Pure 3d gravity in AdS is believed to admit a holographic description in terms of 2d CFT. We introduce a theory of fermionic 3d gravity where we sum over geometries equipped with spin structure, and propose it is holographically described by fermionic 2d CFT data. We evaluate the leading contributions to the gravity path integral with one and two torus boundaries, extracting both the spectrum and its spectral statistics from the torus wormhole. Strikingly, the theory has fermionic black hole microstates, even in the absence of bulk fermionic matter. We then incorporate subtle bulk topological field theories, classified by appropriate cobordism groups, and evaluate the one and two-boundary torus partition functions. The spectral statistics we derive from gravity are shown, in all cases, to be consistent with the pattern of anomalies expected from classifications of fermionic 2d CFT. We also define a version of RMT$_2$, a random-matrix framework compatible with the symmetries of 2d CFTs, which naturally accommodates fermionic spectra and reproduces our gravitational results across all cases we analyze.
In this paper, we regard the $T\bar{T}$/$J\bar{T}$-deformed CFTs as perturbation theories and calculate the first order correction of the correlation functions due to the $T\bar{T}$/$J\bar{T}$-deformation. As applications, we study the Rényi entanglement entropy of excited state in the $T\bar{T}$/$J\bar{T}$-deformed two-dimensional CFTs. We find, up to the perturbation first order of the deformation, the Rényi entanglement entropy of locally excited states will acquire a non-trivial time dependence. The excess of the Rényi entanglement entropy of locally excited state will also be dramatically…
Thermal states of quantum systems with many degrees of freedom are subject to a bound on the rate of onset of chaos, including a bound on the Lyapunov exponent, $λ_L\leq 2π/β$. We harness this bound to constrain the space of putative holographic CFTs and their would-be dual theories of AdS gravity. First, by studying out-of-time-order four-point functions, we discuss how $λ_L=2π/β$ in ordinary two-dimensional holographic CFTs is related to properties of the OPE at strong coupling. We then rule out the existence of unitary, sparse two-dimensional CFTs with large central charge and a set of high…
We compare the behavior of the vacuum free energy (i.e. the Casimir energy) of various $(2+1)$-dimensional CFTs on an ultrastatic spacetime as a function of the spatial geometry. The CFTs we consider are a free Dirac fermion, the conformally-coupled scalar, and a holographic CFT, and we take the spatial geometry to be an axisymmetric deformation of the round sphere. The free energies of the fermion and of the scalar are computed numerically using heat kernel methods; the free energy of the holographic CFT is computed numerically from a static, asymptotically AdS dual geometry using a novel app…
We study operator entanglement measures of the unitary evolution operators of (1+1)-dimensional conformal field theories (CFTs), aiming to uncover their scrambling and chaotic behaviors. In particular, we compute the bi-partite and tri-partite mutual information for various configurations of input and output subsystems, and as a function of time. We contrast three different CFTs: the free fermion theory, compactified free boson theory at various radii, and CFTs with holographic dual. We find that the bi-partite mutual information exhibits distinct behaviors for these different CFTs, reflecting…
We study the Lyapunov exponent $λ_L$ in quantum field theories with spacetime-independent disorder interactions. Generically $λ_L$ can only be computed at isolated points in parameter space, and little is known about the way in which chaos grows as we deform the theory away from weak coupling. In this paper we describe families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow $λ_L$ from weak to strong coupling. We find surprising behaviors in some cases, including a discontinuous transition into chaos. We also derive self-consistency equation…
Recently, the modular linear differential equation (MLDE) for level-two congruence subgroups $Γ_θ, Γ^{0}(2)$ and $Γ_0(2)$ of $\text{SL}_2(\mathbb{Z})$ was developed and used to classify the fermionic rational conformal field theories (RCFT). 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 secto…
We consider renormalization of four-fermion operators in the critical QED and $SU(N_c)$ version of Gross--Neveu--Yukawa model in non-integer dimensions. Since the number of mixing operators is infinite, the diagonalization of an anomalous dimension matrix becomes a nontrivial problem. At leading order, construction of eigen-operators is equivalent to solving certain three-term recurrence relations. We find analytic solutions of these recurrence relations that allows to determine the spectrum of anomalous dimensions and study their properties.
