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MSc Particles, Strings & Cosmology: Dissertation

Higher-Spin Gauge Theories, Vasiliev Theory and Holography

Alexander PeachStudent ID: 00073341

Supervised by Prof. M. Rangamani

September 10, 2013

Abstract

Higher-spin gauge theories are a rich and fascinating topic and in particular the ques-tion of interactions represents a severe technical challenge in light of the many deep no-goresults. We present an introduction to free higher-spin gauge theories in 4d flat space,beginning with a discussion of the most well-known and pertinent of the no-go theoremsconcerning higher-spin interactions and the means to evade them in a physically interestingsetting. The representation theory of massless particles and higher-spin representations ofthe Lorentz group is discussed before the formalism of free higher-spin gauge boson fieldtheory is presented and verified. The formalism of Vasiliev gauge theory is introduced andmany of the salient features of the theory are reviewed, in particular the inherent descriptionof higher-spin gauge fields and a massive scalar field propagating in AdS4 is discussed. Theformalism of the AdS/CFT correspondence is presented and the method to determine, upto normalisation factors, holographic correlation functions is explained. Finally we describehow these holographic methods can be used to compute a tree-level higher-spin three-pointfunction in Vasiliev theory.

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Declaration of Authorship

This piece of work is a result of my own work except where it forms an assessmentbased on group project work. In the case of a group project, the work has been prepared incollaboration with other members of the group. Material from the work of others not involvedin the project has been acknowledged and quotations and paraphrases suitably indicated.

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CONTENTS

Contents

1 Introduction 41.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Plan of the Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 The Spin-2 Barrier 72.1 The Coleman-Mandula Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The Weinberg Low-Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 The Weinberg-Witten Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Summary and Possible Ways Around . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Massless Particles and Higher-Spin Fields 193.1 The Little Group for Massless Particles . . . . . . . . . . . . . . . . . . . . . . . 193.2 Higher-Spin Representations of SO(3, 1) . . . . . . . . . . . . . . . . . . . . . . . 223.3 Gauge Invariance of Massless Tensor Fields . . . . . . . . . . . . . . . . . . . . . 25

4 Free Higher-Spin Gauge Fields 304.1 Linearised Einstein Gravity and the Spin-2 Field. . . . . . . . . . . . . . . . . . . 304.2 The Fronsdal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Philosophy of Higher-Spin Gauge Theories . . . . . . . . . . . . . . . . . . . . . . 364.4 Free Higher-Spin Fields in Anti-de Sitter Space . . . . . . . . . . . . . . . . . . . 37

5 Vasiliev Gauge Theory 395.1 Background: Frame Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Twistors and the Star Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Master Fields and Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 405.4 Physical Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.5 AdS4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.6 Linearised Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.7 The Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.8 The Higher-Spin Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Interactions in Vasiliev Theory: The Holographic 3-point Function 536.1 Overview of AdS/CFT and Holographic Three-Point Correlators . . . . . . . . . 536.2 The Scalar 3-point Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.3 Setting Up The Computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4 Explicit Vertices and Computation of C(s, s; 0) . . . . . . . . . . . . . . . . . . . 616.5 The Dual CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Discussion 64

8 Acknowledgements 65

9 Appendix 66

A Star Products 66

References 68

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1 Introduction

1.1 Motivation

Most introductions to quantum field theory completely forgo a discussion of particles and fieldswith spin greater than two, what are called “higher-spin fields”. The main reason for this mustbe that we have simply not observed any elementary particles with spin greater than 1 in anycollider experiments to date 1. Additionally there are a series of formidable no-go theorems, themost famous of which is probably the Coleman Mandula theorem, that seem to rule out thepossibiliy of realising interacting higher-spin field theories in familiar settings. It is a well knownresult that massless higher-spin field theories in flat space are gauge theories. Therefore ourconcern is the construction of interacting higher-spin gauge theories via a resolution of the so-called “higher-spin problem”. With regard to the no-go results, we know well that any theoremis only as strong as its assumptions. Indeed, there are numerous plausible ways in which we canforego their assumptions and evade the no-go results.

More generally, it is a perfectly natural question to ask: can we write down consistent,interacting theories of higher-spin fields and is it interesting to do so? The ensuing enquirynecessarily requires a deeper understanding of the more fundamental aspects of quantum fieldtheory, which in itself makes the subject worthy of study.

As it turns out, higher-spin fi1eld theories exhibit a number of fascinating exotic proper-ties that represent a potential challenge to the ways in which we describe particle physics.It is even suggested that some aspects of higher-spin gauge theories could hint at a drasti-cally alternative formulation of quantum field theory [1]. The construction of gauge-invararianthigher-spin Lagrangians with unconstrained fields and gauge parameters suggest non-locality ofthe higher-spin Lagrangian. Additionally higher-spin interaction vertices appear to be generi-cally higher-derivative! We can avoid canonical blasphemy by treating such higher-derivativeterms as perturbations around the canonical theory but the alternative suggests the more radicalalternative that the canonical formulation of field theories must be modified.

From the perspective of string theory. Those familiar with the basic aspects know that thestring spectrum contains an infinite tower of massive higher-spin states. Indeed one expects thatit might be possible to describe string theory in terms of some effective higher-spin field theory.Much more intriguing is the idea [2] that the massive tower of higher-spin states and the masslesslow-spin states in string theory are actually the symmetry-broken remnant of fully symmetrictheory of massless higher-spin gauge fields! It is conjectured that higher-spin symmetry breakingvia interactions with a higg-like scalar actually occurs for higher-spin gauge fields in AdS [1]. Ifthe string spectrum can be obtained in this fashion it is interesting to wonder if the symmetricphase might have anything to tell us about the structure of string theory. Additionally, giventhe successes of string theory as a potential realisation of UV finite quantum gravity, it is alsonatural to wonder whether we can go back a step and ask if higher-spin gauge theory can dothe same thing for us with regard to a UV completion of Einstein gravity. Since the algebra ofhigher-spin symmetries turns out to be infinite dimensional, this potentially gives us a handleon the UV divergent behaviour that plagues quantum field theory.

M. Vasiliev, one of the leading proponents of interacting higher-spin gauge theories, realisedthat it is possible to engineer a fully interacting theory of higher-spin gauge fields by realisingthe aforementioned higher-spin algebras in the construction. This eventually lead to the con-struction of Vasiliev’s fully interacting higher-spin gauge theory in AdSd for d = 2, 3, 4. Giventhe immediate osbcurity of the formalism it is quite surprising to see that the theory does allthis for us. The theory is quite complex and rather helpfully much of the literature on thesubject is far from inviting! Probably one of the most interesting aspects of the theory that flysin the face of a standard approach is that the theory currently lacks an action principle; it is

1The detection of gravitational waves, which could happen in the next few decades, would constitute compellingevidence for the existence of a massless spin-2 particle (the graviton)

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1.2 Plan of the Report

currently formulated in terms of the equations of motion only! As a result we don’t know howto covariantly quantise the theory so the question of interactions at the quantum level is stillopen! Nevertheless the very existence of such a the theory in light of the higher-spin problem isa highly non-trivial result.

Potentially much more exciting are the conjectured holographic dualities between higher-spin gauge theories in AdS and conformal field theories on the boundary. In this setting, asremarked by Giombi and Yin [3], it is as physically reasonable to postulate higher-spin gaugetheories as it is to postulate the existence of their dual CFTs. This reasoning seems particularlypowerful in the context of the duality between the type-A Vasiliev theory and a free conformalfield theory!

The aim of this report is to provide a substantial and reasonably self-contained (and hopefullyinteresting!) introduction to the basics of higher-spin gauge theories, complete with an abrasivetour of Vasiliev gauge theory and a review of the holographic methods that have been used in[3, 4] in order to compute tree-level correlation functions in Vasiliev theory.


In chapter 2 we set about discussing the various no-go theorems that prohibit the construction ofinteracting higher-spin gauge theories in flat space. We review the main features of the Colemanmandula theorem, Weinberg’s Low-Energy Theorem and the Weinberg-Witten theorem. Wepresent a summary of the implications of these results and the most obvious ways around them.

Chapter 3 will largely concern a development of the description of massless particles andhigher-spin representations of the Lorentz group in four dimensions that we will need in order toconstruct higher-spin gauge field theories. We then discuss the fact that masslessness of higher-spin fields implies that they transform like gauge fields under general Lorentz transformations.We review the case of the spin-1 field explicitly and discuss the higher-spin generalisation.

In chapter 4 we begin the construction of higher-spin gauge field theories for bosons ofarbitrary spin. We start with the straightforward case of linearised Einstein gravity, which willprovide a segue for the more general construction of the gauge invariant, Fronsdal equations ofmotion for higher-spin gauge fields due to de Wit and Freedman in [5]. We discuss the constraintson the gauge field and gauge parameters that we find necessary in order to derive the Fronsdalequations from a canonical Lagrangian. We then review some of the exotic known properties ofhigher-spin field theory. We move on to discuss the Fronsdal equations in Anti de-Sitter space,since we will want to make contact with this result when we move on to discuss Vasiliev gaugetheory.

In chapter 5 we present Vasiliev gauge theory. We start with a brief discussion of the geo-metric formulation which underlies the construction before presenting the twistor/star productformalism of the theory. We present the equations of motion and master fields and demonstratethat the master fields describe an infinite tower of higher-spin degrees of freedom. We will ex-plicitly demonstrate that the vacuum solution to Vasiliev ’s master field equations describes anAdS4 background. We then write down the equations of motion at the linearised level and setabout extracting the equations for a massive scalar and partially derive how one extracts theequations for an infinite tower of higher-spin fields propagating in an AdS4 background.

Finally in chapter 6 our task is to compute the spin-dependence of a tree-level three-pointfunction for the interaction of two higher-spin particles with the massive scalar particle in Vasilievtheory. The overall form of the amplitude is constrained by conformal invariance and so deter-mining the spin-dependence is essentially the non-trivial part of the computation. This has beendone in [4] using holographic methods, so the first parts of the chapter are devoted to setting upthe relevant machinery of the AdS/CFT correspondence. We set up an interacting toy-modelscalar field theory in AdS and describe how one can obtain the three-point correlation functionsup to their overall normalisation and the coupling constant using the equations of motion only.We describe the main steps needed in order to set up the computation in Vasiliev theory and

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present the main result obtained in [4]. We conclude with a discussion of the conjectured CFTduals of the known types of Vasiliev theory and some positive results in these directions.

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2 The Spin-2 Barrier

In this section we present the main results that constitute a barrier to the existence of inter-acting higher-spin gauge theories with spin greater than 2. The first and most famous resultwe will consider is the beautiful Coleman Mandula theorem, which given its assumptions re-stricts the symmetries of the S-matrix. We then more briefly review the Weinberg Low-Energytheorem which implies that higher-spin couplings vanish in the IR limit. We will also look atthe Weinberg-Witten theorem which rules out cubic couplings between higher-spin particles andconserved currents of spin-1 and spin-2 (the stress tensor) and briefly discuss its generalisation.We then summarise the no-go results discussed and review the means to evade them.

