Higher Spin Gravity and Exact Holography

Higher Spin Gravity and Exact Holography

Kewang Jin
Institute for Theoretical Physics, ETH-Zurich, Switzerland
E-mail: jinke@itp.phys.ethz.ch
Speaker.
Abstract

In this talk, we present some direct evidences of the Higher Spin/Vector Model correspondence. There are two particular examples we would like to address on. The first example concerns a constructive approach of four dimensional higher spin theory from 3d vector model based on a bi-local formulation. These bi-local fields are seen to give a bulk description of the higher spin theory with extra dimension and interactions. The second example is a similar AdS/CFT duality put forward by Gaberdiel and Gopakumar. Specifically, we are interested in black hole solutions carrying nonzero higher spin charges. The partition function of the general higher spin black hole is computed from the CFT side using the symmetry and we found a perfect agreement with the gravity result.

Higher Spin Gravity and Exact Holography

Kewang Jinthanks: Speaker.

Institute for Theoretical Physics, ETH-Zurich, Switzerland

E-mail: jinke@itp.phys.ethz.ch

\abstract@cs

Proceedings of the Corfu Summer Institute 2012 September 8-27, 2012 Corfu, Greece

1 Introduction

The AdS/CFT correspondence [1, 2, 3] represents one of the major tools to understand Quantum Gravity. It is a special example of the Holographic principle [4, 5] which states a quantum theory in dimensions with gravity can be described by a gauge theory on the boundary (with one lower dimension). Therefore, understanding the emergence of the extra dimension plays a central role to unravel the mechanism of the holographic principle. Among various examples of the AdS/CFT correspondence, the best studied one is of course the type IIB string theory on and its dual to Supersymmetric Yang-Mills theory in four dimensions. However, this duality relates two highly non-trivial theories and crucially relies on supersymmetry. Furthermore, this is a strong-weak duality which makes the correspondence hard to prove analytically. Therefore it is very important and meaningful to find some simplified model of the AdS/CFT correspondence where both sides of duality may be exactly solvable. This will provide some direct evidences and definite understanding of the AdS/CFT correspondence.

The Higher Spin/Vector Model correspondence serves such an example. On the field theory side, it is the simplest -component (free) vector model. In the large limit, a special kind of higher spin AdS gravity developed by Vasiliev and collaborators [6, 7] emerges. As pointed out by Klebanov and Polyakov [8], the singlet sector of the vector model is dual to Vasiliev higher spin gravity with minimal coupling to the scalar field (see also [9, 10]). The three-dimensional model has two fixed points: a free (UV) fixed point and an interacting (IR) fixed point. To be more precise, these two fixed points are dual to two boundary conditions of the bulk scalar field.

On the gravity side, the higher spin theory of Vasiliev describes the interaction of an infinite tower of massless higher spin fields with gravity (and other lower spin matter fields). In four and higher dimensions, a simple action principle still remains to be found (see however [11] for an action principle of the “extended” Vasiliev’s system), instead the Vasiliev theory is described in terms of nonlinear (and nonlocal) equations of motion. The higher spin theory is itself a gauge theory with a large class of higher spin gauge symmetry; and the gauge fixing of the full nonlinear theory is far from obvious. Without knowing an explicit action, the calculation of correlation functions through the equations of motion is highly technical. The process usually involves partially gauge fixing combined with partially solving the equations. An impressive comparison of three-point functions was made by Giombi and Yin [12, 13] who were able to show the two fixed points of the vector model are dual to two boundary conditions of the bulk scalar field as conjectured by Klebanov and Polyakov [8].

We have in [14, 15] formulated a constructive approach of bulk AdS higher spin gravity in terms of bi-local fields of the model. These bi-local fields describe (over)-completely the singlet sector of the model and leads to a nonlinear, interacting theory (with as the coupling constant) which was seen to possess all the properties of the dual AdS theory. The interactions are present for both the free and critical fixed points. More importantly, the bi-local formulation automatically reproduces arbitrary-point correlation functions and provides a construction of higher spin theory (in various gauges [16]) based on the CFT. The main idea of the construction is to study how the fields transform under the symmetry and compare directly the generators. A canonical transformation was found between the generators and the conformal generators in 3d, based on which an integral transformation was then found to map the bi-local fields to the higher spin fields with the extra AdS dimension. This map is one-to-one and provides direct evidence and understanding of the emergent AdS spacetime.

