We study conformal conserved currents in arbitrary irreducible representations of the Lorentz group using the embedding space formalism. With the help of the operator product expansion, we first show that conservation conditions can be fully investigated by considering only two- and three-point correlation functions. We then find an explicitly conformally-covariant differential operator in embedding space that implements conservation based on the standard position space operator product expansion differential operator $\partial_μ$, although the latter does not uplift to embedding space covariantly. The differential operator in embedding space that imposes conservation is the same differential operator $\mathcal{D}_{ijA}$ used in the operator product expansion in embedding space. We provide several examples including conserved currents in irreducible representations that are not symmetric and traceless. With an eye on four-point conformal bootstrap equations for four conserved vector currents $\langle JJJJ\rangle$ and four energy-momentum tensors $\langle TTTT\rangle$, we mostly focus on conservation conditions for $\langle JJ\mathcal{O}\rangle$ and $\langle TT\mathcal{O}\rangle$. Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.
We report some recent progress in the computation of the n-point correlation functions of conserved currents in a class of four dimensional conformal field theories with higher spin symmetry. Global conformal invariance leads to very strong constraints on both the general form and the singularity structure of the correlation functions of conserved currents. Namely, all these functions have to be rational functions with at most double pole singularities. We show that this implies that the 4-, 5- and 6-point correlation functions of the (symmetric, conserved and traceless) stress-energy tensor a…
We study the application of our recent holographic entanglement negativity conjecture for mixed states of adjacent subsystems in conformal field theories with a conserved charge. In this context we obtain the holographic entanglement negativity for zero and finite temperature mixed state configurations in $d$-dimensional conformal field theories dual to bulk extremal and non extremal charged $AdS_{d+1}$ black holes. Our results conform to quantum information theory expectations and constitute significant consistency checks for our conjecture.
Irrational conformal field theory (ICFT) includes rational conformal field theory as a small subspace, and the affine-Virasoro Ward identities describe the biconformal correlators of ICFT. We reformulate the Ward identities as an equivalent linear partial differential system with flat connections and new non-local conserved quantities. As examples of the formulation, we solve the system of flat connections for the coset correlators, the correlators of the affine-Sugawara nests and the high-level $n$-point correlators of ICFT.
An analysis of the structure and singularities of the one loop two point function of the higher spin traceless and conserved currents constructed from the single scalar field in $AdS$ space is presented. The detailed renormalization procedure is constructed and the quantum violation of the traceless Ward identity is investigated. The connection with the one loop effective action for higher spin gauge fields is discussed.
We derive a generating function for all the 3-point functions of higher spin conserved currents in four dimensional conformal field theory. The resulting expressions have a rather surprising factorized form which suggest that they can all be realized by currents built from free massless fields of arbitrary (half-)integer spin s. This property is however not necessarily true also for the higher-point functions. As an illustration we analyze the general 4-point function of conserved abelian U(1) currents of scale dimension equal to three and find that apart from the two free field realizations t…
Perturbing a Virasoro minimal model by the (1,3) primary bulk field results in an integrable field theory. In this paper, an infinite set of commuting conserved charges is obtained by considering defects: a one-parameter family of perturbed defect operators is given, and it is shown that these operators mutually commute, that they commute with the Hamiltonian of the perturbed CFT, and that they satisfy a T-system functional relation. The formulation in terms of perturbed defects is a modification of the original prescription for such charges by Bazhanov, Lukyanov and Zamolodchikov.
We study the constraints imposed by the existence of a single higher spin conserved current on a three dimensional conformal field theory. A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. This is an extension of the Coleman-Mandula theorem to CFT's, which do not have a conventional S matrix. We also briefly discuss the case where the higher spi…
We introduce simple group-theoretic techniques for classifying conformally-invariant tensor-structures. With them, we classify tensor structures of general n-point functions of non-conserved operators, and $n\geq 4$-point functions of general conserved currents, with or without permutation symmetries, and in any spacetime dimension d. (The case n=3 for conserved operators will appear in subsequent work.) Our techniques are useful for bootstrap applications. The rules we derive simultaneously count tensor structures for flat-space scattering amplitudes in d+1 dimensions.
