potential GHM-dual CFTs.Whether it is actually a GHM-dual CFT requires more analysis involving meromorphic CFTs [44], [18]; which is beyond the scope of the present work.But all the CFTs in our table 8 are actual GHM-dual CFTs and the meromorphic CFT associated to each and other details can be found in table 2 of [15].
In the last category, from the tables 11-18 we have seven rows corresponding to seven admissible character solutions, which we have gathered in table 9.These are new solutions.They do not seem to fall in any of the known classes of solutions.At this stage of the analysis, one can only claim to have obtained new character-like solutions, also referred to as admissible character solutions.They are not yet CFTs; to establish that we need more work [48].We can only say that we potentially have seven new [3, 0] CFTs.Here, we present the Fourier expansion (a few terms) of the seven potentially new [3,0] RCFTs.

Classifying [3, 2] CFTs
We now study three character rational conformal field theories with a Wronskian index given by 2. The characters of every [3,2] CFT is a solution to the [3, 2] MLDE : -26 -At this stage there are three parameters in the MLDE, but one of them gets fixed: µ −1,1 = 4 and we are in the same situation as the [3, 0] case i.e. two free parameters to scan.The analysis and the strategy follows the [3, 0] case and we will be brief here, highlighting only the contrasts.At the leading order in the q-expansion, the differential equation (3.13) gives the following indicial equation: Note that the sum of the indices is 1  6 which is what is dictated by the Wronskian index being two from (2.13) and which is a direct consequence of choosing µ −1,1 = 4.At the next to leading order in the q-expansion, the differential equation (3.13) gives the following equation We again now evaluate the two equations (3.14) and (3.15) for the identity character and interpret these equations as two equations for the two parameters which give the solution We again have thus determined the two parameters of the MLDE in terms of objects associated with the identity character viz.α 0 and m 1 .We now consider the differential equation (3.13) at the next order in the q-expansion, which when evaluated for the identity character, is an equation for m 2 .After using (3.16) we obtain a polynomial equation in the variables α 0 , m 1 and m 2 .We observe that if we define a new variable the polynomial equation is the following one with indeterminates N , m 1 and m 2 .
We can use the integer-root-theorem to conclude that N has to be an integer.This is again an important turning point in our analysis.N which was just rational is now restricted to be an integer.This means that the central charge, which is given by c = N 210 is a rational number whose denominator (not necessarily in the p q form with p, q coprime) is always 210.Our 4 Potential GHM-dual pairs of [3, 0] CFTs Among the five categories of [3,0] solutions given in section 3.1, we will consider categories I, II and IV as well-understood CFTs.Solutions in category V are new and need to be studied more, as well as solutions in category III, the Franc-Mason vertex operator algebras [43].In this section, we will tabulate some observations about solutions in these two categories and hope to present a more complete study in the future [48].
In the novel coset construction of Gaberdiel-Hampapura-Mukhi [15], the central charges and the scaling dimensions of the denominator [3,0] CFT and the coset [3,0] CFT are related by (3.12).We will define a pair of [3, 0] CFTs (T, T ) to be potentially GHM-dual, or that the theories (T, T ) constitute a potential GHM-dual pair, if their central charges and scaling dimensions follow: We have been able to identify 27 potential GHM-dual pairs of CFTs, which we have tabulated in table 10.There are two kinds of these pairs of solutions.The first kind (17 pairs) is one in which one of the CFTs of the pair is a known WZW CFT and the other is a unknown CFT from category III or category V.This kind of a situation is closer to the GHM construction where one of the CFTs (the denominator CFT) is a WZW CFT.The second kind (10 pairs) is one where both the CFTs of the pair is an unknown CFT from the category III.This seems very interesting.We hope to investigate and report on these potential GHM-dual pairs of CFTs in future work [48].acter solutions in category III and V which are to be considered still unknown, which are also potentially GHM-dual to known WZW CFTs, have better prospects of being consistent CFTs.We hope to report on these ongoing studies.
Note Added: During the final stages of this work we became aware of concurrent work on this subject.The first paper [41] appeared while the first version of our paper was being prepared.The second paper [42] appeared very soon after the first version of our paper and before the present second version of our paper.There is an overlap of our [3,0] results with (small) parts of both these papers but the methods seem to be quite different.

