Higher Spin Currents Research Articles

Higher spin currents in Wolf space: III

The large linear superconformal algebra (generated by four spin- currents, seven spin-1 currents, four spin- currents and one spin-2 current) found by Sevrin, Troost and Van Proeyen (and other groups) was realized in the superconformal coset theory previously. The lowest 16 higher spin currents of spins , , and are obtained by starting with the operator product expansions (OPEs) between the four spin- currents from the above large linear superconformal algebra and the lowest higher spin-1 current which is the same as the one in the Wolf space coset theory. These OPEs determine the four higher spin- currents and the next six higher spin-2 currents are obtained from the OPEs between the above four spin- currents associated with the supersymmetry and these four higher spin- currents. The four higher spin- currents can be determined by calculating the OPEs between the above four spin- currents and the higher spin-2 currents. Similarly, the higher spin-3 current is obtained from the OPEs between the four spin- currents and the higher spin- currents. The explicit relations between the above 16 higher spin currents and the corresponding 16 higher spin currents which were found in the extension of large nonlinear superconformal algebra previously are given. By examining the OPEs between the 16 currents from the large linear superconformal algebra and the 16 higher spin currents, the match with the findings of Beccaria, Candu and Gaberdiel is also given. The next 16 higher spin currents of spins , , and occur from the OPEs between the above lowest 16 higher spin currents.

Higher spin currents in orthogonal Wolf space

For the superconformal coset theory by (that contains an orthogonal Wolf space) with N = 4, the WZW affine current algebra is obtained. The 16 generators (or 11 generators) of the large linear (or nonlinear) superconformal algebra are described by these WZW affine currents explicitly. Along the line of large holography, the extra 16 currents with spins , , , and are obtained in terms of the WZW affine currents. The lowest spin of this multiplet is two rather than one, which is for a unitary Wolf space. The operator product expansions between the above 11 currents and these extra 16 higher spin currents are found explicitly.

Relative entropy in higher spin holography

We examine relative entropy in the context of the higher-spin/CFT duality. We consider 3$d$ bulk configurations in higher spin gravity which are dual to the vacuum and a high temperature state of a CFT with $\mathcal{W}$-algebra symmetries in presence of a chemical potential for a higher spin current. The relative entropy between these states is then evaluated using the Wilson line functional for holographic entanglement entropy. In the limit of small entangling intervals, the relative entropy should vanish for a generic quantum system. We confirm this behaviour by showing that the difference in the expectation values of the modular Hamiltonian between the states matches with the difference in the entanglement entropy in the short-distance regime. Additionally, we compute the relative entropy of states corresponding to smooth solutions in the $SL(2,\mathbb{Z})$ family with respect to the vacuum.

Higher spin currents in Wolf space: Part II

By calculating the operator product expansions (OPEs) between the 16 higher spin currents of spins , , and in the superconformal Wolf space coset realized by WZW affine currents, the next 16 higher spin currents of spins , , and are determined from the right hand sides of these OPEs. Moreover, the composite fields consisting of both the 11 currents in the large nonlinear superconformal algebra and the above 16 lowest higher spin currents also occur in the right hand sides of these OPEs. The latter appears quadratically (and linearly) in the fusion rules together with large nonlinear superconformal family of the identity operator.

Higher spin currents in Wolf space for generic N

We obtain the 16 higher spin currents with spins (1,3/2,3/2,2),(3/2, 2,2, 5/2), (3/2,2,2, 5/2) and (2,5/2,5/2,3) in the N=4 superconformal Wolf space coset SU(N+2)/[SU(N) x SU(2) x U(1)]. The antisymmetric second rank tensor occurs in the quadratic spin-1/2 Kac-Moody currents of the higher spin-1 current. Each higher spin-3/2 current contains the above antisymmetric second rank tensor and three symmetric (and traceless) second rank tensors (i.e. three antisymmetric almost complex structures contracted by the above antisymmetric tensor) in the product of spin-1/2 and spin-1 Kac-Moody currents respectively. Moreover, the remaining higher spin currents of spins 2, 5/2, 3 contain the combinations of the (symmetric) metric, the three almost complex structures, the antisymmetric tensor or the three symmetric tensors in the multiple product of the above Kac-Moody currents as well as the composite currents from the large N=4 nonlinear superconformal algebra.