The second part of this talk concerns the duality between the AdS higher spin theory (coupled to matter [17]) and a 2d minimal model proposed by Gaberdiel and Gopakumar [18]. To be more precise, the 3d massless higher spin theory coupled with two massive scalars is dual to the 2d minimal model in the ’t Hooft limit (defined below). The minimal model can be constructed using the WZW coset

 su(N)k⊕su(N)1su(N)k+1 (1.0)

where the central charge at finite reads

 c=(N−1)[1−N(N+1)(N+k)(N+k+1)] . (1.0)

In the ’t Hooft limit defined by taking and keeping the ’t Hooft coupling constant fixed

 0≤λ≡NN+k≤1 ,\specialhtml:\specialhtml: (1.0)

the central charge scales as . Therefore, these models are vector-like (and the parameter also fixes the mass of the AdS scalar field). The primaries of the minimal model are labelled by where and are respectively the representations of and . The representation of is uniquely determined by the selection rule [19].

Three dimensional higher spin theories are much simpler to study (than higher dimensional theories) because it is consistent to truncate the infinite tower of higher spin fields to a finite set of spins with maximal spin , i.e. the higher spin theory with the spin content is closed. More importantly, this theory has a Chern-Simons formulation with the gauge group , i.e. the action is given by [20]

 SHS=SCS[A]−SCS[¯A] (1.0)

where

 SCS[A]=kCS4π∫Tr(A∧dA+23A∧A∧A) , (1.0)

and similarly for . The Chern-Simons level is related to the AdS radius by

 kCS=ℓ4GN , (1.0)

where is Newton’s constant. The Chern-Simons formulation can be extended to the infinite spin case,111Actually, the original derivation of [20] is for the infinite algebra . in which the relevant Lie algebra is the one-parameter family of higher spin algebra .222The parameter is the same as the ’t Hooft coupling constant (1). It is an infinite dimensional algebra which can be realized as a quotient of the universal enveloping algebra of by a proper ideal [21]

 hs[λ]⊕C=U(sl(2))⟨C2−μ1⟩ , (1.0)

where is the quadratic Casimir of and . The vector corresponding to is the identity generator of the universal enveloping algebra. Setting , an ideal consisting all the higher spin generators with appears. Quotienting out this ideal, one obtains the algebra

 hs[λ=N]/χN≅sl(N) .\specialhtml:\specialhtml: (1.0)

Using the Chern-Simons formulation of higher spin gravity and following the Brown Henneaux analysis for pure gravity [22], the asympotitic symmetry of gravity is the algebra [23, 24, 25]. Similarly, the asymptotic symmetry of the gravity is the one-parameter family of -algebra denoted as [21]. All these classical analyses lead to the same central charge of the -algebra (as the Virasoro algebra for pure gravity)

 c=3ℓ2GN=6kCS .\specialhtml:\specialhtml: (1.0)

Similar to (1), setting and quotienting out an ideal, we obtain

 W∞[λ=N]/χN≅WN . (1.0)

However, the ’t Hooft coupling constant defined in (1) is between 0 and 1; it is not obvious the asymptotic symmetry algebra with is the same as the symmetry algebra of the coset model with . This was clarified in [26] by studying the quantum algebra (with finite values of and ). Specifically, a triality isomorphism of the -algebra (at fixed ) was discovered

 W∞[N]≅W∞[NN+k]≅W∞[−NN+k+1] , (1.0)

which explains the agreement of the symmetry algebra.333The last isomorphism, in the ’t Hooft limit, leads to . This can be easily seen from the commutation relations of algebra [61] that the structure constants only depend on . As a matter of fact, for the bulk higher spin theory, the analytic continuation of gravity to gravity is achieved by [26, 72]. The quantum analysis also reveals that one of the scalar fields in the ’t Hooft limit (which duals to the representation of the CFT) has a non-perturbative origin. This suggests the perturbative Vasiliev theory is not complete at the quantum level (various non-perturbative excitations must be added) [28, 29, 26]. Besides the agreement of the symmetry algebra, the Gaberdiel-Gopakumar conjecture is also supported by the matching of partition functions [27] and certain correlation functions [28, 29, 30, 31, 32, 33], etc.