We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in four dimensions. We prove that every such theory contains an infinite number of higher-spin conserved currents of arbitrarily high spin, and that Ward identities generated by the conserved charges of these currents suffice to completely fix the correlators of the stress tensor and the conserved currents to be equal to one of three free field theories: the free boson, the free fermion, and the free vector field. This is a generalization of the result proved in three dimensions…
Three-point correlators of spinning operators admit multiple tensor structures compatible with conformal symmetry. For conserved currents in three dimensions, we point out that helicity commutes with conformal transformations and we use this to construct three-point structures which diagonalize helicity. In this helicity basis, OPE data is found to be diagonal for mean-field correlators of conserved currents and stress tensor. Furthermore, we use Lorentzian inversion formula to obtain anomalous dimensions for conserved currents at bulk tree-level order in holographic theories, which we compare…
Gauge invariant conserved conformal currents built from massless fields of all spins in 4d Minkowski space-time and $AdS_4$ are described within the unfolded dynamics approach. The current cohomology associated with non-zero conserved charges is found. The resulting list of charges is shown to match the space of parameters of the conformal higher-spin symmetry algebra in four dimensions.
Given a conserved and traceless energy-momentum tensor and a conformal Killing vector, one obtains a conserved current. We generalise this construction to superconformal theories in three, four, five and six dimensions with various amounts of supersymmetry by working in the appropriate superspaces.
We investigate the extension of a holographic construction for the entanglement negativity of two disjoint subsystems in proximity to $CFT_d$s with a conserved charge dual to bulk $AdS_{d+1}$ geometries. The construction involves a specific algebraic sum of the areas of bulk co-dimension two static minimal surfaces homologous to certain appropriate combinations of the subsystems in question. In this connection we compute the holographic entanglement negativity for two disjoint subsystems in proximity, with long rectangular strip geometries in $CFT_d$s dual to bulk non extremal and extremal RN-…
The full list of conserved conformal higher spin currents built from massless scalar and spinor fields is presented. It is shown that, analogously to the relationship between usual conformal and AdS symmetries, the set of the conformal higher spin symmetry parameters associated with the conformal conserved currents in d dimensions is in the one-to-one correspondence with the result of the dimensional reduction of the usual (i.e., non-conformal) higher spin symmetry parameters in d+1 dimensions.
The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted ``spin -1/2'' chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin $+1/2$ chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of…
We find the form of three-point correlation functions of traceless symmetric conserved currents of arbitrary spin in d-dimensional conformal field theory (CFT). These are fixed up to several constants by conformal symmetry and current conservation conditions. We present generating functionals for all structures in arbitrary d. In even dimensions we present an interpretation for each structure in terms of the corresponding free field. In odd dimensions d>3 an infinite number of structures is found which are not generated by known CFTs.
The implications of conformal invariance, as relevant in quantum field theories at a renormalisation group fixed point, are analysed with particular reference to results for correlation functions involving conserved currents and the energy momentum tensor. Ward identities resulting from conformal invariance are discussed. Explicit expressions for two and three point functions, which are essentially determined by conformal invariance, are obtained. As special cases we consider the three point functions for two vector and an axial current in four dimensions, which realises the usual anomaly simp…
Couch-Torrence conformal inversion, supersymmetry and conserved charges for D3-branes
Sezione INFN Roma “Tor Vergata”（INFN罗马“托尔维加塔”分部）；Dipartimento di Fisica, Universita di Roma “Tor Vergata”（罗马“托尔维加塔”大学物理系）；Dipartimento di Fisica, Università di Roma “La Sapienza”（罗马“萨皮恩扎”大学物理系）；Sezione INFN Roma “La Sapienza”（INFN罗马“萨皮恩扎”分部）
We study the Abelian Thirring Model when the fermionic fields have non-conserved chiral charge: $Δ{\cal Q}_5 =N$. One of the main features we find for this model is the dependence of the Virasoro central charge on both the Thirring coupling constant and $N$. We show how to evaluate correlation functions and in particular we compute the conformal dimensions for fermions and fermionic bilinears, which depend on the fermionic chiral charge. Finally we build primary fields with arbitrary conformal weight.
We first prove that, in Vasiliev's theory, the zero-form charges studied in 1103.2360 and 1208.3880 are twisted open Wilson lines in the noncommutative $Z$ space. This is shown by mapping Vasiliev's higher-spin model on noncommutative Yang--Mills theory. We then prove that, prior to Bose-symmetrising, the cyclically-symmetric higher-spin invariants given by the leading order of these $n$-point zero-form charges are equal to corresponding cyclically-invariant building blocks of $n$-point correlation functions of bilinear operators in free conformal field theories (CFT) in three dimensions. On t…
We compute current correlators of the CP^{N-1} field theory in 2+1 dimensions, both at the critical point and in the phase with spontaneously broken SU(N) symmetry. Universal constants are obtained to next-to-leading order in the 1/N expansion. Implications are noted for quantum critical points of antiferromagnets, and their vicinity.