Gauge-noninvariant higher-spin currents in four-dimensional Minkowski space

We find the conserved currents of any spin t > 0 constructed from massless gauge fields of any integer spin s ≥ t in four-dimensional Minkowski space. In particular, we construct the stress-energy tensor for a field of any higher spin. Analogously to the spin-two stress (pseudo)tensor, the currents we consider are not gauge invariant, but we show that they generate gauge-invariant charges. In addition to the expected even-parity higher-spin currents, we find unexpected odd-parity currents that generate fewer symmetries than the even currents. We argue that these odd currents most likely do not admit a consistent AdS deformation.

On higher spin extension of the Jackiw–Teitelboim gravity model

We formulate two-dimensional anti-de Sitter (AdS2) higher spin gravity as BF theory with fields taking values in algebra treated as higher spin algebra. The theory is topological and naturally extends the Jackiw–Teitelboim gravity model so as to include higher spin fields. The BF equations linearized about the background are interpreted as describing higher spin partially massless fields of maximal depth along with dilaton fields. It is shown that there are dual metric-like formulations following from the original linearized BF higher spin theory. The duality establishes a dynamical equivalence of the metric-like field equations that can be given either as massive scalar field equations or as conservation conditions for higher spin currents.

Holographic representation of local operators in de Sitter space

Assuming the existence of the dS/CFT correspondence, we construct local scalar fields with ${m}^{2}>{(\frac{d}{2})}^{2}$ in de Sitter space by smearing over conformal field theory operators on the future or past boundary. To maintain bulk microcausality and recover the bulk Wightman function in the Euclidean vacuum, the smearing prescription must involve two sets of single-trace operators with dimensions $\mathrm{\ensuremath{\Delta}}$ and $d\ensuremath{-}\mathrm{\ensuremath{\Delta}}$. Thus the local operator prescription in de Sitter space differs from the analytic continuation from the prescription in anti--de Sitter space. Pushing a local operator in the global patch to future or past infinity is shown to lead to an operator relation between single-trace operators in conformal field theories at ${\mathcal{I}}^{\ifmmode\pm\else\textpm\fi{}}$, which can be interpreted as a basis transformation, also identified as the relation between an operator in CFT and its shadow operator. Construction of spin$\text{\ensuremath{-}}s$ gauge field operators is discussed, and it is shown that the construction of higher spin gauge fields in de Sitter space is equivalent to constructing scalar fields with specific values of mass parameter ${m}^{2}<{(\frac{d}{2})}^{2}$. An acausal higher spin bulk operator that matches onto boundary higher spin current is constructed. Implementation of the scalar operator constructions in AdS and dS with embedding formalism is briefly described.

Comments on Rényi entropy in AdS3/CFT2

We extend and refine recent results on Renyi entropy in two-dimensional conformal field theories at large central charge. To do so, we examine the effects of higher spin symmetry and of allowing unequal left and right central charges, at leading and subleading order in large total central charge. The result is a straightforward generalization of previously derived formulae, supported by both gravity and CFT arguments. The preceding statements pertain to CFTs in the ground state, or on a circle at unequal left- and right-moving temperatures. For the case of two short intervals in a CFT ground state, we derive certain universal contributions to Renyi and entanglement entropy from Virasoro primaries of arbitrary scaling weights, to leading and next-to-leading order in the interval size; this result applies to any CFT. When these primaries are higher spin currents, such terms are placed in one-to-one correspondence with terms in the bulk 1-loop determinants for higher spin gauge fields propagating on handlebody geometries.