2 Direct Construction of Higher Spin Gravity from the O(n) Model

Starting with the AdS/CFT case, the Lagrangian of the 3d model is given by

 L=∫d3x(12(∂μϕa)(∂μϕa)+g4N(ϕ⋅ϕ)2),a=1,...,N. (2.0)

As mentioned before, this model has two fixed points: the UV fixed point with zero coupling constant () and the IR fixed point with non-zero coupling constant. On the bulk side, the mass of the scalar field is which gives two possible boundary conditions with scaling dimensions and . The Klebanov-Polyakov conjecture states that the boundary condition is dual to the free model, whereas the boundary condition duals to the critical model. These two fixed points are related by a Legendre transformation [34, 35] (see also [36, 37, 38]). Therefore, the duality between Vasiliev’s theory and the critical model follows, order by order in , from the duality with free model [39]. While for collective field theory (reviewed in the next subsection), the critical model can be treated on equal footing as the free model. In the following, we will focus on the free model whereas trying to keep some formulae valid also for the critical model.

2.1 Collective field theory of the O(n) model

The collective field theory we employ here was developed by Jevicki and Sakita [40] for large- quantum field theory. The main idea is to reformulate the theory using the invariant (under the symmetry group) collective fields. After a non-trivial change of variables, a collective action (for the Euclidean case) or Hamiltonian (for the Minkowski case) can be derived in terms of the collective fields (and their conjugate momenta). A subsequent expansion can be carried out in a rather straightforward way. This method had great success in applying to the matrix model [40], which leads to a field theoretic formulation of string as a massless scalar field in two dimensions [41]. There, the emergence of extra dimension comes from the large- color index of the matrix model.

The constructive approach for the AdS/CFT duality formulated in [14] is based on the invariant bi-local fields

 Φ(x,y)≡N∑a=1ϕa(x)⋅ϕa(y) (2.0)

which close under the Schwinger-Dyson equations in the large limit. Through a series of chain rules of the type

 ∂∂ϕa(x)=∫dy∫dz∂Φ(y,z)∂ϕa(x)∂∂Φ(y,z)=∫dyϕa(y)[∂∂Φ(y,x)+∂∂Φ(x,y)] ,\specialhtml:\specialhtml: (2.0)

the collective action which evaluates the complete invariant partition function can be derived as [42]

 Z=∫[∏adϕa(x)]e−S[ϕ]=∫[dΦ(x,y)]μ(Φ)e−Sc[Φ] (2.0)

where the measure is given by with the volume of space and the volume of momentum space (where is the momentum cutoff). Explicitly the collective action can be computed as in [42, 14] to be

 Sc[Φ]=∫dx[−12limy→x∂2xΦ(x,y)+g4NΦ2(x,x)]−N2∫dxlnΦ(x,x) ,\specialhtml:\specialhtml: (2.0)

where the first term is a direct rewriting of the original action in terms of bi-local fields; while the second interaction term arises from the Jacobian of the change of variables

 ∫∏adϕa(x)=∫dΦ(x,y)J[Φ] . (2.0)

We stress that the Jacobian here gives both the measure and the interaction term .

The collective action (2.1) is nonlinear, with appearing as the expansion parameter. The perturbative expansion in this bi-local theory proceeds in the standard way. The nonlinear equation of motion specified by gives the background

 Φ∂Sc∂Φ∣∣∣Φ=Φ0=0⟹−∂2xΦ0(x,y)+gNΦ20(x,y)−2Nδ(x−y)=0 . (2.0)

Expanding around the background gives an infinite sequence of interaction vertices [43]

 Sc[Φ]=Sc[Φ0]+Tr[14Φ−10ηΦ−10η+g4N2η2+∑n≥312nN1−n2(Φ−10η)n] , (2.0)

where the trace is defined to be . The nonlinearities built into are precisely such that all invariant -point correlators of the singlet fields

 ⟨Φ(x1,y1)⋯Φ(xn,yn)⟩=⟨ϕ(x1)⋅ϕ(y1)⋯ϕ(xn)⋅ϕ(yn)⟩ (2.0)

are reproduced through the Witten diagrams with vertices as shown in Figure 1. We stress that this nonlinear structure is there for both the interacting and the free () fixed points.