2.1 The Coleman-Mandula Theorem

The Coleman-Mandula theorem is one of the classic ”no-go” S-matrix theorems that tells us thatif we try to mix bosonic generators with the Poincare group then the S-matrix for the resultingtheory will be trivial. The theorem is immediately quite formidable, however, fortunately forus, the Coleman Mandula theorem certainly doesn’t represent an insurmountable difficulty.Throughout this section we will closely follow the original proof as presented in [6]. Somedefinitions are in order before we present the theorem.

The multi-particle Hilbert space is a direct sum of all integer-particle subspaces, i.e. H =∑i⊕H(i), where H(i) is the subspace spanned by all i-particle states. We shall denote by D all

one-particle states H(1) whose momentum space wave-functions are test functions, this is thecase for all physical states. The S-matrix S is defined in the usual way; S = 1+ iT where T isthe part of S which connects ingoing and outgoing eigenstates of different momenta and thusT 6= 0 describes scattering of wavepackets. Throughout this chapter, S will always refer to someS-matrix and P will denote the Poincare group.

Definition 2.1. G is a symmetry group of S if given U unitary ∈ G:

1. ∀|p〉 ∈ H(1), U |p〉 ∈ H(1)

2. ∀|p1, ..., pn〉 ∈ H(n), U |p1, ..., pn〉 = (U |p1〉) ⊗ ... ⊗ |pn〉 ⊕ |p1〉 ⊗ (U |p2〉) ⊗ ... ⊗ |pn〉 ⊕ ... ⊕|p1〉 ⊗ ...⊗ (U |pn〉)

3. [S,U ] = 0

the second property just states that unitary transformations act on multi-particle states asif they were products of single-particle states.

An S-matrix S is Poincare invariant if given G a symmetry group of S, there is a subgroup Hwhich is locally isomorphic to the Poincare group. Poincare invariance of the S-matrix ensuresthat it corresponds to a theory of objects living on spacetime.

Definition 2.2. Given a symmetry group G of S, G is an internal symmetry group of S if[g,P] = 0, ∀P ∈ P and ∀g ∈ G

With these definitions in place we are now we are ready for the theorem.

Theorem 2.1. The Coleman Mandula Theorem: Suppose ∃ a connected group G which isa symmetry group of S = 1+ iT and suppose:

1. S is Poincare Invariant.

2. Particle Finiteness: H(1) carries only positive energy rep.s of P and given some arbitrarymass M there are a finite number of particle species with mass m < M .

3. Weak Elasticity Analyticity: Elastic scattering amplitudes are analytic functions of thecentre of mass momentum s and the transfer momentum t.

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4. Scattering: Given |p〉, |q〉 ∈ H(1) and |p, q〉 ∈ H(2) then T |p, q〉 6= 0 except for isolatedvalues of s = (p+ q)2 and t = (p− q)2.

5. All elements of G in a neighbourhood around the identity live in a one parameter familyof functions g(t) parameterised by t. Additionally, given |p〉, |q〉 ∈ H(1)

−i ddt〈p|g(t)q〉 = 〈p|Aq〉. (2.1)

is well-defined at t=0 and defines a continuous linear function of |p〉 and antilinear functionof |q〉

Then G ∼= P ⊗ F locally, where F is an internal symmetry group of S.

Some remarks are in order. Weak elastic analyticity is expected to be a property of allphysically reasonable theories. We assume non-zero scattering because in general we want toconstruct interacting theories. Assumption 5 is just a technical means of stating that the objectsin the theory of interest are the ”usual kind” we deal with in a quantum field theory. Namelythat we can expand operators (such as fields) as momentum space integrals of operator-valueddistributions.

Equation (2.1) can be easily generalised to describe the action of the generator A on twoparticle states. Given |pi〉, |qi〉 ∈ H(1) with i = 1, 2 and the tensor product states |p1〉⊗|p2〉, |q1〉⊗|q2〉 ∈ H(2) and using definition 2.1 we find that [7],

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i

d

dt(p1 ⊗ p2, g(t)[q1 ⊗ q2]) = (p1 ⊗ p2, A[q1 ⊗ q2])

= (p1 ⊗ p2, [Aq1 ⊗ q2] + [q1 ⊗Aq2])

= (p1, Aq1)(p2, q1) + (p1, q1)(p2, Aq2)

(2.2)

Additionally, since by defintion 2.1 [A,S] = 0 then equation (2.2) implies that,

(S[p1 ⊗ p2], SA[q1 ⊗ q2]) = (S[p1 ⊗ p2], AS[q1 ⊗ q2]) = (p1 ⊗ p2, A[q1 ⊗ q2]) (2.3)

We will denote by α the set of all distributions that obey equations (2.2) and (2.3).Roughly the proof of the theorem works by starting with the generator A of an infinitesimal

symmetry transformation. We then postulate a number of possible properties of A towards aseries of contradictions which continually narrow down the possible forms of A until we finallyobtain the infinitesimal version of the theorem.

First consider the infinitesimal symmetry transformation generator A. We can use the states|p〉, |q〉 ∈ H(1) to define the following distibution in momentum space.

A(p, q) = 〈p|A|q〉 (2.4)

where we have suppressed other labels such as helicty and particle species, since we won’t needthem for the proof. First we would like to show that the generator A cannot connect states ondifferent mass-shells, which equivalently implies that it cannot change one particle species intoanother. We consider a test-function f(p) with support in an open region R, not including theorigin in momentum space, with which we can define the following distibution:

(f ·A)(p, q) =

∫d4a U †(1, a)A(p, q)U(1, a)f(a) (2.5)

where U(1, a) = e−ipµaµ

is a unitary representative of a Poincare transformation correspond-ing to the constant translation xµ → xµ + aµ and f(a) is the Fourier transform of f(p).

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Now using (2.4) and (2.5) we find that,

(f ·A)(p, q) =

∫d4a U †(1, a)〈p|A|q〉U(1, a)f(a) =

∫d4ae−ia(p−q)〈p|A|q〉f(a) = f(p− q)A(p, q).

(2.6)But because f(p) is non-zero only in R, clearly f(p− q) = 0 when the vector (p− q) lies outsideof R. Together with (2.6) this implies that (f · A)(p, q) annihilates the state |q〉 with (p − q)lying outside of R. So we have proved the following lemma,

Lemma 2.1. (f ·A)(p, q) is only non-zero for momenta p, q connected by a vector in R.

Since the mass-shell is a co-dimension 1 subspace then given R sufficiently small and notcontaining the zero vector, then there are regions on the mass shell such that adding any vectorin R to points in these regions takes us off the mass-shell. In this case f(p) is non-zero bydefinition, but since A(p, q) only connects on-shell momenta it must be zero by definition whenone of its arguments is off-shell. Looking at (2.6) then implies that states in D with support inthese regions are annihilated by f ·A.

By particle finiteness there are a finite number of mass-shells in momentum space. For givenf supported in a small region R not including the origin, then for each mass-shell there is aregion in momentum space of all points connected to the mass shell by vectors in R (see figure1). Because of lemma 2.1, particles with supports outside these regions are annihilated by (f ·A).

Figure 1: Illustration of a mass-shell (curved line) in momentum space and a small region Rwhich does not contain the origin. The point P is an arbitrary point on the mass-shell. Thecircle (dark-grey) contains all points connected to the point P by a vector in R. The shadedband is traced out as P moves along the mass-shell and therefore contains all points which areconnected to the mass shell by vectors in R.

Now we can show that A only connects equal momentum eigenstates by showing that f ·Aannihilates all states on all mass-hyperboloids. Suppose towards a contradiction that there existsome set x ∈ H1 comprised of non-null states which are orthogonal to all those that areannihilated by f ·A. Let |x〉 ∈ x be one of these states with support at p in momentum space.Now if |x〉 is annihilated by (f · A) then it is clearly orthogonal to itself by definition. Butthis contradicts the definition that |x〉 is not null. Therefore |x〉 is not annihilated by (f · A).But lemma 2.1 now implies that the of |x〉 must lie on one of the shaded regions . Now let usset up some 2→2 scattering procress where the initial particle momenta are p, q and the finalmomenta are p′, q′. Let f be a test-function supported in R not including the origin and let

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|x〉 ∈ X have support localised around p which is connected to the lowest mass-shell by avector in R. Additionally let the other three states have supports localised around q, p′ and q′

which lie outside the region of points connected to the mass-shell by vectors in the support of f(see figure 2) .

Figure 2: Illustration of a shaded region in momentum space connected to the lowest mass shellby vectors in the support of a test function f . In our hypothetical scattering experiment, thestate |x〉 ∈ X has support localised around the point p which lies in the shaded band. Theother particle states have supports localised around the points q, p′ and q′ which are not in theshaded band.

Clearly we have p + q = p′ + q′. As usual, the centre of mass and transfer momentum aredefined to be s = (p+ q)2 and t = (p− q)2 respectively. We are not interested in the case wherethe collision creates a pair of heavier particles, so we choose s to lie below the threshold for thenext-heaviest species pair production. Because f ·A obeys equations (2.2) and (2.3) and becausef ·A annihilates the states with momentum q, p′ and q′, we find that,

(S[p⊗ q], S[f ·A

][p′ ⊗ q′]) = (p, [f ·A]p′)(q, q′) + (p, p′)(q, [f ·A]q′) = 0

whch implies that the scattering vanishes for these values of s and t. But now by Lorentzinvariance if we rotate around the axis of the spatial part of p, changing q, p′, q′ then the aboveholds so that for at least one particle type, scattering to any other particle type is zero forthese values of s and t. Now we can change the values of s and t continuously by changing theconfiguration. For example we could make the shift q → q+ε, with ε infinitesimal which changesthe values of s and t. But we can now repeat the same arguments as before and we find exactlythe same result for these new values s and t. But now by the analyticity forces the scatteringto vanish for all values of s and t. This contradicts the assumption of the non-triviality of theS-matrix, so we obtain the required contradiction which shows that |x〉 = 0. This implies thatx is the empty set, i.e all states with support on this mass-hyperboloid are annihilated byf ·A. Going on to the next mass-shell the same arguments used here imply that all states withsupports on any of the mass-shells are annihilated by f ·A. Overall this implies that,

f ·A = 0 (2.7)

It is now easy to see that A only connects states with equal momenta. Recall that f is arbitraryup to the condition that it is not supported at the origin i.e f(0) = 0. This, together with (2.7)implies that,

(f ·A)pp = f(0)A(p, p) = 0

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only in this case do we find that A 6= 0 is a solution. So we have proved the following lemma:

Lemma 2.2. The support of A(p, q) is restricted to the set p = q.