On the next-to-leading holographic entanglement entropy in AdS 3 /CFT 2

We reconsider the one-loop correction to the holographic entanglement entropy in $AdS_{3}/CFT_{2}$ by analysing the contributions due to a bulk higher spin $s$ current or a scalar field with scaling dimension $\Delta$. We consider the two-interval case and work perturbatively in their small cross ratio $x$. We provide various results for the entanglement entropy due to the so-called CDW elements of the associated Schottky uniformization group. In particular, in the higher spin current case, we obtain a closed formula for all the contributions of the form $\mathcal O(x^{2s+p})$ up to $\mathcal O(x^{4s})$, where 2-CDW elements are relevant. In the scalar field case, we calculate the similar contributions for generic values of $\Delta$. The terms up to $\mathcal O(x^{2\Delta+5})$ are compared with an explicit CFT calculation with full agreement. The analysis exploits various simplifications which are valid in the strict entanglement limit of the R\'enyi entropy. This allows to identify in a clean way the relevant operators that provide the gravity result. The 2-CDW contributions are also analysed and a closed formula for the leading $\mathcal O(x^{4s})$ coefficient is presented as a function of the generic spin $s$. As a specific application, we combine the CDW and 2-CDW calculations and present the complete $\mathcal O(x^{4s+2})$ entanglement entropy for a spin $s=2,3,4$ higher spin current.

Higher spin currents in Wolf space. Part I

For the N=4 superconformal coset theory described by SU(N+2)/SU(N) (that contains a Wolf space) with N=3, the N=2 WZW affine current algebra with constraints is obtained. The 16 generators of the large N=4 linear superconformal algebra are described by those WZW affine currents explicitly. By factoring out four spin-1/2 currents and the spin-1 current from these 16 generators, the remaining 11 generators (spin-2 current, four spin-3/2 currents, and six spin-1 currents) corresponding to the large N=4 nonlinear superconformal algebra are obtained. Based on the recent work by Gaberdiel and Gopakumar on the large N=4 holography, the extra 16 currents, with spin contents (1, 3/2, 3/2, 2), (3/2, 2, 2, 5/2), (3/2, 2, 2, 5/2), and (2, 5/2, 5/2, 3) described in terms of N=2 multiplets, are obtained and realized by the WZW affine currents. As a first step towards N=4 W algebra (which is NOT known so far), the operator product expansions (OPEs) between the above 11 currents and these extra 16 higher spin currents are found explicitly. It turns out that the composite fields with definite U(1) charges, made of above (11+16) currents (which commute with the Wolf space subgroup SU(N=3) x SU(2) x U(1) currents), occur in the right hand sides of these OPEs.

AdS/CFT for 3D higher-spin gravity coupled to matter fields

New holographic prescription for the model of 3d higher-spin gravity coupled to real matter fields Bμν and C, which was introduced in Fujisawa and Nakayama (2014 Class. Quantum Grav. 31 015003), is formulated. By using a local symmetry, two of the components of Bμν are eliminated, and gauge-fixing conditions are imposed such that the non-vanishing component, Bϕρ, satisfies a covariantly-constancy condition in the background of Chern-Simons gauge fields Aμ, . In this model, solutions to the classical equations of motion for Aμ and are non-flat due to the interactions with matter fields. The solutions for the gauge fields can, however, be split into two parts, flat gauge fields , , and those terms that depend on the matter fields. The equations for the matter fields then coincide with covariantly-constancy equations in the flat backgrounds and , which are exactly the same as those in linearized 3d Vasiliev gravity. The two- and three-point correlation functions of the single-trace operators and the higher-spin currents in the boundary CFT are computed by using an on-shell action tr (Bϕρ C). This term does not depend on coordinates due to the matter equations of motion, and it is not necessary to take the near-boundary limit ρ → ∞. Analysis is presented for SL(3,R) × SL(3,R) as well as higher-spin gravity. In the latter model, scalar operators with scaling dimensions Δ+ = 3/2 and Δ− = 1/2 appear in a single quantization.

