Recently, the modular linear differential equation (MLDE) for level-two congruence subgroups $Gamma_theta, Gamma^{0}(2)$ and $Gamma_0(2)$ of $text{SL}_2(mathbb{Z})$ was developed and used to classify the fermionic rational conformal field theories (R
CFT). Two character solutions of the second-order fermionic MLDE without poles were found and their corresponding CFTs are identified. Here we extend this analysis to explore the landscape of three character fermionic RCFTs obtained from the third-order fermionic MLDE without poles. Especially, we focus on a class of the fermionic RCFTs whose Neveu-Schwarz sector vacuum character has no free-fermion currents and Ramond sector saturates the bound $h^{text{R}} ge frac{c}{24}$, which is the unitarity bound for the supersymmetric case. Most of the solutions can be mapped to characters of the fermionized WZW models. We find the pairs of fermionic CFTs whose characters can be combined to produce $K(tau)$, the character of the $c=12$ fermionic CFT for $text{Co}_0$ sporadic group.
The WZW models describe the dynamics of the edge modes of Chern-Simons theories in three dimensions. We explore the WZW models which can be mapped to supersymmetric theories via the generalized Jordan-Wigner transformation. Some of such models have s
upersymmetric Ramond vacua, but the others break the supersymmetry spontaneously. We also make a comment on recent proposals that the Read-Rezayi states at filling fraction $ u=1/2,~2/3$ are able to support supersymmetry.
We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups $Gamma_vartheta$, $Gamma^0(2)$ and $Gamma_0(2)$ of $text{SL}_2(mathbb Z)$. Each subgroup corresponds to one of the spin structures on the torus. The pole st
ructures of the fermionic MLDEs are investigated by exploiting the valence formula for the level-two congruence subgroups. We focus on the first and second order holomorphic MLDEs without poles and use them to find a large class of `Fermionic Rational Conformal Field Theories, which have non-negative integer coefficients in the $q$-series expansion of their characters. We study the detailed properties of these fermionic RCFTs, some of which are supersymmetric. This work also provides a starting point for the classification of the fermionic Modular Tensor Category.
We study the spectrum of pure massless higher spin theories in $AdS_3$. The light spectrum is given by a tower of massless particles of spin $s=2,cdots,N$ and their multi-particles states. Their contribution to the torus partition function organises
into the vacuum character of the ${cal W}_N$ algebra. Modular invariance puts constraints on the heavy spectrum of the theory, and in particular leads to negative norm states, which would be inconsistent with unitarity. This negativity can be cured by including additional light states, below the black hole threshold but whose mass grows with the central charge. We show that these states can be interpreted as conical defects with deficit angle $2pi(1-1/M)$. Unitarity allows the inclusion of such defects into the path integral provided $M geq N$.
The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vecto
rs of smaller central charge; for example, it admits $48$ commuting $c=frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups ${}^2{E}_6(2)$ and ${F}_4(2)$ of Lie type. Many of these examples are naturally associated to McKays $widehat{E_8}$ correspondence, and we use the structure of Nortons monstralizer pairs more generally to organize our presentation.
A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field the
ories, with no extended chiral algebra and $c>1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j eq 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.
We study the twisted index of 4d $mathcal{N}$ = 2 class S theories on a closed hyperbolic 3-manifold $M_3$. Via 6d picture, the index can be written in terms of topological invariants called analytic torsions twisted by irreducible flat connections o
n the 3-manifold. Using the topological expression, we determine the full perturbative 1/N expansion of the twisted index. The leading part nicely matches the Bekestein-Hawking entropy of a magnetically charged black hole in the holographic dual $AdS_5$ with $AdS_2times M_3$ near-horizon.
We investigate the two-dimensional conformal field theories (CFTs) of $c=frac{47}{2}$, $c=frac{116}{5}$ and $c=23$ `dual to the critical Ising model, the three state Potts model and the tensor product of two Ising models, respectively. We argue that
these CFTs exhibit moonshines for the double covering of the baby Monster group, $2cdot mathbb{B}$, the triple covering of the largest Fischer group, $3cdot text{Fi}_{24}$ and multiple-covering of the second largest Conway group, $2cdot 2^{1+22} cdot text{Co}_2$. Various twined characters are shown to satisfy generalized bilinear relations involving Mckay-Thompson series. We also rediscover that the `self-dual two-dimensional bosonic conformal field theory of $c=12$ has the Conway group $text{Co}_{0}simeq2cdottext{Co}_1$ as an automorphism group.
We constrain the spectrum of $mathcal{N}=(1, 1)$ and $mathcal{N}=(2, 2)$ superconformal field theories in two-dimensions by requiring the NS-NS sector partition function to be invariant under the $Gamma_theta$ congruence subgroup of the full modular
group $SL(2, mathbb{Z})$. We employ semi-definite programming to find constraints on the allowed spectrum of operators with or without $U(1)$ charges. Especially, the upper bounds on the twist gap for the non-current primaries exhibit interesting peaks, kinks, and plateau. We identify a number of candidate rational (S)CFTs realized at the numerical boundaries and find that they are realized as the solutions to modular differential equations associated to $Gamma_theta$. Some of the candidate theories have been discussed by Hohn in the context of self-dual extremal vertex operator (super)algebra. We also obtain bounds for the charged operators and study their implications to the weak gravity conjecture in AdS$_3$.
We study constraints coming from the modular invariance of the partition function of two-dimensional conformal field theories. We constrain the spectrum of CFTs in the presence of holomorphic and anti-holomorphic currents using the semi-definite prog
ramming. In particular, we find the bounds on the twist gap for the non-current primaries depend dramatically on the presence of holomorphic currents, showing numerous kinks and peaks. Various rational CFTs are realized at the numerical boundary of the twist gap, saturating the upper limits on the degeneracies. Such theories include Wess-Zumino-Witten models for the Delignes exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by $mathcal{W}$-algebras of various type and observe that the bounds on the gap depend on the choice of $mathcal{W}$-algebra in the small central charge region.