In particular this lemma implies that the symmetry transformation generated by A cannotconnect eigenstates of different momenta and therefore cannot connect states on different mass-shells. For example, in a theory with multiple species of particle, A cannot transform one particleinto another species.

Now any distribution with support at just a single point is just a finite sum of delta functionsand their derivatives. In particular this implies that A is a differential operator on the mass-shells, it is a polynomial in the tangential differential operator [7]:

∇µ =∂

∂pµ− pµpν

m2

∂

∂pν

this is just the directional derivative along the mass-shell. This equates to the following lemma.

Lemma 2.3. Any element A is a polynomial in ∇µ and thus it can be written,

A =N∑n=0

A(n)µ1...µn(p)

∂

∂pµ1· ... · ∂

∂pµn

and in particular, since [ ∂∂pµ1

, pµpµ] = [ ∂

∂pµ1,−M2] = 0 we have,

[A, pµpµ] = 0.

This is one of the two lemmas we need to prove the theorem. What we’ll now do is look atthe parts of our hypothetical symmetry group which commute with translations. By seperatelyconsidering the trace and traceless parts we will be able to show that they generate infinitesimaltranslations and internal symmetry transformations respectively. We will first need to set up anumber of definitions.

Recall that the non-triviality assumption permits T |p, q〉 to vanish for isolated values ofs = (p+ q)2 and t only. We call the pair (p, q) a null pair if T |p, q〉 = 0.

Now let β be the subset of α consisting of Hermitian distributions that commute with Pµ andlet β∞ be the subset of β for which B(p) ∈ β∞ is an infinitely differentiable function. Clearlyelements of β∞ can the be thought of as operating on momentum eigenstates, because we candiagonalise both elements of β∞ and Pµ in some basis. Presently we have a handle on elementsof β∞, but later we can easily prove results also hold for elements of β with a simple definition.

We’ll consider an operator B(p) ∈ β∞. Let B∗(p) be the traceless part of B. For given pwe define the set K(p) of all B ∈ β∞ such that their traceless part vanishes, i.e B∗(p) = 0.Definition 2.1 allows us to define the operator B(p, q) which acts on two-particle states,

B(p, q) = [B(p)⊗ I]⊕ [I ⊕B(q)] (2.8)

We further define K(p, q) as the set B ∈ β∞ such that B∗(p, q) = 0. From (2.8) it is easy to seethat,

K(p, q) = K(p) ∩K(q) (2.9)

Now given (p, q) not a null pair, let J be the generator of rotations around an arbitrary axis inthe centre-of-mass frame. Suppose towards a contradiction that there is a B in K(p, q) so that[B, J ] is not in K(p, q) we see that,

[B, J ](p, q) = BJ(p, q)− JB(p, q) = B(p′, q′)− JB(p, q)⇔[B, J ]∗(p, q) = B∗(p′, q′)− [JB]∗(p, q) 6= 0⇔ B∗(p′, q′) 6= 0

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where we have defined (p′, q′) as the pair obtained by action of J on (p, q). Simple linear algebranow implies that B(p, q) has two distinct eigenvalues [6]. Clearly at least one of these eigenvaluesis not equal to the single eigenvalue of B(p, q). The scattering of the eigenvector corresponding tothis eigenvalue to a pair (p,q) must be zero [6], but this contradicts the analyticity assumption,and so [B, J ] is in K(p, q). This implies that elements of K(p, q) commute with J ; they areinvariant under rotations in the centre of mass frame of (p, q). The total momentum (p+q) isobviously conserved under rotations. In fact, up to a factor it is the only linear function of pand q which is invariant under rotations. But since K(p, q) is polynomial in p,q then this impliesthe following lemma.

Lemma 2.4. Elements of K(p, q) are functions of (p+q) only.

Consider some 2→ 2 scattering from (p, q) to (p′, q′) so that p+ q = p′+ q′. By lemma (2.4)and total momentum conservation we must have that,

K(p, q) = K(p′, q′)

Now by (2.9) it is easy to see that K(p) ⊃ K(p, q) and K(p′) ⊃ K(p′, q′) = K(p, q). But cruciallythese imply that

K(p, p′) = K(p) ∩K(p′) ⊃ K(p, q).

But now since p+ p′ 6= p+ q then we have shown that K(p, q) contains elements K(k) where kis an arbitrary momenta on the mass-hyperboloid! So if K(p, q) is non-trivial, then all momentaon the mass-hyperboloid are contained in K(p, q) so we establish the following lemma.

Lemma 2.5. Let (p, q) be a non-null pair on any mass-hyperboloid and suppose we have B ∈ β∞.Then if,

B∗(p, q) = 0

then B∗ vanishes everywhere on the mass-hyperboloid.

To proceed we require arguments based on group theory as originally presented in [6]. Wewill briefly summarise their approach. We consider the traceless parts of all B in β∞ on aparticular mass-shell. These furnish an algebra B that, using Lemma 2.5 can be shown to bethe direct sum of an Abelian Lie algebra and a compact semisimple Lie algebra. It can then beshown that elements of these subalgebras must commute with Lorentz transformations! Therforewe prove the following lemma.

Lemma 2.6. For all B ∈ β∞, B∗(p) is an infinitesimal internal symmetry transformation.

Note that this result implies that, in fact B∗ does not depend on p. This result has somewhatmore of the flavour of the final proof. Indeed all that remains is to consider the trace of B ∈ β∞.

The hypothetical scattering experiment we used to establish Lemma 2.4 now easily impliesthat for p+ q = p′ + q′ we have [6],

TrB(p) + TrB(q) = TrB(p′) + TrB(q′) (2.10)

By hypothesis, TrB(p) is polynomial in p so by (2.10),

TrB(p) + TrB(q)− TrB(p′)− TrB(q′) =

∞∑1

an[(pn + qn)− (p′n + q′n)] = 0

which, without imposing further constraints on the momentum requires setting an = 0 for n > 1.This implies the following Lemma,

Lemma 2.7. TrB(p) is a linear function of p for all B ∈ β∞.

12


Now we can finally lift the restriction that the distributions we are considering are infinitelydifferentiable functions by using the following definition. Given B ∈ β and an arbitrary testfunction f(Λ) of elements of the homogeneous Lorentz group, then,

Bf =

∫dΛf(Λ)U(Λ)BU †(Λ) (2.11)

defines a distribution in β∞ [6]. Now the lemmas 2.6 and 2.7 apply to elements of β∞. Howeverfor any B ∈ β, (2.11) gives us the corresponding Bf ∈ β∞. But since Bf obeys all the lemmasfor arbitrary f(Λ) then B must also obey the lemmas. This, together with the lemma 2.6 and2.7 implies the following lemma [6].

Lemma 2.8. All elements of β have the form,

B(p) = aµpµ + b

where aµ is a constant four-vector and b is a constant matrix.

We are now finally able to prove the theorem using the lemmas (2.3) and (2.8). CommutingA in (2.3) with pµ, N times this gives,

[pµ1 , [pµ2 , ...[pµnA]...] = n!A(n)µ1...µn (2.12)

This is simply because [ ∂∂pµ

, pν ] = δµν and [A,BC] = [A,B]C +B[A,C], so that, for example,

[∂µ[∂ν , A(0) +A(1)

ρ pρ +A(2)ρσ p

ρpσ]] = [∂µ, A(1)ρ [∂ν , p

ρ]] + [∂µ, A(2)ρσ [∂ν , p

ρpσ]]

= [∂µ, A(1)ρ δρν ] + [∂µ, A

(2)ρσ [∂ν , p

ρ]pσ + pρ[∂ν , pσ]]

= A(2)ρσ (δρν [∂µ, p

σ] + δσν [∂µ, pρ])

= A(2)ρσ (δρνδ

σµ + δσν δ

ρµ)

= A(2)µν +A(2)

νµ

= 2A(2)µν

Now the right-hand side of (2.12) is in α and commutes with pµ, therefore is it in β. By lemma(2.8) this then implies that,

A(n)µ1...µn = aνµ1...µnp

ν + bµ1...µn (2.13)

Recall that lemma 2.3 implies that [A, p2] = 0 so that if we commute A given by (2.13) withN − 1 p s and p2 once then the result must be zero,

[pµpµ, [pµ2 , ...[pµN , A]...]] = N ![pµpµ, A(N)µ2...µN

]

= N ![pµpµ, aνµ2...µNpν + bµ2...µN ]

= aνµ1...µnpνpµ1 + bµ1...µnp

µ1

= 0

(2.14)

but equating coefficients in pµ1 we see that bµ1...µn = 0 unless N = 0. But in the latter case wecannot do the commutation. Plugging this back into (2.14) we see that,

aνµ1...µnpνpµ1 =

1

2(aνµ1...µn + aµ1ν...µn)pνpµ1 ⇔

aνµ1...µn = −aµ1ν...µn

unless N = 0 which would not allow us to perform commutation. But by definition (andbecause derivatives commute) a is symmetric in the indices µi, i = 1, ..., N , so the above result

13

2.2 The Weinberg Low-Energy Theorem

is inconsistent with this and sets a = 0 unless N = 1. Therefore we find that either N = 0, 1.In the former case N = 0, (2.13) implies that,

A = A0(p) = aµpµ + b

this is just an infinitesimal translation plus internal symmetry transformation. Finally, for thecase when N = 1 we have,

A = A(0)(p) +A(1)µ ∂µ = aµpµ + aµνp

ν∂µ + b

But the first term on the right-hand side is an infinitesimal translation, the second term is aninfintesimal Lorentz transformation, and the last term is an infinitesimal internal symmetrytransformation. Thus we have found the most general form of the infinitesimal generator A ofthe symmetry group of S, establishing the following lemma:

Lemma 2.9. Any A ∈ α is the sum of an infinitesimal translation, an infinitesimal Lorentztransformation and an internal symmetry transformation.

Finally, the finite version of this lemma prooves the theorem!The Coleman Mandula theorem is certainly very beautiful in that, given a set of relatively

basic and very general sounding assumptions, we have determined, in the authors’ own words,“allpossible symmetries of the S-matrix”! Clearly this is a formidable result 2. However the as-sumptions, however compelling they might seem in their generality, are by no means strictrequirements of all physical theories, for many simple reasons. We defer a discussion of theconsequences of the theorem and possible ways around until later.


The Weinberg Low-Energy theorem [8] can be regarded as a low-energy constraint on the cubiccouplings of the type s-s′-s′ with s,s′ arbitrary. The proof of the theorem works by demandingLorentz invariance of some arbitrary scattering amplitude which is deformed by the emissionof a massless spin-s particle in the limit in which its momentum tends to zero, what is calledthe ”soft limit”. This implies a conservation law involving the coupling constants of the theoryand the momenta. The point is that for higher-spin scattering this amounts to a quadratic (orhigher order) constraint on the momenta, which, without trivialising the momenta-exchange,implies that the higher-spin couplings vanish in the soft limit. A straightforward corollary ofthis is that higher-spin particles cannot mediate low-energy, i.e long-range interactions in flatspace [9].