This bi-local theory is expected to represent a covariant-type gauge fixing of Vasiliev’s gauge invariant theory. A large number of degrees of freedom are removed in fixing a gauge and this happens in higher spin gravity too. In section 2.4, we will make a connection of this covariant bi-local theory to a symmetric gauge of the Vasiliev theory. We can show this reduced bulk system (after further solving some components of the equations of motion) has the same dimensionality as the covariant bi-local theory.

A one-to-one relationship between bi-local fields and AdS higher spin fields are demonstrated in a physical gauge with a single time [15]. The existence of such a gauge is not a priori obvious (because this is a reduction of two-time physics to a single-time physics where nontrivial issues of unitarity should be addressed). In [16], we have shown such a gauge fixing and a discussion of the collective dipole underlying the collective construction was given.

The single-time formulation involves the equal-time bi-local fields

 Ψ(t,→x,→y)=∑aϕa(t,→x)ϕa(t,→y)\specialhtml:\specialhtml: (2.0)

which are local in time but bi-local in spatial dimensions. These observables (collective fields) are characterized by the fact that they represent a complete set of invariant canonical variables. Writing the conjugate momenta as

 Π(→x,→y)=−i∂∂Ψ(→y,→x) , (2.0)

the collective Hamiltonian is of the form [44, 45]

 H=2∫d→xd→yd→zΠ(→x,→y)Ψ(→y,→z)Π(→z,→x)+V[Ψ]\specialhtml:\specialhtml: (2.0)

where the potential term reads

 V[Ψ]=∫d→x[−12lim→y→→x∇2→xΨ(→x,→y)+g4NΨ2(→x,→x)]+N28∫d→xΨ−1(→x,→x) .\specialhtml:\specialhtml: (2.0)

Similar to the Euclidean case, the first term is a direct rewriting of the original potential term in terms of the bi-local fields. The appearance of the last term needs an explanation. In general, a direct reformulation of the original Hamiltonian in terms of the bi-local fields (after applying the chain rules (2.1)) is not hermitian. There exists a similarity transformation after which the final effective Hamiltonian is hermitian. The last term in (2.1) arises precisely from such a similarity transformation and this is in some sense analogous to the Jacobian of the Euclidean case.

The Hamiltonian (2.1) again has a natural expansion, after a background shift

 Ψ=Ψ0+1√Nη ,Π=√Nπ , (2.0)

where is the saddle point of , the quadratic Hamiltonian reads

 H(2)=Tr(πΨ0π)+18Tr(Ψ−10ηΨ−10ηΨ−10) (2.0)

where we have set the coupling constant for simplicity.444There is an overall constant we omit here (and subsequently for all the higher vertices), which states the coupling constant of the collective field theory is . Fourier transforming the fluctuations as well as the background field

 Ψ0→x→y = ∫d→kei→k⋅(→x−→y)ψ0→k ,ψ0→k=12√→k2 (2.0) η→x→y = ∫d→k1d→k2e−i→k1⋅→x+i→k2⋅→yη→k1→k2 , (2.0) π→x→y = ∫d→k1d→k2e+i→k1⋅→x−i→k2⋅→yπ→k1→k2 , (2.0)

one finds the quadratic Hamiltonian in momentum space

 H(2)=12∫d→k1d→k2π→k1→k2π→k1→k2+18∫d→k1d→k2η→k1→k2(ψ0−1→k1+ψ0−1→k2)2η→k1→k2 (2.0)

which correctly describes the (singlet) spectrum of the theory with . Higher vertices representing interactions can be found similarly, in particular, the cubic and quartic interactions are given explicitly as

 H(3) = 2√NTr(πηπ)−18√NTr(Ψ−10ηΨ−10ηΨ−10ηΨ−10) , (2.0) H(4) = 18NTr(Ψ−10ηΨ−10ηΨ−10ηΨ−10ηΨ−10) . (2.0)

We note that the form of these vertices is the same for both the free (UV) and the interacting (IR) conformal theories. The only difference is induced by different background shifts in these two cases.

