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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$_3$.
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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 programming. 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.
We propose a graph-theoretic description to determine and characterize 5d superconformal field theories (SCFTs) that arise as circle reductions of 6d $mathcal{N} = (1,0)$ SCFTs. Each 5d SCFT is captured by a graph, called a Combined Fiber Diagram (CFD). Transitions between CFDs encode mass deformations that trigger flows between SCFTs. In this way, the complete set of descendants of a given 6d theory are obtained from a single marginal CFD. The graphs encode key physical information like the superconformal flavor symmetry and BPS states. As an illustration, we ascertain the aforementioned data associated to all the 5d SCFTs descending from 6d minimal $(E_6, E_6)$ and $(D_k, D_k)$ conformal matter for any $k$. This includes predictions for thus far unknown flavor symmetry enhancements.
Building on recent progress in the study of compactifications of $6d$ $(1,0)$ superconformal field theories (SCFTs) on Riemann surfaces to $4d$ $mathcal{N}=1$ theories, we initiate a systematic study of compactifications of $5d$ $mathcal{N}=1$ SCFTs on Riemann surfaces to $3d$ $mathcal{N}=2$ theories. Specifically, we consider the compactification of the so-called rank 1 Seiberg $E_{N_f+1}$ SCFTs on tori and tubes with flux in their global symmetry, and put the resulting $3d$ theories to various consistency checks. These include matching the (usually enhanced) IR symmetry of the $3d$ theories with the one expected from the compactification, given by the commutant of the flux in the global symmetry of the corresponding $5d$ SCFT, and identifying the spectrum of operators and conformal manifolds predicted by the $5d$ picture. As the models we examine are in three dimensions, we encounter novel elements that are not present in compactifications to four dimensions, notably Chern-Simons terms and monopole superpotentials, that play an important role in our construction. The methods used in this paper can also be used for the compactification of any other $5d$ SCFT that has a deformation leading to a $5d$ gauge theory.
We explore the connection of anti-de-Sitter supergravity in six dimensions, based on the exceptional F(4) superalgebra, and its boundary superconformal singleton theory. The interpretation of these results in terms of a D4-D8 system and its near horizon geometry is suggested.
We compute correlation functions of chiral primary operators in N=2 superconformal theories at large N using a construction based on supersymmetric localization recently developed by Gerchkovitz et al. We focus on N=4 SYM as well as on superconformal QCD. In the case of N=4 we recover the free field theory results as expected due to non-renormalization theorems. In the case of superconformal QCD we study the planar expansion in the large N limit. The final correlators admit a simple generalization to a finite N formula which exactly matches the various small $N$ results in the literature.
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