This theorem has very powerful consequences when we consider the result for spin-2 particles,which can be understood as a restatement of Einstein’s Equivalence Principle for gravitation inquantum field theory [1].

We start by considering an arbitrary N -particle scattering process for particles with momen-tum pi, i = 1, ..., N , also with the emission of a soft massless spin-s particle with momentum q,as shown in figure 3.

The extra emission of the spin-s particle is determined by the cubic vertex of the type s′-s-s′

where s′ is arbitrary. The amplitude for this process is of the form [8],

A(p1, ..., pn; q, e) = eµ1...µs(q)Aµ1...µs(p1, ..., pN ; q) (2.15)

where e is the polarisation tensor for the outcoming spin-s field. However the polarisation tensorfor the massless spin-s particle field is not Lorentz covariant! We will say much more about thislater. Under general Lorentz transformations the polarisation tensor in fact transforms as [1],

eµ1...µs(q)→ eµ1...µs(q) + q(µ1ζµ2...µs) (2.16)

2Interestingly, before the advent of the theorem, there were many (not surprisingly!) futile attempts atformulating interacting field theories in flat-space with internal symmetry Lie groups that commuted non-triviallywith the Poincare group.

14


Figure 3: The emission of an additional soft-massless particle of momentum q after a scatteringevent. [8]

where ζ is an totally symmetric spin (s − 1) field. Therefore under this transformation theamplitude (2.15) goes like,

A(p1, ..., pN ; q)→ eµ1...µs(q)Aµ1...µs(p1, ..., pN ; q) + ζ(µ2...µsqµ1)Aµ1...µs(p1, ..., pN ; q)

So Lorentz invariance of the amplitude requires that the second term on the right vanishes.Since ζ is arbitrary then this requires that the amplitude is transverse to the spin-s particlemomenta,

qνAνµ2...µs(p1, ..., pN ; q) = 0 (2.17)

But Weinberg explained in [8] that the amplitud has the following form in the soft-limit,

Aµ1...µs(p, q) =

∑Ni=1 g

(s)i pµ1i ...p

µsi

(pi · q)

Therefore (2.17) implies the fascinating condition,

N∑i=1

g(s)i pµ1i ...p

µs−1

i = 0 (2.18)

Now let’s look at what this implies for particles of various spins. For the emission of a soft-photon we have s = 1 and (2.18) becomes simply the conservation of the coupling constants∑N

i=1 g(1)i = 0. The first case that implicates the momenta is the case of the emission of a soft

massless spin-2 particle, for which (2.18) gives,

N∑i=1

g(2)i pµ1i = 0

However, momentum conservation simply reads that∑N

i=1 pi = 0. Therefore we can solve

the above without requiring an additional constraint on the momenta with g(2)i = g

(2)j ∀ i, j.

Therefore the massless spin-2 particle couples in the same way to all particles at low energies.In particular this also implies that any particle which couples to another which couples tothe graviton, must itself also couple to the graviton [1]. The coupling constant g(2) can beidentified with Newton’s constant. According to Weinberg this is the expression of Einstein’sequivalence principle in quantum field theory! This result is referred to as Weinberg Low-EnergyEquiavlence. Moving on to the case of spin greater than two. We can already see that for spin-3(2.18) implies that,

N∑i=1

g(3)i pµ1i p

µ2i = 0

15

2.3 The Weinberg-Witten Theorem.

Without imposing further constraints on the momentum, this requires that g(3)i = 0. But

for higher spins the situation is identical; the end result being that in order to avoid further

momentum constraints we require that g(s)i = 0, s > 2. In other words, the couplings of all

massless higher-spin fields vanish in the soft-limit!

2.3 The Weinberg-Witten Theorem.

Weinberg and Witten laid out in their “Limits On Massless Particles” [10] a powerful andsuprisingly simple-to-prove pair of theorems.

Theorem 2.2. The Weinberg-Witten Theorem 1: A theory with a Lorentz covariant,conserved four current Jµ and associated conserved charge Q =

∫dd~xJ0 does not admit massless

particles with helicity σ > 12 with non-zero charge under Q.

Theorem 2.3. The Weinberg-Witten Theorem 2: A theory with a Lorentz covariant,conserved energy momentum tensor Tµν and associated conserved momenta Pµ =

∫dd~xT 0ν

does not admit massless particles with helicity σ > 1.

The original proofs of the theorems involves considering amplitudes which involve a sandwichof massless, arbitrary-helicity, single-particle states and a conserved 4-vector or conserved energymomentum tensor. We find the crucial result that in the limit in which the momenta of theparticles coincide these amplitudes cannot vanish. Since the proof of each theorem relies on thesame analysis we can consider them both at once.

We prove the theorems by supposing towards a contradicition that massless particles ofarbitrary helicity exist in a theory with Jµ or Tµν with Lorentz covariant and conserved andfurther that the particles have a non-zero value of the charge Q =

∫dd~xJ0. We find that

Lorentz invariance forces the aforementioned amplitudes to vanish for sufficiently high spin ineither case and thus we obtain a contradiction. We need to consider the following amplitudesfor a conserved spin-1 and spin-2 current Jµ and Tµν respectively. In the limit when p′ → p wefind the amplitudes must take the following form [8].

〈p′,±σ|Jµ|p,±σ〉 → gpµ

E(2π)3

〈p′,±σ|Tµν |p,±σ〉 → fpµpν

E(2π)3

(2.19)

Crucial to the first result is the assumption that the particles are charged, otherwise the firstamplitude vanishes. Firstly we go to a Lorentz frame in which the null particle momenta are ,

p = (|p|, ~p), p′ = (|p|,−~p) (2.20)

We can do this because for generic null momenta we find that,

(p+p′)2 = p2 +p′2 +2pµp′µ = 2pµp

′µ = 2(~p · ~p′−|~p||~p|) =2

|~p||~p|(~p · ~p′|~p||~p|

−1) =2

|~p||~p|(cos θ−1) ≤ 0

where θ is the angle between the space parts of p and p′. Therefore (p + p′) is always timelikeand we can transform to a frame where the space part of (p + p′) is zero, as we have done. Inthe frame (2.20) we find that under a rotation around the ~p axis, the states transforms as [11],

U [R(θ)]|p,±σ〉 = e±iθσ|p,±σ〉, U [R(θ)]|p′,±σ〉 = e∓iθσ|p′,±σ〉

The minus sign appeared on the right because a rotation of θ around the ~p axis is the same asperforming a rotation of −θ around the −~p = ~p′ axis. Now the currents simply transform as,

U(θ)JµU−1(θ) = Rµ ν(θ)Jν

U(θ)TµνU−1(θ) = Rµ ρ(θ)Rνσ(θ)T ρσ

(2.21)

16

2.4 Summary and Possible Ways Around

So overall Lorentz invariance of the amplitudes (2.19) requires that,

〈p′,±σ|U−1JµU |p,±σ〉 = e±2iσ〈p′,±σ|Jµ|p,±σ〉 = Rµ ν(θ)〈p′,±σ|Jν |p,±σ〉〈p′,±σ|U−1TµνU |p,±σ〉 = e±2iσ〈p′,±σ|Tµν |p,±σ〉 = Rµ ρ(θ)R

νσ(θ)〈p′,±σ|T ρσ|p,±σ〉

But since R(θ) only has Fourier modes 1, and e±iθ then these results are only possible for, inthe first case for σ = 0, 1

2 and in the second case for 0, 12 , 1 [10]. Lorentz invariance then implies

that the amplitudes vanish for σ > 12 in the first instance, and σ > 1 in the second. But in these

cases we contradict (2.19). This completes the proof of the theorems.Some remarks are in order to clarrify the consequences of the theorems. Naively one might

worry whether the second theorem has a problem with gravity. Generally, both theorems implythat the existence of massless higher-spin particles and Lorentz covariant conserved currentsare simply not compatible [1]. However, concerning general relativity, the energy-momentumtensor is simply not Lorentz covariant, and the second theorem doesn’t take issue with this.Additionally we find that the way is clear for QED and QCD. In QED, the photon carries zeroelectromagnetic charge and we avoid the first theorem. Alternatively, in QCD the gluon carriesnon-zero charge, but the charge is associated to a current which is not Lorentz covariant [12]and so again we avoid the first theorem.

A recent generalisation of the Weinberg-Witten theorem [13] concerns “Lorentz minimalcoupling”. The Lorentz mininal spin-2 coupling is obtained [14] from the free spin-s Lagrangian

L(0)s via the Lorentz covariantisation of derivatives ∂ → ∂+g(2)Γ. This genarates a cubic coupling

of the type 2 − s− s which has the form hµνΘµν(φs) where hµν is the graviton and Θµν(φs) isbilinear in φs, the spin-s field. Under linearised diffeomorphisms one has that δhµν = s∂(µεν).Therefore, the variation of the vertex is,

δhµνΘµν = ∂(µεν)Θµν = ∂µ(...) + ∂ν(...)− 2ε(µ∂ν)Θ

µν

Ignoring total derivatives, setting the above to zero requires that, for arbitrary ε that ∂µΘµν = 0,i.e that Θ is a conserved spin-2 current. The cubic coupling is minimal if and only if Θ is equal

to the energy-momentum tensor obtained from L(0)s via the usual Noether procedure [1]. 3.

Now the minimal coupling procedure implies a non-trivial cubic vertex in flat space of thetype 2-s-s which is governed by the term hµνΘµν(φs) . Then the matrix element for the scatteringof a massless higher-spin particle of momentum p with helicity ±σ off of a single graviton withmomentum q is [13],

〈p+ q,±σ|Tµν |p,±σ〉The limit in which the graviton is soft is q → 0, in which case this matrix element is exactlythe second line of (2.19)! which is forced to vanish for σ > 1 given Tµν Lorentz covariant.However, we cannot conclude that the Weinberg-Witten theorem forbids Lorentz minimal cou-pling of massless higher-spin particles because, as we have already discussed, the Noether energymomentum tensor is not Lorentz covariant. Nevertheless a generalisation of the theorem wasrecently found [13] where one can weaken the requirement that the energy-momentum tensor isLorentz covariant by introducing unphysical polarisations of the higher-spin particles, so thatthe resulting amplitude shown above is Lorentz covariant. The conclusion is indeed then thatmassless higher-spin particles cannot couple minimally to the graviton in a flat background!