It has been known for some time that the (1,3) perturbations of the (2k+1,2) Virasoro minimal models have conserved currents which are also singular vectors of the Virasoro algebra. This also turns out to hold for the (1,2) perturbation of the (3k+-1,3) models. In this paper we investigate the requirement that a perturbation of an extended conformal field theory has conserved currents which are also singular vectors. We consider conformal field theories with W3 and (bosonic) WBC2 = W(2,4) extended symmetries. Our analysis relies heavily on the general conjecture of de Vos and van Driel relatin…
Operator algebra of (not necessarily free) higher-spin conformal conserved currents in generalized matrix spaces, that include 3d Minkowski space-time as a particular case, is shown to be determined by an associative algebra $M$ of functions on the twistor space. For free conserved currents, $M$ is the universal enveloping algebra of the higher-spin algebra. Proposed construction greatly simplifies computation and analysis of correlators of conserved currents. Generating function for $n$-point functions of 3d (super)currents of all spins, built from $N$ free constituent massless scalars and sp…
It is shown how to obtain conformal blocks from embedding space with the help of the operator product expansion. The minimal conformal block originates from scalar exchange in a four-point correlation functions of four scalars. All remaining conformal blocks are simple derivatives of the minimal conformal block. With the help of the orthogonality properties of the conformal blocks, the analytic conformal bootstrap can be implemented directly in embedding space, leading to a Jacobi-like definition of conformal field theories.
We construct conformal three-point functions in momentum space with a general tensor and conserved currents of spin $1$ and $2$. While conformal correlators in momentum space have been studied especially in the connection with cosmology, correlators involving a tensor of general spin and scaling dimension have not been studied very much yet. Such a direction is unavoidable when we go beyond three-point functions because general tensors always appear as an intermediate state. In this paper, as a first step, we solve the Ward-Takahashi identities for correlators of a general tensor and conserved…
We consider the perturbations of the 3-state Potts conformal field theory introduced by Cardy as a description of the chiral 3-state Potts model. By generalising Zamolodchikov's counting argument and by explicit calculation we find new inhomogeneous conserved currents for this theory. We conjecture the existence of an infinite set of conserved currents of this form and discuss their relevance to the description of the chiral Potts models.
We analyse the general structure of the three-point functions involving conserved bosonic and fermionic higher-spin currents in three-dimensional conformal field theory. Using the constraints of conformal symmetry and conservation equations, we use a computational formalism to analyse the general structure of $\langle J^{}_{s_{1}} J'_{s_{2}} J''_{s_{3}} \rangle$, where $J^{}_{s_{1}}$, $J'_{s_{2}}$ and $J''_{s_{3}}$ are conserved currents with spins $s_{1}$, $s_{2}$ and $s_{3}$ respectively (integer or half-integer). The calculations are completely automated for any chosen spins and are limited…
Using unfolded formulation of free equations for massless fields of all spins we obtain explicit form of higher-spin conformal conserved charges bilinear in 4d massless fields of arbitrary spins.
We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involu…
We develop techniques useful for obtaining conformal blocks in embedding space. We construct a unique differential operator in embedding space and use it to construct a function that will be an important ingredient in assembling conformal blocks. We show a number of relations that the components of conformal blocks satisfy and find invariance of our expressions under the dihedral group.
We discuss conserved currents and operator product expansions (OPE's) in the context of a $O(N)$ invariant conformal field theory. Using OPE's we find explicit expressions for the first few terms in suitable short-distance limits for various four-point functions involving the fundamental $N$-component scalar field $ϕ^α(x)$, $α=1,2,..,N$. We propose an alternative evaluation of these four-point functions based on graphical expansions. Requiring consistency of the algebraic and graphical treatments of the four-point functions we obtain the values of the dynamical parameters in either a free theo…
We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary…
We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff's law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one dimensional spatial manifold exhibits con…
Infinite set of higher spin conserved charges is found for the $sp(2M)$ symmetric dynamical systems in $\f{1}{2} M (M+1)$-dimensional generalized spacetime $\M_M$. Since the dynamics in $\M_M$ is equivalent to the conformal dynamics of infinite towers of fields in $d$-dimensional Minkowski spacetime with $d=3,4,6,10,...$ for $M=2,4,8,16, ...$, respectively, the constructed currents in $\M_M$ generate infinite towers of (mostly new) higher spin conformal currents in Minkowski spacetime. The charges have a form of integrals of $M$-forms which are bilinear in the field variables and are closed as…
In this paper we study the three-point correlation functions of two scalar operators with large conformal dimensions and the R-current or stress-energy tensor at strong coupling with the help of the AdS_4/CFT_3 correspondence. The scalar operators are dual to semiclassical strings in AdS_4 x CP^3, which are point-like in AdS. We establish thorough concordance between string theory results at large coupling constant and general predictions coming from Ward identities in the dual three-dimensional superconformal gauge theory.