The OPEs of spin-4 Casimir currents in the holographic SO(N) coset minimal models

The operator product expansion (OPE) between the spin-4 current and itself in the WD4 coset minimal model was calculated using SO(8) current algebra. The right-hand side of this OPE contains the spin-6 Casimir current, which is also a generator of the WD4 coset minimal model. Based on this N = 8 result, the above OPE was generalized for the general N (in the coset minimal model) using the two N-generalized coupling constants initiated by Hornfeck, which is the simplest OPE for the lowest higher spin currents. The similar OPE in the WB3 (and ) coset minimal model was also analyzed with the SO(7) current algebra. The large N 't Hooft limits are discussed. The results in two-dimensional conformal field theory provide asymptotic symmetry at the quantum level of the higher spin AdS3 gravity reported by Chen et al.

A note on CFT correlators in three dimensions

In this note we present a simple method of constructing general conformally invariant three point functions of operators of various spins in three dimensions. Upon further imposing current conservation conditions, we find new parity violating structures for the three point functions involving either the stress-energy tensor, spin one currents, or higher spin currents. We find that all parity preserving structures for conformally invariant three point functions of higher spin conserved currents can be realized by free fields, whereas there is at most one parity violating structure for three point functions for each set of spins, which is not realized by free fields.

Constraining conformal field theories with a slightly broken higher spin symmetry

We consider three-dimensional conformal field theories that have a higher spin symmetry that is slightly broken. The theories have a large-N limit in the sense that the operators separate into single trace and multitrace and obey the usual large-N factorization properties. We assume that the spectrum of single trace operators is similar to the one that one obtains in the Vasiliev theories. Namely the only single trace operators are the higher spin currents plus an additional scalar. The anomalous dimensions of the higher spin currents are of the order 1/N. Using the slightly broken higher spin symmetry, we constrain the three-point functions of the theories to a leading order in N. We show that there are two families of solutions. One family can be realized as a theory of N fermions with an O(N) Chern–Simons gauge field, the other as an N bosons plus the Chern–Simons gauge field. The family of solutions is parametrized by the 't Hooft coupling. At special parity preserving points, we obtain the critical O(N) models: the Wilson–Fisher one and the Gross–Neveu one. Our analysis also fixes the on-shell three-point functions of Vasiliev's theory on AdS4 or dS4.

Higher spin currents in the $ \mathcal{N} $ = 1 stringy coset minimal model

This study reconsidered the $ \mathcal{N} $ = 1 supersymmetric extension of the W 3 algebra which was studied previously. This extension consists of seven higher spin supercurrents (fourteen higher spin currents in the components) as well as the $ \mathcal{N} $ = 1 stress energy tensor of spins $ \left( {\frac{3}{2},2} \right) $ . Thus far, the complete expressions for the higher spin currents have not been derived. This paper constructs them explicitly in both the c = 4 eight free fermion model and the supersymmetric coset model based on $ \left( {A_2^{(1)}\oplus A_2^{(1) },A_2^{(1) }} \right) $ at level (3, k). By acting with the above $ \mathrm{spin}\hbox{-}\frac{3}{2} $ current on the higher spin-3 Casimir current, its fermionic partner, the higher $ \mathrm{spin}\hbox{-}\frac{5}{2} $ current, can be generated and combined as a first higher spin supercurrent with spins $ \left( {\frac{5}{2},3} \right) $ . By calculating the operator product expansions (OPE) between the higher spin supercurrent and itself, the next two higher spin supercurrents can be generated with spins $ \left( {\frac{7}{2},4} \right) $ and $ \left( {4,\frac{9}{2}} \right) $ . Moreover, the other two higher spin supercurrents with spins $ \left( {4,\frac{9}{2}} \right) $ and $ \left( {\frac{9}{2},5} \right) $ can be generated by calculating the OPE between the first higher spin supercurrent with spins $ \left( {\frac{5}{2},3} \right) $ and the second higher spin supercurrent with spins $ \left( {\frac{7}{2},4} \right) $ . Finally, the higher spin supercurrents, $ \left( {\frac{11 }{2},6} \right) $ and $ \left( {6,\frac{13 }{2}} \right) $ , can be extracted from the right hand side of the OPE between the higher spin supercurrents, $ \left( {\frac{5}{2},3} \right) $ and $ \left( {4,\frac{9}{2}} \right) $ .