The most powerful no-go theorem with regard to the construction of interacting massless higher-spin field theories is the Coleman-Mandula theorem. 4 We will argue later that massless higher-spin field theories are necessarily gauge invariant. In fact, the higher-spin gauge transformations

3Actually, is it possible that this only needs to hold on-shell [13].4The most well-known escape method is to abandon the requirement that the symmetry groups of the S-

matrix are Lie algebras. By considering graded-Lie algebras instead we find that the new generators may mix,via anticommutation, with the Poincare generators which leads directly to supersymmetry.

17


generate non-trivial “large gauge transformations” at infinity which are generated by asymp-totically conserved charges [15]. These charges generate global symmetry transformations andcrucially they do not commute with Lorentz transformations [1]. For this reason, massless higher-spin field theories fall deeply under the spell Coleman Mandula theorem.

Together, the generalised Weinberg-Witten theorems and Weinberg Low Energy Equivalenceconstitute a particularly poisonous result for interacting higher-spin gauge theories, at least froma phenonenological perspective! If higher-spin fields couple minimally to lower-spin fields, thenby the Weinberg Low-Energy theorem, they must also couple minimally to the graviton [1].However, by the generalised Weinberg-Witten theorem, massless higher-spin fields cannot coupleto minimally to the graviton in flat space, so we obtain a contradiction. Therefore, higher-spinfields cannot couple to lower-spin fields at low energies. This implies that we cannot constructa theory of the standard model with the addition of non-trivial interactions with a higher-spinsector [1]. In particular, if the associated Lagrangian described physics in the IR and UV limits,then higher-spin fields cannot couple to lower spin fields!

All of the no-go results that we have considered rely heavily on a flat background. Thesimplest possibility in order to circumvent the theorems then is to pass to curved backgrounds.Indeed, the Coleman Mandula theorem relies on the very existence of an S-matrix, which in turnrelies on a well-defined notion of asymptotic states. In flat spacetime we can talk about singleparticle states in general and define aymptotic states as those states which are either in the farpast or the far future (if you like, at the ”boundary” of Minkowski space!) 5. However in curvedspace the very notion of single-particle states immediately becomes ambiguous, since we couldswitch to a coordinate system where a single-particle state in one set of coordinates becomes amulti-particle state in another! Consequently there is no hope of defining an S-matrix in curvedspace. However, this is not to say that we are without any kind of boundary observables incurved space. Indeed, in the light of the AdS/CFT correspondence we may simply resort to adual CFT. We will say much more about this later!

Perhaps surprisingly, many consistent cubic vertices in flat space have been found [17]. Hereby consistency of higher-spin vertices we mean whether or not they are both gauge invariant andLorentz covariant. We will not discuss how these results were achieved. For us it is sufficient tonote that the existence of non-trivial tree-level amplitudes does not imply that a fully interactingquantum theory exists; a fact that we will have to acknowledge later.

Key Points:

• The Coleman Mandula theorem rules out the existence of higher-spin conserved chargesin flat-space and therefore forbids interacting higher-spin gauge field theories in flat-space.

• The Weinberg Low-Energy theorem implies that higher-spin cubic couplings all vanish inthe soft-limit in flat-space.

• The generalised Weinberg Witten theorem prevents higher-spin fields from Lorentz-minimallycoupling to the graviton in flat-space.

• The generalised Weinberg Witten theorem and the Weinberg Low Energy theorem implythat higher-spin fields cannot couple to any of the standard model fields at low energies.

• The no-go results presented all concern flat-space theories. A natural and sensible wayout is to pass to curved backgrounds, where an S-matrix does not exist.

5As pointed out by Witten in [16], given some S-matrix which is defined for on-shell momenta, there is a veryreal sense in which the capicity to reconstruct a QFT in a bulk spacetime from this S-matrix is tantamount tothe most basic realisation of the holographic principle, with the S-matrix comprising the to the boundary-datafor the bulk QFT!

18

3 Massless Particles and Higher-Spin Fields

By considering the representation theory of the single particle states in quantum field theory wewill find that their transformation properties are conveyed entirely by their little group. Thisis sufficient to determine that massless particle states carry exactly 2 degrees of freedom. Itwill be useful for us to see explicitly how to construct fields from generally covariant tensorswhich transform under particular spin representations of the Lorentz group, so that we knowexactly what we mean by the spin-s field in general. To this end it is sufficient to ensure that thethe representation under which the field transforms is irreducible. A divergence-free conditioncondition ensures the energy is bounded from below by killing the lower-spin invariant subspacesof the rank-s generally covariant tensor. With this representation theory we then explain whymassless fields with spin s ≥ 1 transform like gauge fields under general Lorentz transformations,showing explicitly how this arises in the spin-1 case. Finally, using the known inhomogeneoustransformation properties of the higher-spin polarisation tensor we will derive the form of thespin-s gauge transformation in flat space.

3.1 The Little Group for Massless Particles

When we refer to one-particle states we are of course talking about the single-particle momentumeigenstates of the free Hamiltonian. By considering the invariants of the Lorentz group we findequivalence classes of momenta that we may use to classify one-particle states. The first Casimirof the Lorentz group is [11],

C1 = pµpµ = p2 = −M2 (3.1)

and in general we know that sign(p0) is not affected by Lorentz transformations. Overall wefind that p2 = pµpµ = −M2 all momenta may be classified by the sign of their time componentand their mass squared. We find that the following are possible momenta classes.

1. p2 = −M2 > 0, p0 < 0: massive, negative energy

2. p2 = −M2 > 0, p0 > 0: massive, positive energy

3. p2 = 0, p0 < 0: massless, negative energy

4. p2 = 0, p0 > 0: massless, positive energy

5. p2 = N2 > 0

6. pµ = 0

While these momenta classes exist, only the massive and massless positive energy classes areevidently physical. The unphysical cases won’t conern us in this report. Let’s consider somesingle particle state which we will denote |p, a〉, where p labels the four momenta (spacetimeindices omitted) and a labels the particle species. A general element of the Poincare groupcorresponding to the Lorentz transformation Λ and a translation by a is represented in H byU(Λ, a) unitary. Pure Lorentz transformations are represented by U(Λ, 0) := U(Λ). Now let’scompute the momentum of a lorentz-transformed single particle state:

PµU(Λ)|p, a〉 = U(Λ)U−1(Λ)PµU(Λ)|p, a〉 = U(Λ)Λµ νPν |p, a〉 = U(Λ)Λµ νp

ν |p, a〉= U(Λ)p′µ|p, a〉

i.e, as expected, U(Λ)|p, a〉 is an eigenstate with momenta Λp. Now this implies that U(Λ)|p, a〉is a linear combination of |Λp, a〉 eigenstates, i.e:

U(Λ)|p, a〉 =∑a′

Daa′ |Λp, a′〉 (3.2)

19


Suppose we choose a standard momenta kµ for each class. Now a sub-group of the Lorentztransformations leave the standard momentum invariant:

Λµ νkν = kν . (3.3)

The subgroup of Λ ∈ SO(3, 1) satisfying (3.3) comprises the so-called little group of the Lorentzgroup for a given momenta class for which k is the standard momenta. Using the result for(3.2), a representative of the little group U(Λ) acts on a state with the standard momenta asfollows:

U(Λ)|k, a〉 =∑a′

Waa′ |Λk, a′〉 =∑a′

Waa′ |k, a′〉. (3.4)

The Waa′ then furnish a representation of the little group. Now suppose that a generic momentain a particular class is obtained via a lorentz transformation of the standard momenta as follows,

pν = L(p)µ νkν , (3.5)

Then a single particle state with generic momenta p is obtained by acting on the state with thestandard momenta k with the representative U(L(p)):

|p, a〉 = U(L(p))|k, a〉

Now we can go back and use (3.2) to study the transformation of the single particle states. Wefind:

U(Λ)|p, a〉 = U(Λ)U(L(p))|k, a〉

and multiplying by the identity in the form 1 = U(L(Λp))U−1(L(Λp)) we obtain,

U(Λ)|p, a〉 = U(L(Λp))U−1(L(Λp))U(Λ)U(L(p))|k, a〉= U(L(Λp))U(L−1(Λp)ΛL(p))|k, a〉.

We passed to the second line above using the fact that since the U form a representation wemust have that U(A)U(B) = U(AB) and we also used a result which follows, that U−1(L(p)) =U(L(p)−1) [18]. Let’s look at the matrix U(L−1(Λp)ΛL(p)) seen above. Looking at the argument,L(p) first takes k to p, Λ then takes p to Λp. Now using (3.5) for L−1(Λp) we have,

(Λp)µ = L(Λp)µ νkν ⇔ kµ = L−1(Λp)µ ν(Λp)ν ,

so that L−1(Λp) takes Λp to k. The end result is that L−1(Λp)ΛL(p) takes k to k, i.e it is amember of the little group, which means that U(L−1(Λp)ΛL(p)) is a representative of the littlegroup! Now using this result and recalling (3.4) we see that,

U(Λ)|p, a〉 = U(L(Λp))U(L−1(Λp)ΛL(p))|k, a〉 =∑a′

Waa′U(L(Λp))|k, a〉

=∑a′

Waa′ |Λp, a〉(3.6)

so that under general Lorentz transformations, the single particle states only transform non-trivially under some representation of the little group! A discerning of the representations ofthe little group for each of the various classes of momenta is enough to competely charaterisethe transformation of their single particle states! Since we know that spin is characterised bythe way the states transform under rotations we understand from this general result that spinfor particles in one class of momenta is not the same as for others, i.e spin for massless particlesis not the same as spin for massive particles [11].

20


So what is the little group for massless particles? We consider that the standard momentais kµ = (1, 1, 0, 0), which is invariant under the little group transformations Wµ

νkν = kµ. Now

we consider the unit timelike four-vector tµ = (1, 0, 0, 0) [9]. by Lorentz invariance we have,

tµtµ = −1 = WµνtνWµ

ρtρ = (Wt)µ(Wt)µ (3.7)

and alsotµkµ = −1 = Wµ

νtνWµ

ρkρ = (Wt)µkµ = (Wt)0 − (Wt)1 (3.8)

which is solved by setting (Wt)0 = 1 + (Wt)1. Without loss of generality we can consider that(Wt)µ = (1 + γ, γ, α, β). Plugging this into (3.7) then leads to the constraint on γ:

(Wt)µ(Wt)µ = −1 = α2 + β2 + γ2 − (1 + γ)2 ⇔ γ =α2 + β2

2(3.9)

Now (Wt)µ = [S(α, β)t]µ where,

S(α, β)µ ν =

1 + γ −γ β αγ 1− γ β αβ −β 1 0α −α 0 1

(3.10)

is a combined rotation and boost in the y-z plane [9]. Overall then we have that,

[S(α, β)t]µ = (Wt)µ ⇔ [S−1(α, β)Wt]µ = tµ

So S−1(α, β)W leaves the timelike four-vector invariant, so it must be a pure rotation. But nowwe can easily see that S(α, β) leaves the standard momenta k invariant:

S(α, β)µ νkν =

1 + γ −γ β αγ 1− γ β αβ −β 1 0α −α 0 1

1100

=

1 + γ − γγ + 1− γβ − βα− α

=

1100

= kµ

But the only pure rotations that leave k invariant are clearly rotations by θ arbitrary aroundthe x-axis. These are,

Rµ ν(θ) =

1 0 0 00 1 0 00 0 cos θ − sin θ0 0 sin θ cos θ

(3.11)

So we have that in general S−1(α, β)W = R(θ), or equivalently that,

W (α, β, θ) = S(α, β)R(θ) (3.12)

So we have determined the most general form of the elements of the little group for masslessparticles. It can be shown that the product rules for the seperate cases θ = 0 and α = β = 0give respectively [9],

S(α, β)S(α′β′) = S(α+ α′, β + β′), R(θ)R(θ′) = R(θ + θ′)

But since [S,R] = 0, we find that the resulting product laws are those for adding vectors androtations in R2 respectively. Therefore the little group W is the group ISO(2) of translationsand rotations in R2 [12]. Representations of ISO(2) are exhausted by totally symmetric andtraceless tensor fields. This implies that fields with massless quanta are described by totallysymmetric and traceless tensor fields.