. 29 )
We thus have an infinite number of admissible character solutions, parametrised by A 2 , in(2.29).All members of this infinite family have the same indices and hence the same c, h 1 , h 2 and also they have the same Wronskian.But they are different solutions (they differ in the first character).
and 70.More significantly, our problem has become a "finite" problem in the sense that if one is searching for CFTs in a certain range of central charge, say 0 < c ≤ 1, one only needs to do 70 computations for c = 1 70 , 2 70 , 3 70 , . . . 6970 , 70 70 .Furthermore, (3.7) is a equation with all indeterminates being positive integers, hence a Diophantine equation.We pick a value of N , or equivalently c, and look for positive integer solutions to (3.7).The considerations that will give us the rest of the procedure for solving (3.2) are as follows.The two parameters of the MLDE given in (3.5), can now ((3.6)) can be thought of as functions of N and m 1 .The indicial equation (3.3) becomes a cubic equation whose coefficients are functions of N and m 1 only.Furthermore we readily know one of the roots viz.α 0 or more pertinently − N 1680 .Thus the cubic indicial equation factorises into linear and quadratic equations:

Table 4 :
Simultaneous solutions to Diophantine equations for 0 < N ≤ 70 At this stage, we have the three indices: α 0 , α 1 and α 2 from equations (3.6) and

[ 3 ,
0] solutions should have A 1 η 12 as the Wronskian (see section 2.3), if A 1 = 0 (which is also a possibility).It is not clear to us if we should consider this as a new [3, 0] admissible character or just derived from an old [2, 0] solution.The solution in S. No.15 is similar, except that -21 - [2,3)and are appropriate powers of the Dedekind-eta function, so as to have weight k.We give the Wronskians for a few MLDEs in table 2, which includes all the particular MLDEs we study in this paper.TheA 1 that appears in the last column is the single integration constant resulting from solving the first order differential equation(2.22).A 1 can take different values for different CFTs; we illustrate this in table3where we give the A 1 value for the MMS[2, 0] Solving(2.22)iseffectively one integration.And we are left with n − 1 integrations more to be performed.These are done by solving the following n − 1-order MLDE, which is

Table
in the first five rows.The third row of table 5 is not there in(3.11).This is a known CFT, corresponding to two decoupled copies of the Ising model.It is also a WZW CFT, ( ŝo(2)) 1 , and can perhaps be included in the ( Dr ) 1 series.We performed computations up to N ≤ 6720 or for c ≤ 96.But we are reporting only computations up to N ≤ 3920 or for c ≤ 56, see table17.In the range 56 < c ≤ 96, we have 80 admissible character solutions and all of them are in category I.They are ( Br ) 1 for 56 ≤ r ≤ 95 snd ( Dr ) 1 for 57 ≤ r ≤ 96.It is tempting to conjecture that for all c > 96 also there are only category I solutions (the B and D series WZW CFTs ) and attempt to find a proof for the same.

Table 6
contains the other 9 of the 122 solutions of category I.Each of them corresponds to the tensor product of two copies of a [2, 0] character-like solution (see table1).Eight of these can be located in the partial classification of CFTs(3.11)in the last two rows.The ninth E 7 1 2

table 7
[1,4]rs these questions.The solutions of category II are related to [n, l] CFTs or [n, l] admissible character solutions with n ≤ 2 and l ≤ 4. Two solutions viz.S. No. 9 and 10 of table 7, have two "unstable" characters and the stable character is that of a[1,2]and[1,4]CFT, respectively.18solutions in table 7 have one "unstable" character with the "stable" characters that of known two-character CFTs.The [2, 0] CFT M(5, 2) is not there in table7.It turns out that

Table 7 :
Category II