2.2 Scattering matrix

For the UV fixed point of the vector model, there exists an infinite sequence of exactly conserved higher spin currents. Consequently one has a higher symmetry with infinite number of generators. In such a theory, the Coleman-Mandula theorem would imply the matrix should be 1. In general, there is a question regarding the existence of an matrix in CFT (and also in AdS gravity). Maldacena and Zhiboedov [46] considered the implication of this theorem on correlation functions. In particular, they have shown the correlation functions of a CFT in the presence of higher spin symmetry can be written as that of free fields (free bosons or free fermions). However, the correlators themselves are nonzero and nontrivial for all .

We consider the scattering of “collective dipoles” just introduced and calculate the collective -matrix defined by the LSZ-type reduction formula

 S=lim∏i(E2i−(|→ki|+|→ki′|)2)⟨~Ψ(E1,→k1,→k1′)~Ψ(E2,→k2,→k2′)⋯⟩ (2.0)

where is the energy-momentum transform of the bi-local field (2.1). The limit implies the on-shell specification for the energies of the dipoles: .

Our evaluation of the -matrix proceeds as follows. Using the time-like quantization we will evaluate the 3 and 4-point scattering amplitude corresponding to the associated Witten diagrams. In momentum space, in terms of the bi-local fields

 η(t;→x1,→x2) = ∫d→k1d→k21√2ωk1ωk2(e+i(→k1⋅→x1+→k2⋅→x2)α→k1→k2+h.c.) (2.0) π(t;→x1,→x2) = i∫d→k1d→k2√ωk1ωk22(e−i(→k1⋅→x1+→k2⋅→x2)α†→k1→k2−h.c.) (2.0)

the cubic (2.1) and quartic (2.1) interactions take the form

 H(3) = √2√N∫3∏i=1d→ki[−ωk1k2k33α→k1→k2α−→k2→k3α−→k3−→k1+ωk2α→k1→k2α−→k2→k3α†→k3→k1+h.c.] (2.0)
 H(4) = 1N∫4∏i=1d→kiωk1k2k3k44[α→k1→k2α−→k2→k3α−→k3→k4α−→k4−→k1+4α→k1→k2α−→k2→k3α−→k3→k4α†→k4→k1+h.c. (2.0) +4α→k1→k2α−→k2→k3α†→k3→k4α†−→k4→k1+2α→k1→k2α†→k2→k3α→k3→k4α†→k4→k1]

where we have used the notation and means the hermitian conjugate of only the terms ahead of it.

For the three-dipole scattering (), the amplitude is given by

 ⟨0|α→p3→p3′Texp[−i∫∞−∞dtH(3)(t)]α†→p2→p2′α†→p1→p1′|0⟩ (2.0)

where means time-ordered. Graphically, this corresponds to evaluating the Feynman diagram as shown in Figure 2a. The evaluation involves using the bi-local propagator symmetrized over the momenta

 ⟨0|Tα→p1→p1′(t1)α†→p2→p2′(t2)|0⟩=∫dEie−iE(t1−t2)E−ωp1−ωp1′ 12[δ(→p1−→p2)δ(→p1′−→p2′) (2.0) +δ(→p1−→p2′)δ(→p1′−→p2)] . (2.0)

The on-shell three-point scattering amplitude is obtained by amputating the leg poles and putting the external states on-shell which gives the final result (see [47] for more details)

 S(1+2→3)= −√28√N(E1+E2−E3)δ(E1+E2−E3) (2.0) ×[δ(→p1−→p3)δ(→p2′−→p3′)δ(→p1′+→p2)+7 more % terms] . (2.0)

The seven more terms in the end are due to the symmetrization over . The final result (2.2) is of the form , therefore follows.

Next for the four-dipole scattering (), the calculation is similar. The scattering amplitude is given by

 ⟨0|α→p3→p3′α→p4→p4′Texp[−i∫∞−∞dt(H(3)(t)+H(4)(t))]α†→p1→p1′α†→p2→p2′|0⟩ (2.0)

where is explicitly given in (2.2). Summing all the -channel diagrams in Figure 2b and the cross-shaped diagrams in Figure 2c, the final result is [47]

 S(1+2→3+4)=i16N(E1+E2−E3−E4)δ(E1+E2−E3−E4) (2.0) ×[δ(→p1−→p3)δ(→p1′+→p2)δ(→p2′−→p4′)δ(→p3′+→p4)+15 more terms (2.0) +δ(→p2−→p3)δ(→p1+→p2′)δ(→p1′−→p4′)δ(→p3′+→p4)+15 more terms] , (2.0)

which implies . We need to stress that the vanishing of the -matrix is due to some genuine cancellations between different Feynman diagrams which are not individually zero.