21

3.2 Higher-Spin Representations of SO(3, 1)

The little group ISO(2) is furnished by the following generators with commutation relations[12]

[J,A] = iB, [J,B] = −iA, [A,B] = 0

Clearly A and B can be simultaneously diagonalised. We find that non-zero eigenvalues of Aand B imply that massless particles carry continuous-spin degrees of freedom. We know thatcontinuous-spin degrees of freedom have not been observed so we must elimate these spuriousdegrees of freedom by setting A = B = 0. The remaining eigenvalue under J is the projection ofthe angular momentum in the direction of propagation, which is the well-known helicity of theparticle, which we do observe! Including parity [11] implies that a massless particle may carryeither a positive or negative helicity state, so that massless particle states carry exactly twodegrees of freedom. It is not obvious why the helicity is integral or half-integral at this point.There are topological considerations that one must consider in order to establish this, but wewill not look into this.


The Lorentz algebra is furnished by the generators Mµν satisfying the well-known commutationrelations,

[Mµν ,Mρσ] = Mρνησµ +Mµρηνσ −Mσνηρµ −Mµσηνρ

which furnish the lie algebra so(3, 1). In terms of these the generators of boosts and rotationsare,

Ji = iεijkMjk, Bi = M0i (3.13)

which satisfy the commutation relations,

[Ji, Jj ] = iεijkJk, [Ji, Bj ] = iεijkBk, [Bi, Bj ] = −iεijkJk (3.14)

We may seperate these into two commuting sets by introducing the operators,

Ai =1

2(Ji + iBi), Bi =

1

2(Ji − iBi) (3.15)

In terms of these the commutation relations (3.14) become,

[Ai,Aj ] = iεijkAk, [Bi,Bj ] = iεijkBk, [Ai,Bj ] = 0 (3.16)

so the A,B each generate the su(2) lie algebra and we establish the isomorphism SO(3, 1) ∼=SU(2)× SU(2) which is the well-known result that the Lorentz group in four dimensions is thedouble cover of SU(2) [11]. This implies that all reps. of SO(3, 1) are a product rep. of twoSU(2) reps. These are labelled by the eigenvalue of their highest weight state [18], so all rep.sof SO(3, 1) are labelled by the highest weights of A and B and can be denoted (a, b), wherea,b are their respective highest weights. Representation theory of SU(2) is particularly simple.A representation of highest weight a is spanned by 2a + 1 states. The three generators can becombined into a pair of ladder operators and an operator which gives the weight of a given state,what we will call the ”J3” operator of the rep. following frequent conventions in the literature.

It is easy to see that the highest weight in the rep. (a, b) is a+ b, because taking the productof two SU(2) rep.s is a direct sum of irrep.s where (a + b) is the weight of the largest SU(2)irrep. [18]. Naively then, the spin-s rep.s all take the form (s+k, s−k). Clearly there are manychoices for any given spin, but these are in general not rep.s of the SO(3, 1). To see why we canconsider spacetime inversion. A rep. of SO(3, 1) must, by definition, contain a representative ofspacetime inversion xµ → −xµ which flips the sign of odd-rank tensors while leaving even-ranktensors invariant, i.e for an arbitrary Lorentz tensor T ,

V Tµ1...µ2n+1V−1 = −Tµ1...µ2n+1 V Tµ1...µ2nV

−1 = Tµ1...µ2n

22


For n integer and where V represents spacetime inversion [9]. But acting on the generators(3.15) this gives,

VAV −1 =1

2(V JiV

−1 + iV BiV−1) =

1

2(Ji − iBi) = B

V BV −1 =1

2(V JiV

−1 − iV BiV −1) =1

2(Ji + iBi) = A

(3.17)

This implies that the J3 operators in the rep.s of A and of B have identical spectra. In particularthis implies that the eigenvalue of the highest weight state in the rep. of A is equal to the valueunder the“J3” of the rep. of B. This can be satisfied for (a, b) if a = b, i.e the rep.s is of the form(a, a). This is the whole story for integer spin (bosonic) rep.s and in particular we find that thebosonic spin-s irrep. is ( s2 ,

s2). However (3.17) is also satisfied by rep.s of the form (a, b)⊕ (b, a)

because then action of the J3 of A or B gives a+ b. These are the half-integer (fermionic) rep.s.The fermionic spin (s+ 1

2) rep is ( s2 ,s+1

2 )⊕ ( s+12 , s2).

We can show that the rep.s ( s2 ,s2) correspond to totally symmetric tensor. First we note that

in four dimensions, because of the isomorphism SO(3, 1) ∼= SU(2)⊗SU(2), then the fundamentalrep. of SO(3, 1) (the vector representation) transforms in the same way as the fundamental rep.of SU(2) ⊗ SU(2) which is the rep. (1

2 ,12) with index structure φαα. The well-known Pauli

matrices then furnish a winding dictionary [3] which relates tensors in the vector rep. to thoseof the spinor rep. as follows,

V µ = σµααVαα, (3.18)

Now the tensor structures for the ( s2 ,s2) field are of the form φα1...αnα1...αn for n ≤ s. We can use

the winding dictionary (3.18) to relate the spinorial fields to rank-n generally covariant tensors,

φµ1...µn = σµ1α1α1...σµnαnαnφ

α1...αnα1...αn

Because φα1...αnα1...αn is totally symmetric in αi and αi, then swapping any two vector indiceswe find that for example,

φµ2µ1...µn = σµ2α1α1σµ1α2α2

...σµαnαnφα1α2...αnα1α2...αn

= σµ2α1α1σµ1α2α2

...σµαnαnφα2α2...αnα1α1...αn

= φµ1µ2...µn

Therefore we find that the fields in the rep. ( s2 ,s2) are totally symmetric. SU(2) representation

theory tells us that ( s2 ,s2) contains invariant subspaces with all integer spins up to and including

s [18]. Fierz and Pauli showed that in order to ensure positivity of the energy it is necessary toeliminate all of the invariant subspaces with spin less than s [19]. Roughly, using (3.18) the rep.( s2 ,

s2) has the tensor structure,

φµ1...µss = ψµ1...µss ⊕ ψµ1...µs−1

s−1 + ...+ 1

= ψµ1...µss ⊕ φµ1...µs−1

(s−1)

where the fields φi are totally symmetric fields and we have written the sum of the spin ≤ ssubspaces as the field ( s−1

2 , s−12 ). Removing the lower-spin invariant subspaces requires setting

the part of φs which transforms like φ(s−1) to zero. This can be achieved by, in addition toimposing the Klein-Gordon equation, setting to zero the divergence [20],

∂ρφρµ2...µss = 0 (3.19)

which transforms like the φ(s−1) field. This is the well-known transversailty condition obtainedby Fierz and Pauli [19].

To demonstrate how to construct fermionic rep.s from rank-s tensors and Dirac spinors wecan look at the simple case of the Rarita-Schwinger field which lives in the spin-3

2 rep. We can

23


start by considering the field ψµ,α which is the product of a vector (µ index) and a dirac spinor(α index) [9].

(1

2,1

2)⊗ [(

1

2, 0)⊕ (0,

1

2)] = (1,

1

2)⊕ (

1

2, 1)⊕ [(

1

2, 0)⊕ (0,

1

2)]

The above product is the direct sum of the Rarita-Schwinger field and the Dirac spinor. Weneed to isolate and set-to-zero the subspace which transforms like a Dirac spinor. Clearly, usingthe gamma matrix γµαβ we see that γµαβψµ,α = (γµψµ)β transforms like a Dirac spinor, so we findthat setting,

(γµψµ)α = 0

implies that ψµ,α is the Rarita-Schwinger field. It can be shown in analogous fashion that thethe fermionic ( s+1

2 , s2)⊕ ( s2 ,s+1

2 ) field is ψµ1...µsα satisfying the ”γ tracelessness condition” [21],

(γρψρµ2...µs)α = 0

To summarise, the little group for massless particles is ISO(2) whose representation areexhausted by totally symmetric, traceless tensor fields. This implies that the physical field istotally symmetric and traceless rank-s tensor field. Further, positivity of the energy requiresthe transversality condition (3.19). Including the mass-shell condition we obtain a series ofphysicality conditions:

φµ1...µs = 0, ∂ρφρµ2...µs = 0, gµ1µ2φµ1µ2µ3...µs = 0

i 6 ∂ψ = 0, ∂ρψρµ2...µsα = 0, (γρψρµ2...µs)α = 0.(3.20)

for spin-s bosons (top line) and spin-(s + 12) fermions (bottom line). These conditions are the

mass-shell, transversality, traceless\γ-traceless conditions respectively.One might have trouble with the idea that the transversality condition is somehow “nec-

essary” for gauge fields, because in the case where we likely first met is as the Lorentz gaugecondition it is merely a ”convinient choice of gauge”. In the case of free non-gauge fields theoriesthe fields carry no redundant degrees of freedom and so the transversality condition must beimplied by the equations of motion. This is nicely illustrated by considering the Maxwell Procaequations for the spin-1 field. The massive spin-1 particle carry both helicity states of the mass-less particles plus the zero helicty state for a total of three degrees of freedom [9]. The massivespin-1 field is not a gauge field because the on-shell field carries three degrees of freedom, sonone of its components are physically redundant. The Maxwell-Proca equations for the massivespin-1 field are [11],

Aµ − ∂µ∂νAν −m2Aµ = 0

Now taking the divergence of the above implies that,

∂µAµ −∂νAν −m2∂µAµ = 0 ⇔ m2∂µA

µ = 0

which implies the transversality condition on-shell. Further, plugging this back into the theMaxwell-Proca equations just implies the mass-shell condition. But since the on-shell field is fullyphysical we could not merely choose the transverse condition, rather the physicality conditionsmust have followed from the equations of motion. But because this is the case for all non-gaugefield theories then in order to satisfy the physicality conditions, we must construct lagrangianswhose equations of motion imply the physicality conditions on-shell. We can accomodate this byconstructing the free field theory with appropriate Lagrange multipliers. This is the approachthat was originally adopted by Singh and Hagen in [22] in order to construct the Lagrangiansfor massive higher-spin fields. Now for gauge fields the on-shell field carries redundant degreesof freedom, so we have some freedom to simply pick a gauge where the physicality conditionsholds without requiring that they follow from the equations of motion. To construct the massless,spin-s field Lagrangian it is sufficient to construct it from the totally symmetric, rank-s tensor

24

3.3 Gauge Invariance of Massless Tensor Fields

and then require that transversality and tracelessness are allowed gauge choices, because if thephysical degrees of freedom can be determined a suitable gauge, then the conditions are clearlysatisfied by the physical field. We will refer to this gauge condition as the “transverse-tracelessgauge”.


