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In the Mathur-Mukhi-Sen (MMS) classification scheme for rational conformal field theories (RCFTs), a RCFT is identified by a pair of non-negative integers $\mathbf{[n, \ell]}$, with $\mathbf{n}$ being the number of characters and $\mathbf{\ell}$ the Wronskian index. The modular linear differential equation (MLDE) that the characters of a RCFT solve are labelled similarly. All RCFTs with a given $\mathbf{[n, \ell]}$ solve the modular linear differential equation (MLDE) labelled by the same $\mathbf{[n, \ell]}$. With the goal of classifying $\mathbf{[3,3]}$ and $\mathbf{[3,4]}$ CFTs, we set-up and solve those MLDEs, each of which is a three-parameter non-rigid MLDE, for character-like solutions. In the former case, we obtain four infinite families and a discrete set of $15$ solutions, all in the range $0 < c \leq 32$. Amongst these $\mathbf{[3,3]}$ character-like solutions, we find pairs of them that form coset-bilinear relations with meromorphic CFTs/characters of central charges $16, 24, 32, 40, 48, 56, 64$. There are six families of coset-bilinear relations where both the RCFTs of the pair are drawn from the infinite families of solutions. There are an additional $23$ coset-bilinear relations between character-like solutions of the discrete set. The coset-bilinear relations should help in identifying the $\mathbf{[3,3]}$ CFTs. In the $\mathbf{[3,4]}$ case, we obtain nine character-like solutions each of which is a $\mathbf{[2,2]}$ character-like solution adjoined with a constant character., Comment: 64 pages, 20 tables

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., Comment: 81 pages, 4 figures, 2 tables. v2: minor corrections, references updated, typos fixed. v3: minor corrections, main results unchanged, final version, to appear in JHEP

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 $\textbf{[n,l]}$. $\mathbf{n}$ is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and $\mathbf{l}$, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study $\textbf{[3,0]}$ and $\textbf{[3,2]}$ MLDEs in order to classify the corresponding CFTs. We reduce the problem to a "finite" problem: to classify CFTs with central charge $ 0 < c \leq 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 $\textbf{[3,0]}$ case, for $ 0 < c \leq 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 $\textbf{[3,2]}$ CFTs for $ 0 < c \leq 96$: each CFT in this case is obtained by adjoining a constant character to a $\textbf{[2,0]}$ CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago., Comment: Added references and explanations and tables. an added section on possible GHM-dual pairs of CFTs. unchanged section three which contains the classification. 4 pages

The classification scheme for rational conformal field theories, given by the Mathur-Mukhi-Sen (MMS) program, identifies a rational conformal field theory by two numbers: $(n, l)$. $n$ is the number of characters of the rational conformal field theory. The characters form linearly independent solutions to a modular linear differential equation (which is also labelled by $(n, l)$); the Wronskian index $l$ is a non-negative integer associated to the structure of zeroes of the Wronskian. In this paper, we compute the $(n, l)$ values for three classes of well-known CFTs viz. the WZW CFTs, the Virasoro minimal models and the $\mathcal{N} = 1$ super-Virasoro minimal models. For the latter two, we obtain exact formulae for the Wronskian indices. For WZW CFTs, we get exact formulae for small ranks (upto 2) and all levels and for all ranks and small levels (upto 2) and for the rest we compute using a computer program. We find that any WZW CFT at level 1 has a vanishing Wronskian index as does the $\mathbf{\hat{A}_1}$ CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with $\mathbf{\hat{A}_2}$ and $\mathbf{\hat{G}_2}$ have the same $(n,l)$ values, (ii) for the same level CFTs with $\mathbf{\hat{B}_r}$ and $\mathbf{\hat{D}_r}$ have the same $(n,l)$ values for all $r \geq 5$. Classifying all rational conformal field theories for a given $(n, l)$ is one of the aims of the MMS program. We can use our computations to provide partial classifications. For the famous $(2, 0)$ case, our partial classification turns out to be the full classification (achieved by MMS three decades ago). For the $(3, 0)$ case, our partial classification includes two infinite series of CFTs as well as seven ``discrete'' CFTs; except two all others have Kac-Moody symmetry., Comment: 48 pages, 5 tables, partial classification of (3,0) CFTs expanded to include tensor product CFTs

We study the tricritical Ising universality class using conformal bootstrap techniques. By studying bootstrap constraints originating from multiple correlators on the CFT data of multiple OPEs, we are able to determine the scaling dimension of the spin field $\Delta_\sigma$ in various non-integer dimensions $2 \le d \le 3$. $\Delta_{\sigma}$ is connected to the critical exponent $\eta$ that governs the (tri-)critical behaviour of the two point function via the relation, $\eta = 2 - d + 2 \Delta_{\sigma}$. Our results for $\Delta_\sigma$ match with the exactly known values in two and three dimensions and are a conjecture for non-integer dimensions. We also compare our CFT results for $\Delta_\sigma$ with $\epsilon$-expansion results, available up to $\epsilon^3$ order. Our techniques can be naturally extended to study higher-order multi-critical points., Comment: version published in Physical Review D, improved and additional figures, expanded version