We study some of the novel properties of conformal field theories with noncompact target spaces as applied to string theory. Standard CFT results get corrected by boundary terms in the target space in a way consistent with the expected spacetime physics. For instance, one-point functions of general operators on the sphere and boundary operators on the disk need not vanish; we show that they are instead equal to boundary terms in spacetime. By applying this result to vertex operators for spacetime gauge transformations with support at infinity, we derive formulas for conserved gauge charges in …
We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in d>3 dimensions. We prove that every such theory contains an infinite number of higher-spin conserved currents of arbitrarily high spin, and that Ward identities generated by the conserved charges of these currents imply that the correlators of the stress tensor and the conserved currents of the theory must coincide with one of the following three possibilities: a) a theory of n free bosons (for some integer n), b) a theory of n free fermions, or c) a theory of n (d-2)/2-forms…
The $Z_N$-invariant chiral Potts model is considered as a perturbation of a $Z_N$ conformal field theory. In the self-dual case the renormalization group equations become simple, and yield critical exponents and anisotropic scaling which agree with exact results for the super-integrable lattice models. Although the continuum theory is not Lorentz invariant, it respects a novel type of space-time symmetry which allows for the observed spontaneous breaking of translational symmetry in the ground state. The continuum theory is shown to possess an infinite number of conserved charges on the self-d…
In Minkowski spacetime it is well-known that the canonical energy-momentum current is involved in the construction of the globally conserved currents of energy-momentum and total angular momentum. For the construction of conserved currents corresponding to (approximate) scale and proper conformal symmetries, however, an improved energy-momentum current is needed. By extending the Minkowskian framework to a genuine metric-affine spacetime, we find that the affine Noether identities and the conformal Killing equations enforce this improvement in a rather natural way. So far, no gravitational dyn…
We construct a set of conserved charges for asymptotically deSitter spacetimes that correspond to asymptotic conformal isometries. The charges are given by boundary integrals at spatial infinity in the flat cosmological slicing of deSitter. Using a spinor construction, we show that the charge associated with conformal time translations is necessarilly positive and hence may provide a useful definition of energy for these spacetimes. A similar spinor construction shows that the charge associated with the time translation Killing vector of deSitter in static coordinates has both positive and neg…
It is well-known that conformal embeddings can be used to construct non-diagonal modular invariants for affine lie algebras. This idea can be extended to construct infinite series of non-diagonal modular invariants for coset CFTs. In this paper, we systematically approach the problem of identifying higher-spin bulk duals for these kind of non-diagonal invariants. In particular, for a special value of the 't Hooft coupling, there exist a class of partition functions that have enhanced supersymmetry, which should be reflected in a bulk dual. As a illustration of this, we show that a partition fu…
In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension $Δ$ of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the recursion relation introduced in 1307.6856 for conformal blocks of external scalar operators. We then generalize…
We revisit the thermodynamic aspects of the scalar-tensor theory of gravity in the Jordan and in the Einstein frame. Examining the {\it missing links} of this theory carefully, we establish the thermodynamic descriptions from the conserved currents and potentials by following both the Noether and the Abbott-Deser-Tekin (ADT) formalism. With the help of conserved Noether current and potential, we define the thermodynamic quantities, which we show to be {\it conformally invariant}. Moreover, the defined quantities are shown to fit nicely in the laws of (the first and the second) black hole therm…
We revisit a newfound construction of rational electromagnetic knots based on the conformal correspondence between Minkowski space and a finite $S^3$-cylinder. We present here a more direct approach for this conformal correspondence based on Carter-Penrose transformation that avoids a detour to de Sitter space. The Maxwell equations can be analytically solved on the cylinder in terms of $S^3$ harmonics $Y_{j;m,n}$, which can then be transformed into Minkowski coordinates using the conformal map. The resultant "knot basis" electromagnetic field configurations have non-trivial topology in that t…
We consider $N=1$ supersymmetric Toda theories which admit a fermionic untwisted affine extension, i.e. the systems based on the $A(n,n)$, $D(n+1,n)$ and $B(n,n)$ superalgebras. We construct the superspace Miura trasformations which allow to determine the W-supercurrents of the conformal theories and we compute their renormalized expressions. The analysis of the renormalization and conservation of higher-spin currents is then performed for the corresponding supersymmetric massive theories. We establish the quantum integrability of these models and show that although their Lagrangian is not her…
Gravitational currents and charges from conformal Killing-Yano tensors
The Blackett Laboratory, Imperial College London（帝国理工学院布莱克特实验室）；Department of Physics and Astronomy, Uppsala University（乌普萨拉大学物理与天文学系）；Centre for Geometry and Physics, Uppsala University（乌普萨拉大学几何与物理中心）