In flat space we cannot add a mass term for a gauge field to a Lagrangian without breaking gaugeinvariance explicitly because such a term is simply not gauge invariant. Therefore we cannothave massive higher-spin gauge fields without explicitly broken gauge symmetry. Though, as itwell-known, it is fine for the field to acquire mass via interaction with a self-interacting field likethe Higgs field. What is frequently overlooked in most introductions to quantum field theoryis that masslessness of the spin-1 and higher-spin bosonic fields implies that it transforms upto a gauge transformation under general Lorentz transformations. In other words theories ofmassless particles of spin greater-than-one are gauge field theories [9]. This should not really beso surprising because given that some arbitrary higher-spin field has more than the two degreesof freedom necessary to describe massless quanta, then all but two of its components must beredundant. Gauge invariance is nothing other than the manifestation of this redundancy. This isrealised because we are not, in general, able to construct fields that transform tensorially undergeneral irrep.s of the Lorentz group using the creation and annihilation operators for masslessparticles.

Suppose we want to construct some field φl which transforms under some rep. l of theLorentz group and is constructed using the ladder operators for massless particles. The field iswritten in the usual way, by introducing coefficient functions, as follows,

φl(x) =

∫d3p

(2π3)Aul(p, σ)a(p, σ)eipx +Bvl(p, σ)a†(p, σ)e−ipx (3.21)

where ul and vl are the coefficient functions for the field φl and A and B are constants determinedby causality. Now the transformation of the massless particle states is fixed, and massless ladderoperators transform in the same way as single particle states,

U(Λ)a†(p, σ)U−1(Λ) = eiσθa(p′, σ) ⇔ U(Λ)a(p, σ)U−1(Λ) = e−iσθa(p′, σ) (3.22)

This is because if a single particle state is roughly |p〉 = a†p|0〉, with |0〉 being the invariantvacuum, the creation operator must transform exactly like the single-particle state. Overall thetransformation properties of the representation l must be conveyed to φl by the transformationproperties of the coefficient functions. In particular φl transform under some rep. of the Lorentzgroup, which is true if,

U(Λ)φl(x)U−1(Λ) =∑l′

Dll′(Λ−1)φl′(Λx) (3.23)

where Dll′ furnish the representation l. But using the transformation properties of the ladderoperators and the definition (3.21) we find that,

Uφl(x)U−1 =

∫dp3

(2π3)AUul(p, σ)a(p, σ)U−1eipx +BUvl(p, σ)a†(p, σ)U−1e−ipx

=

∫dp3

(2π3)AUul(p, σ)U−1Ua(p, σ)U−1eipx +BUvl(p, σ)U−1Ua†(p, σ)U−1e−ipx

=

∫dp3

(2π3)Aul(Λp, σ)e−iσθa(Λp, σ)eipx +Bvl(Λp, σ)eiσθa(Λp, σ)e−ipx

25


where we denoted U(Λ) = U and where we inserted the identity I = U−1U on the secondline above. Requiring that (3.23) holds implies the following transformation property of thecoefficient functions:

ul(Λp, σ)eiσθ =∑l′

Dll′(Λ)ul′(p, σ)

vl(Λp, σ)e−iσθ =∑l′

Dll′(Λ)vl′(p, σ)(3.24)

Now we let k be the standard momenta again and L(p) be the Lorentz transformation whichtake the standard momenta to arbitrary momenta p as in (3.5). Then (3.24) is satisfied if wehave that,

ul(p, σ) = N∑l′

Dll′(L(p))ul′(k, σ)

vl(p, σ) = N∑l′

Dll′(L(p))vl′(k, σ)(3.25)

where N is a suitable normalisation constant. This just implies that the coefficient functionswith arbitrary momenta p can be obtained by acting on the functions at the standard monentak with the representative in l of the Lorentz transformation which takes k to p. But by (3.24)the coefficient functions must satisfy,

ul(k, σ)eiσθ =∑l′

Dll′(W )ul′(k, σ)

vl(k, σ)e−iσθ =∑l′

Dll′(W )vl′(k, σ)(3.26)

Where W is an element of the little group (3.12) Therefore we then find that in general,

Dll′(W ) = Dll′ [R(θ)S(α, β)] =∑k

Dlk[R(θ)]Dkl′ [S(α, β)]

where the standard momenta is kµ = (1, 1, 0, 0) and S(α, β) and R(θ) are given by (3.10) and(3.11) respectively. Plugging the above result into (3.26) we find that,

ul(k, σ)eiσθ =∑l′,k

Dlk[R(θ)]Dkl′ [S(α, β)]ul′(k, σ) (3.27)

Setting θ = 0 sets R(θ) = 1 and the above result implies that,

ul(k, σ) =∑l′,k

δlkDkl′ [S(α, β)]ul′(k, σ) =∑l′

Dll′ [S(α, β)]ul′(k, σ) (3.28)

Now plugging this back into (3.27), for general θ this gives,

ul(k, σ)eiσθ =∑k

Dlk[R(θ)]uk(k, σ) (3.29)

Altogether the two equations (3.28) and (3.29) specify conditions that the coefficient functionsof the standard momenta must satisfy, in order that the field (3.21) transforms tensorially inthe representation l.

So how does the field transform under general Lorentz transformations? The case of themassless spin-1 field provides the simplest example, so we will try and construct a four-vectorfield from massless particle ladder operators. The operators in the vector representation aresimply,

D(Λ)µν = Λµ

ν

26


It is useful to define the polarisation tensor,

eµ(p,±1) =√

2E uµ(p,±1)

Plugging this into equation (3.29) this simply implies that,

eµ(k,±1)e±iθ = Rµ ν(θ)eν(k,±1) (3.30)

We can explicitly solve four equations for the components of the polarisation vector by plugging(3.11) into the above:

e0(k,±1)e±iθ = R00(θ)e0(k,±1) = ε0(k,±1)

e1(k,±1)e±iθ = R11(θ)e1(k,±1) = ε1(k,±1)

e2(k,±1)e±iθ = R22(θ)e2(k,±1) +R2

3(θ)e3(k,±1) = cos θe2(k,±1)− sin θe3(k,±1)

e3(k,±1)e±iθ = R32(θ)e2(k,±1) +R3

3(θ)e3(k,±1) = sin θe2(k,±1) + cos θe3(k,±1)

Satisfying the first two lines above for general θ implies that ε0 = ε1 = 0. However we cansatisfy the third line by choosing ε3 = ∓iε2 and the fourth line by setting ε2 = ±iε3 which areequivalent conditions. So (3.30) is satisfied by the unit polarisation vector,

eµ =1√2

00±i1

(3.31)

Now we can find the transformation of the polarisation vector under the full little group byconsidering the second condition (3.28), this implies that,

eµ(k,±1) = Sµ ν(α, β)eν(k,±1) (3.32)

Now plugging in (3.31) into (3.32) and recalling (3.10) we find that,

eµ(k,±1) = Sµ ν(α, β)eν(k,±1) =1√2

α± iβα± iβ±i1

=1√2

00±i1

+α± iβ√

2

1100

= eµ(k,±1) +

α± iβ√2|k|

kµ

(3.33)

and we see explicitly that we cannot simultaneously satisfy (3.29) and (3.28) since the secondterm on the right above does not vanish for general real α,β (because then α±iβ 6= 0). Thereforeit is impossible to build a four-vector using the ladder operators for massless particles! Insteadusing the above result and (3.31) satisfying (3.30), altogether we have that under the full littlegroup the polarisation vector transforms as,

Dµν [W (θ, α, β)]eν(k,±1) = S(α, β)µ λR

λν(θ)eν(k,±1) = e±iθS(α, β)µ λe

λ(k,±1)

= e±iθ[eµ(k,±1) +

α± iβ√2|k|

kµ] (3.34)

The standard momentum appears explicitly in the transformation law! Under general Lorentztransformations of the whole field (3.21) this term generates a transformation up to the derivativeof some scalar function which is exactly the spin-1 gauge transformation. Let’s see this by first

27


considering the transformation of the polarisation tensor evaluated for general momenta. Now(3.34) implies that for general momenta p and a general Lorentz transformation Λ that,

eµ(Λp,±1)e±iθ = Λµ νeν(p,±1) + pµΩ±(p,Λ)

instead of (3.24), where Ω±(p,Λ) is a scalar function [9]. Finally, plugging this result into (3.23)we find that ,

Uφµ(x)U−1 =

∫dp3

(2π3)AUuµ(p,±1)a(p,±1)U−1eipx +BUvµ(p,±1)a†(p,±1)U−1e−ipx

=

∫dp3

(2π3)AUuµ(p,±1)U−1Ua(p,±1)U−1eipx +BUvµ(p,±1)U−1Ua†(p,±1)U−1e−ipx

=

∫dp3

(2π3)Auµ(Λp,±1)e±iθa(Λp,±1)eipx +Bvµ(Λp,±1)e∓iθa(Λp,±1)e−ipx

=

∫dp3

(2π3)AΛµ

νuν(p,±1)a(Λp,±1)eipx +BΛµνvν(p,±1)a(Λp,±1)e−ipx

+ApµΩ±(p,Λ)a(Λp,±1)eipx +BpµΩ±(p,Λ)a(Λp,±1)e−ipx

= Λµν

∫dp3

(2π3)Auν(p,±1)a(Λp,±1)eipx +Bvν(p,±1)a(Λp,±1)e−ipx

+ ∂µ

∫dp3

(2π3)− iAΩ±(p,Λ)a(Λp,±1)eipx + iBΩ±(p,Λ)a(Λp,±1)e−ipx

= Λµνφν(Λx) + ∂µΩ(x,Λ) (3.35)

where Ω(x,Λ) is the scalar function which is equal to the integral in the penultimate lineabove. This is exactly the spin-1 gauge transformation! Therefore we have established thatinvariance of a theory containing the massless spin-1 field under general Lorentz transformationsimplies that the theory must also be gauge invariant.