The operator product expansion of the lowest higher spin current at finite N

For the N=2 Kazama-Suzuki(KS) model on CP^3, the lowest higher spin current with spins (2, 5/2, 5/2,3) is obtained from the generalized GKO coset construction. By computing the operator product expansion of this current and itself, the next higher spin current with spins (3, 7/2, 7/2, 4) is also derived. This is a realization of the N=2 W_{N+1} algebra with N=3 in the supersymmetric WZW model. By incorporating the self-coupling constant of lowest higher spin current which is known for the general (N,k), we present the complete nonlinear operator product expansion of the lowest higher spin current with spins (2, 5/2, 5/2, 3) in the N=2 KS model on CP^N space. This should coincide with the asymptotic symmetry of the higher spin AdS_3 supergravity at the quantum level. The large (N,k) 't Hooft limit and the corresponding classical nonlinear algebra are also discussed.

Chern–Simons theory with vector fermion matter

We study three dimensional conformal field theories described by U(N) Chern-Simons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger-Dyson equation in lightcone gauge, we compute the exact planar free energy of the theory at finite temperature on R^2 as a function of the 't Hooft coupling lambda=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |lambda|=1; the conformal theory does not exist for |lambda|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three point functions up to two loops. We also discuss a lightcone Hamiltonian formulation of this theory where a W-infinity algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.

The large N ’t Hooft limit of Kazama-Suzuki model

We consider N=2 Kazama-Suzuki model on CP^N=SU(N+1)/SU(N)xU(1). It is known that the N=2 current algebra for the supersymmetric WZW model, at level k, is a nonlinear algebra. The N=2 W_3 algebra corresponding to N=2 was recovered from the generalized GKO coset construction previously. For N=4, we construct one of the higher spin currents, in N=2 W_5 algebra, with spins (2, 5/2, 5/2, 3). The self-coupling constant in the operator product expansion of this current and itself depends on N as well as k explicitly. We also observe a new higher spin primary current of spins (3, 7/2, 7/2, 4). From the behaviors of N=2, 4 cases, we expect the operator product expansion of the lowest higher spin current and itself in N=2 W_{N+1} algebra. By taking the large (N, k) limit on the various operator product expansions in components, we reproduce, at the linear order, the corresponding operator product expansions in N=2 classical W_{\infty}^{cl}[\lambda] algebra which is the asymptotic symmetry of the higher spin AdS_3 supergravity found recently.

The primary spin-4 Casimir operators in the holographic SO(N ) coset minimal models

Starting from SO(N) current algebra, we construct two lowest primary higher spin-4 Casimir operators which are quartic in spin-1 fields. For N is odd, one of them corresponds to the current in the WB_{\frac{N-1}{2}} minimal model. For N is even, the other corresponds to the current in the WD_{\frac{N}{2}} minimal model. These primary higher spin currents, the generators of wedge subalgebra, are obtained from the operator product expansion of fermionic (or bosonic) primary spin-N/2 field with itself in each minimal model respectively. We obtain, indirectly, the three-point functions with two real scalars, in the large N 't Hooft limit, for all values of the 't Hooft coupling which should be dual to the three-point functions in the higher spin AdS_3 gravity with matter.