It is believed that thermalization drives the reduced density matrix of a subsystem to approach a short-range entangled operator. If the initial state is also short-range entangled, it is possible that the reduced density matrix remains low-entangled throughout thermalization; or there could exist a barrier with high operator entanglement between the initial and thermalized reduced density matrix. In this paper, we study such dynamics in three classes of models: the rational CFTs, the random unitary circuit, and the holographic CFTs, representing systems of increasing quantum chaoticity. We sh…
We apply the method of the large spin bootstrap to analyse fermionic conformal field theories with weakly broken higher spin symmetry. Through the study of correlators of composite operators, we find the anomalous dimensions and OPE coefficients in the Gross-Neveu model in $d=2+\varepsilon$ dimensions and the Gross-Neveu-Yukawa model in $d=4-\varepsilon$ dimensions, based only on crossing symmetry. Furthermore a non-trivial solution in the $d=2+\varepsilon$ expansion is found for a fermionic theory in which the fundamental field is not part of the spectrum. The results are perturbative in $\va…
We study fermionic conformal field theories on surfaces with spin structure in the presence of boundaries, defects, and interfaces. We obtain the relevant crossing relations, taking particular care with parity signs and signs arising from the change of spin structure in different limits. We define fermionic classifying algebras for boundaries, defects, and interfaces, which allow one to read off the elementary boundary conditions, etc. As examples, we define fermionic extensions of Virasoro minimal models and give explicit solutions for the spectrum and bulk structure constants. We show how th…
In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of Z_2 orbifold CFT on a torus. We also find an expression of Z_N cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi en…
We study two novel approaches to efficiently encoding universal constraints imposed by conformal symmetry, and describe applications to quantum chaos in higher dimensional CFTs. The first approach consists of a reformulation of the shadow operator formalism and kinematic space techniques. We observe that the shadow operator associated with the stress tensor (or other conserved currents) can be written as the descendant of a field ${\cal E}$ with negative dimension. Computations of stress tensor contributions to conformal blocks can be systematically organized in terms of the "soft mode" ${\cal…
We study constraints on the space of $d=2$ fermionic CFTs as a function of non-perturbative anomalies exhibited under a fermionic discrete symmetry group $G^f$, focusing our attention also on cases where $G^f$ is non-abelian or presents a non-trivial twist of the $\mathbb{Z}^f_2$ subgroup. For the cases we selected, among our results we find that modular bootstrap consistency bounds predict the presence of relevant/marginal operators only for some groups and anomalies. From this point of view, the appearance in the analysis of several kinks around irrelevant operators with $Δ>2$ means that for…
We extend the non-relativistic AdS/CFT correspondence to the fermionic fields. In particular we study the two point function of a fermionic operator in non-relativistic CFTs by making use of a massive fermion propagating in geometries with Schrodinger group isometry. Although the boundary of the geometries with Schrodinger group isometry differ from that in AdS geometries where the dictionary of AdS/CFT is established, using the general procedure of AdS/CFT correspondence, we see that the resultant two point function has the expected form for fermionic operators in non-relativistic CFTs, thoug…
We study the O(N) vector model and the U(N) Gross-Neveu model with fixed total fermion number, in three dimensions. Using non-trivial polylogarithmic identities, we calculate the large-N renormalized free-energy density of these models, at their conformal points in a ``slab'' geometry with one finite dimension of length L. We comment on the possible implications of our results.
We compute out-of-time-order correlators (OTOCs) in two-dimensional holographic conformal field theories (CFTs) with different left- and right-moving temperatures. Depending on whether the CFT lives on a spatial line or circle, the dual bulk geometry is a boosted BTZ black brane or a rotating BTZ black hole. In the case when the spatial direction is non-compact, we generalise a computation of Roberts and Stanford and show that to reproduce the correct bulk answer a maximal channel contribution needs to be selected when using the identity block approximation. We use the correspondence between g…
It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator $T \bar T$, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that posess a conserved $U(1)$ current, $J$. The deformation takes the schematic form $J \bar T$ and is interesting because it preserves an $SL(2,\mathbb{R}) \times…
We study conformal field theories with Yukawa interactions in dimensions between 2 and 4; they provide UV completions of the Nambu-Jona-Lasinio and Gross-Neveu models which have four-fermion interactions. We compute the sphere free energy and certain operator scaling dimensions using dimensional continuation. In the Gross-Neveu CFT with $N$ fermion degrees of freedom we obtain the first few terms in the $4-ε$ expansion using the Gross-Neveu-Yukawa model, and the first few terms in the $2+ε$ expansion using the four-fermion interaction. We then apply Pade approximants to produce estimates in $d…
Bulk localized excited states in an AdS spacetime can be constructed from Ishibashi states with respect to the global conformal symmetry in the dual CFT. We study boundary two point functions of primary operators in the presence of bulk localized excitations in two dimensional CFTs. From two point functions in holographic CFTs, we observe causal propagations of radiations when the mass of dual bulk scalar field is close to the BF bound. This behavior for holographic CFTs is consistent with the locality and causality in classical gravity duals. We also show that this cannot be seen in free ferm…