It is clear that the direct evaluation can be continued to higher points with the conjectured result . One can describe the nonlinear collective field theory in the following way: its nonlinearity, and higher point vertices are precisely such that they reproduce the boundary correlators through the bi-local (Witten) diagrams. The sum of these diagrams however give vanishing results in the on-shell evaluation as described above. implies triviality. Consequently, using the equivalence theorem, the nonlinearities built into the model can be transformed away under field redefinitions. We have in [47] described such a field transformation that leads to a quadratic Hamiltonian.

2.3 One-to-one mapping in the light-cone gauge

Our goal is to demonstrate that the collective field contains all the necessary information and is in a one-to-one map with the physical fields of the higher-spin theory in AdS. For this comparison to be done it is advantageous to work in the light-cone gauge, where the physical degrees of freedom are most transparent. Our strategy is to compare directly the action of the conformal group of the field theory with that of the AdS higher spin fields. In this direct comparison we will see as expected a very different set of spacetime variables and a different realization of . The number of canonical variables however will be shown to be identical and one can search for a (canonical) transformation to establish a one-to-one relation between the two representations.

The conformal generators of the model in the null-plane quantization () can be worked out as in [15] using Noether’s theorem. Denoting the transverse coordinates using the index and the conjugate momenta of as

 p+1=∂∂x−1 ,p+2=∂∂x−2 ,pi1=∂∂xi1 ,pi2=∂∂xi2 , (2.0)

in the case of , the 10 conformal generators (acting on the bi-local field ) are listed as follows

 P− = p−1+p−2=−(pi1pi12p+1+pi2pi22p+2) (2.0) P+ = p+1+p+2 (2.0) Pi = pi1+pi2 (2.0) M+− = tP−−x−1p+1−x−2p+2 (2.0) M+i = tPi−xi1p+1−xi2p+2 (2.0) M−i = x−1pi1+x−2pi2+xi1pj1pj12p+1+xi2pj2pj22p+2 (2.0) D = tP−+x−1p+1+x−2p+2+xi1pi1+xi2pi2+2dϕ (2.0) K− = xi1xi1pj1pj14p+1+xi2xi2pj2pj24p+2+x−1(x−1p+1+xi1pi1+dϕ) (2.0) +x−2(x−2p+2+xi2pi2+dϕ) K+ = t2P−+t(xi1pi1+xi2pi2+2dϕ)−12xi1xi1p+1−12xi2xi2p+2 (2.0) Ki = −t(xi1pj1pj12p+1+xi2pj2pj22p+2+x−1pi1+x−2pi2)−12xj1xj1pi1−12xj2xj2pi2 (2.0) +xi1(x−1p+1+xj1pj1+dϕ)+xi2(x−2p+2+xj2pj2+dϕ)

where is the scaling dimension of a single boson.

Turing to the AdS side, the gauge invariant equations of motion for free higher spin theory was first worked out by Fronsdal [48], which takes the form [49]

 ∇ρ∇ρhμ1...μs−s∇ρ∇μ1hρ μ2...μs+12s(s−1)∇μ1∇μ2hρ ρμ3...μs (2.0) +2(s−1)(s+d−2)hμ1...μs=0 (2.0)

where is the covariant derivative in AdS space. This can also be derived from linearization of Vasiliev’s full nonlinear theory. The higher spin fields satisfy the double traceless condition

 gμ1μ2gμ3μ4hμ1μ2μ3μ4...μs=0 (2.0)

which becomes important when . In order to fix the light-cone gauge, it is more convenient to use the tangent space tensor fields defined by

 HA1...As=eA1μ1⋯eAsμshμ1....μs\specialhtml:\specialhtml: (2.0)

where is the frame one-form. Furthermore, one can construct the Fock space vector using the creation and annihilation operators as

 |H⟩=HA1...AsαA1⋯αAs|0⟩ ,¯αA|0⟩=0 , (2.0)

where the commutators are , . The light-cone gauge is fixed by solving the following set of constraints [50]

 ¯α+|H⟩=0 (2.0) αI¯αI|H⟩=s|H⟩ (2.0) ¯αI¯αI|H⟩=0 (2.0)