Abstract:We consider the holographic duality where the CFT side is given by SU(N ) adjoint free scalar field theory. Compared to the vector models, the set of single trace operators is immensely extended so that the corresponding AdS theory also contains infinitely many massive higher spin fields on top of the massless ones. We compute the one-loop vacuum energy of these AdS fields to test this duality at the subleading order in large N expansion. The determination of the bulk vacuum energy requires a proper scheme to sum up the infinitely many contributions. For that, we develop a new method and apply it first to calculate the vacuum energies for the first few 'Regge trajectories' in AdS 4 and AdS 5 . In considering the full vacuum energy of AdS theory dual to a matrix model CFT, we find that there exist more than one available prescriptions for the one-loop vacuum energy. Taking a particular prescription, we determine the full vacuum energy of the AdS 5 theory, whereas the AdS 4 calculation still remains technically prohibitive. This result shows that the full vacuum energy of the AdS 5 theory coincides with minus of the free energy of a single scalar field on the boundary. This is analogous to the O(N ) vector model case, hence suggests an interpretation of the positive shift of the bulk coupling constant, i.e. from N 2 − 1 to N 2 .

Supersymmetric field theories can be studied exactly on off-shell "localizing" supergravity backgrounds. We show that these supergravity configurations can be identified with BRST invariant configurations of background topological gravity coupled to background topological gauge multiplets. We apply this topological point of view to two-dimensional N = (2, 2) supersymmetric matter theories to obtain, in a simple and straightforward way, a complete classification of localizing supersymmetric backgrounds in two dimensions. We recover all known localizing backgrounds and (infinitely) many more that have not been explored so far. The newly found localizing backgrounds are characterized by quantized fluxes for both graviphotons of the N = (2, 2) supergravity multiplet. The BRST invariant topological backgrounds are parametrized by both Killing vectors and S 1 -equivariant cohomology of the two-dimensional spacetime. We completely reconstruct the supergravity backgrounds from the topological data: some of the supergravity fields are twisted versions of the topological backgrounds, but others are composite, in that they are nonlinear functionals of topological fields. Moreover, we show that the supersymmetric Ω-deformation is nothing but the background value of the ghost-for-ghost of topological gravity, a result which holds for higher dimensions too.

We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups Γθ, Γ0(2) and Γ0(2) of SL2(ℤ). Each subgroup corresponds to one of the spin structures on the torus. The pole structures 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 constrain the spectrum of N = (1, 1) and N = (2, 2) superconformal field theories in two-dimensions by requiring the NS-NS sector partition function to be invariant under the Γ θ congruence subgroup of the full modular group SL(2, 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 Γ θ . Some of the candidate theories have been discussed by Höhn 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 . #1 Throughout this paper, we only consider parity invariant CFTs so that left-moving and rightmoving fermions have the same boundary condition.

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 Deligne's exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by W-algebras of various type and observe that the bounds on the gap depend on the choice of W-algebra in the small central charge region.