In this work we extensively study the dynamics of excited states created by instantaneous local quenches at two different points, i.e., double local quenches. We focus on setups in two dimensional holographic and free Dirac fermion CFTs. We calculate the energy stress tensor and entanglement entropy for double joining and splitting local quenches. In the splitting local quenches we find an interesting oscillating behaviors. Finally, we study the energy stress tensor in double operator local quenches. In all these examples, we find that, in general, there are non-trivial interactions between th…
We consider two-dimensional chiral, first-order conformal field theories governing maps from the Riemann sphere to the projective light cone inside Minkowski space -- the natural setting for describing conformal field theories in two fewer dimensions. These theories have a SL(2) algebra of local bosonic constraints which can be supplemented by additional fermionic constraints depending on the matter content of the theory. By computing the BRST charge associated with gauge fixing these constraints, we find anomalies which vanish for specific target space dimensions. These critical dimensions co…
We propose to use a two-species Fermi gas with the interspecies s-wave Feshbach resonance to realize p-wave superfluidity in two dimensions. By confining one species of fermions in a two-dimensional plane immersed in the background three-dimensional Fermi sea of the other species, an attractive interaction is induced between two-dimensional fermions. We compute the pairing gap in the weak-coupling regime and show that it has the symmetry of p_x+ip_y. Because the magnitude of the pairing gap increases toward the unitarity limit, it is possible that the critical temperature for the p_x+ip_y-wave…
In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Renyi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient $σ$ for the EE and $σ_n$ for the Renyi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for $σ$ and $σ_n$ for all $n=2,3,\dots$. The results for $σ$ agr…
We consider higher derivative gravity and obtain universal relations for the shape coefficients $(f_a, f_b, f_c)$ of the shape dependent universal part of the Rényi entropy for four dimensional CFTs in terms of the parameters $(c, t_2, t_4)$ of two-point and three-point functions of stress tensors. As a consistency check, these shape coefficients $f_a$ and $f_c$ satisfy the differential relation as derived previously for the Rényi entropy. Interestingly, these holographic relations also apply to weakly coupled conformal field theories such as theories of free fermions and vectors but are viola…
The Gordon-Andrews identities, which generalize the Rogers-Ramanujan-Schur identities, provide product and fermionic forms for the characters of the minimal conformal field theories (CFTs) M(2,2k+1). We discuss/conjecture identities of a similar type, providing two different fermionic forms for the characters of the models SM(2,4k) in the minimal series of N=1 super-CFTs. These two forms are related to two families of thermodynamic Bethe Ansatz (TBA) systems, which are argued to be associated with the $\hatϕ_{1,3}^{\rm top}$- and $\hatϕ_{1,5}^{\rm bot}$-perturbations of the models SM(2,4k). Ce…
We study properties of strongly coupled CFT's with non-zero background electric charge in 1+1 dimensions by studying the dual gravity theory - which is a charged BTZ black hole. Correlators of operators dual to scalars, gauge fields and fermions are studied at both T=0 and $T\neq 0$. In the $T=0$ case we are also able to compare with analytical results based on $ AdS_2$ and find reasonable agreement. In particular the correlation between log periodicity and the presence of finite spectral density of gapless modes is seen. The real part of the conductivity (given by the current-current correlat…
In recent work we computed the path integral of three-dimensional gravity with negative cosmological constant on spaces which are topologically a torus times an interval. Here we employ a modular bootstrap to show that the amplitude is completely fixed by consistency conditions and a few basic inputs from gravity. This bootstrap is notably for an ensemble of CFTs, rather than for a single instance. We also compare the 3d gravity result with the Narain ensemble. The former is well-approximated at low temperature by a random matrix theory ansatz, and we conjecture that this behavior is generic f…
The massless three dimensional Gross-Neveu-Yukawa (GNY) and Nambu--Jona-Lasinio--Yukawa (NJLY) models at finite temperatures are analyzed within the mean field framework considering all coupling values. When the number of Dirac fermions is taken to be $N_f=1/4$ (GNY) and $N_f=1/2$ (NJLY) these models relate to the supersymmetric Wess-Zumino (WZ) theory with cubic superpotential and one superfield. In this case the results show that the strong-weak entropy density ratio decreases from the Stefan-Boltzmann value, in the weak limit, to $s/s_{free}=31/35$ at strong couplings. This value agrees wit…
We examine the correspondence between the anti-de Sitter (AdS) description of conformal field theories (CFTs) and the unparticle description of CFTs. We show how unparticle actions are equivalent to holographic boundary actions for fields in AdS, and how massive unparticles provide a new type of infrared cutoff that can be simply implemented in AdS by a soft breaking of conformal symmetry. We also show that processes involving scalar unparticles with dimensions d_s<2 or fermion unparticles with dimensions d_f<5/2 are insensitive to ultraviolet cutoff effects. Finally we show that gauge interac…
We study, by using tools of the dynamical system theory, a fermionic string streched from a non-commutative D2-brane (stack of D0-branes in the BFSS model) to a probe D0-brane as a quantum system driven by a chaotic system, the (classical and quantum) chaos being induced in the D2-brane dynamics by quantum fluctuations. We show that this dynamics with fluctuations induces a decoherence phenomenon on the reduced density matrix of the fermionic string which is characteristic of the chaotic behaviour since it presents an horizon of coherence. Moreover we show that this dynamics is associated with…