Turning now to the case of the spin-s field φµ1...µs , we have seen already that the analogousresult for the spin-s field is that the polarisation tensor eµ1...µs transforms according to (2.16),

eµ1...µs(k,±s)e±isθ = eµ1...µs(k,±s) + sk(µ1εµ2...µs−1)(k,±s)

where εµ1...µs−1 is a totally symmetric rank - (s− 1) tensor field. Instead of (3.24) we find thatthe correct spin-s transformation is generated by the inhomogeneous transformation [26],

eµ1...µs(Λp,±s)e±isθ = Λν1 µ1 ...Λνs

µseν1...νs(p,±s) + s p(µ1Ω±µ2...µs)(p,±s)

Plugging this into (3.23) the transformed spin-s field is,

Uφµ1...µsU−1 =

∫d3p

(2π3)AUuµ1...µs(p,±s)a(p,±s)U−1eipx +BUvµ1...µs(p,±s)a†(p,±s)U−1e−ipx

=

∫d3p

(2π3)Auµ1...µs(Λp,±s)e±isθa(Λp,±s)eipx +Bvµ1...µs(Λp,±s)e∓isθa†(Λp,±s)e−ipx

=

∫d3p

(2π3)Λν1 µ1 ...Λ

νsµs [Auµ1...µs(Λp,±s)a(Λp,±s)eipx

+Bvµ1...µs(Λp,±s)a†(Λp,±s)e−ipx]

+A s p(µ1Ω±µ2...µs)(p,±s)eipx +B s p(µ1Ω±µ2...µs)(p,±s)e

−ipx

= Λν1 µ1 ...Λνs

µsφν1...νs + ∂(µ1ζµ2...µs)

Therefore the spin-s gauge theory must be invariant under the spin-s gauge transformation law,

δφµ1...µs = s∂(µ1ζµ2...µs) (3.36)

28


where ζµ1...µs−1 a totally symmetric tensor field. The spin-1 result gives the correct gaugetransformation law,

δφµ = ∂µζ

and the spin-2 case gives,δφµν = ∂µζν + ∂νζµ

This is exactly the transformation of the metric under a diffeomorphism. This implies that anytheory of massless spin-2 particles exhibits something like invariance under diffeomorphisms!Indeed, in the next chapter we will see that linearised Einstein gravity corresponds to a theoryof massless spin-2 particles, where diffeomorphism invariance of the full theory plays the role ofspin-2 gauge invariance at the linearised level.

Key Points:

• Particles transform as reps. of their little group. The little group for massless particles isISO(2) whose reps. are exhausted by totally symmetric and traceless tensors.

• The spin-s rep. of SO(3, 1) contains invariant subspaces of all spins < s. The energy ispositive if and only if the lower-spin fields are eliminated, which implies that the field isdivergenceless.

• In a fully-fixed gauge, the gauge fields must obey the mass-shell, transverse and traceless(or γ-traceless for Fermions) conditions.

• We cannot construct fields that transform as arbitrary rep.s of the Lorentz group using theladder operators for massless particles. In particular, the polarisation tensors transformnon-tensorially under under general Lorentz transformations, which implies that a fullyLorentz covariant theory of massless higher-spin particles must be gauge invariant.

29

4 Free Higher-Spin Gauge Fields

With the aim of constructing higher-spin gauge theories, our first priority is to show that the freetheory is well-defined. To this end we now present the formalism required to obtain the gaugeinvariant second-order wave equation, the Fronsdal equation for higher-spin gauge fields. We willessentially review the construction of higher-spin gauge theories due to de Wit and Freedman[5]. Their approach utilises generalisations of the connections and curvature tensors of Einsteingravity. It will be instructive for us to review the case of linearised Einstein gravity first, wherewe see that this properly describes free massless spin-2 gauge bosons in a flat background.This motivates us to seek a generalisation of this construction to fields of arbitrary spin. Wewill demonstrate that the Fronsdal equation is gauge invariant and that it properly describesmassless bosons of arbitrary spin. We will then review some of the known generic features ofhigher-spin gauge fields and interactions before we conclude by describing how one generalisesour formulation to the case of AdS4, which will lead us into Vasiliev theory.

4.1 Linearised Einstein Gravity and the Spin-2 Field.

The dynamical object in Einstein gravity is the metric gµν . In the linearised theory we approx-imate the metric dynamics as small fluctuations around the flat Minkowski background i.e wetake,

gµν = ηµν + hµν (4.1)

where hµν is some small perturbation around the flat background. The linearised Einsteinequations describe, as we will show, massless spin-2 particles propagating in a flat background.At this level we will demonstrate that we can regard dynamical gravity as a classical fieldtheory of gravitons on flat spacetime. Trying to canonically quantise this, we get the naivenon-renormalizable quantum general relativity.

Now the Christoffel symbols of Einstein gravity are [23],

Γρ;µν =1

2(∂µgνρ + ∂νgµρ − ∂ρgµν) =

1

2(∂µhνρ + ∂νhµρ − ∂ρhµν) (4.2)

where we used (4.1) to find the linearised form. Using the above result we find that the linearisedRiemann Tensor is [23],

2Rµνρσ = ∂ρΓµ;σν − ∂σΓµ;ρν + Γµ;ρλΓλσν − Γµ;σλΓλρν = ∂ν∂ρhµσ + ∂µ∂σhνρ − ∂µ∂ρhνσ − ∂ν∂σhµρ

Where we ignored terms quadratic in Γ since they are of order h2. The Ricci tensor is thecontraction Rµν = Rµρνσg

ρσ. Since the linearised Riemann tensor is of order h, we find that thelinearised Ricci tensor is,

Rµν = Rµρνσ(η + h)ρσ =1

2

(∂ν∂ρhµ

ρ + ∂µ∂ρhνρ − ∂µ∂νh−hµν

)(4.3)

Now the vacuum Einstein equations are simply Einstein’s equations with zero energy-momentumtensor [21],

Gµν = Rµν −1

2Rgµν = 0 (4.4)

Contracting with gµν gives, gµνRµν− 12Rg

µνgµν = R− 12R = 0⇔ R = 0. Plugging this back into

the vacuum Einstein equations we find they reduce simply to the Ricci flat condition Rµν = 0.Using (4.3) we see that the linearised Ricci flat condition implies a second-order equation forthe metric perturbation hµν ,

hµν − (∂ν∂ρhµρ + ∂µ∂

ρhνρ) + ∂µ∂νh = 0 (4.5)

30

4.1 Linearised Einstein Gravity and the Spin-2 Field.

After a suitable choice of gauge we will see that this reduces to a wave equation for the linearisedfield. The required choice is the Lorentz or De Donder gauge in which,

∂µhµν −1

2∂νh = 0 (4.6)

Substituting this into (4.5) the divergence terms become trace terms which precisely cancel theoriginal trace term leaving just the wave term:

hµν − (∂ν∂ρhµρ + ∂µ∂

ρhνρ) + ∂µ∂νh = hµν −1

2(∂ν∂µh+ ∂µ∂νh) + ∂µ∂νh = hµν

So with this choice of gauge the equation (4.5) indeed reduces to the wave equation,

hµν = 0 (4.7)

for a massless, symmetric tensor field. To see that this equation genuinely describes the masslessspin-2 field it remains for us to check that the physical field is traceless, transverse and containsexactly 2 degrees of freedom, so we need to completely fix the gauge, ensuring these physicalityconditions hold. We can tell that further gauge transformations are possible because the gaugevariation of the left-hand side of (4.6) is,

∂µ∂µεν + ∂µ∂νεµ −1

2· 2∂ν∂µεµ = εν

which implies that (4.6) holds (is gauge invariant) in any gauge for which the gauge parametersatisfies a wave equation εν = 0, so we have enough gauge freedom remaining to pick such agauge. We will try to use our gauge freedom to gauge away the trace. Therefore we want tosolve for ζ the equation δh′ = h+ δh = 0. Explicitly this is,

h = −ηµνδhµν = −2∂ρερ (4.8)

By choosing the plane wave solutions for hµν and εµ,

hµν = Hµνeikx, εµ = Cµe

ikx

where Hµν and Cµ are constant tensors and k2 = 0, since for hµν , the solution should describepropagating massless particles. For ε we choose the same k for convinience. We may furtherchoose the standard momenta in light-cone coordinates to be k = (k+, 0, 0, 0). With thesechoices the condition (4.8) becomes,

Heikx = −2∂ρCρeikx = −(k · C)eikx = −k+C+e

ikx

where H = Hµµ. Therefore we can gauge away the trace with C+ = − H

k+. Gauging away the

trace, the De-Donder gauge condition (4.6) reads,

∂µhµν = 0

which is exactly the transversality condition! Further gauge transformations are possible if theydo not change the trace, i.e if δφ′ = −2∂ρερ = 0 which implies that the regauge parameter isdivergenceless. Such a parameter has 4 − 1 = 3 independent components. We started with 10degrees of freedom, the gauge condition (4.6) fixes 4, gauging away the trace h removes 1 degreeof freedom and the remaining regauge parameter fixes 3 degrees of freedom. Therefore the fullygauge-fixed field contains exactly 2 degrees of freedom. The transverse-traceless gauge can bereached and the correct physical degrees of freedom for massless particles remain in a fully fixedgauge. Therefore linearised Einstein gravity rightly admits an interpretation as the classicalfield theory of massless spin-2 particles propagating in a flat background. 6

6Moreover, because the Weinberg Low-Energy theorem implies that the massless spin-2 particle couples to allparticles in the same way at low energies, then together with the conclusion of this subsection this demonstratesthat the massless spin-2 particle is the graviton.

31

4.2 The Fronsdal Equation


Having shown that linearised Einstein gravity properly describes the spin-2 quantum field theory,we now seek to generalise the formalism to the case of arbitrary spin. We can easily constructhigher-spin generalisations of the linearised Christoffel symbols (4.2) and use these to constructa gauge invariant second order equation for higher-spin fields which is the analog of the vacuumequation (4.5) for the spin-2 field. This is the method that we will follow that was considered byde Wit and Freedman [5]. The gauge-invariant, second order equation they found is exactly theequation that was found by Fronsdal in [24]. The Fronsdal equation reduces to the wave equationfor massless fields upon gauge fixing, though the resulting free theory requires double-tracelessfields and traceless gauge parameters.

Inspired by the form of the linearised Christoffel symbols of ordinary Einstein gravity (4.2)we construct the generalised Christoffel symbols for the spin-s gauge field [5],

Γ(1)ν;µ1...µs = ∂νφµ1...µs − s∂(µ1φµ2...µs)ν (4.9)

we can check that for example in the case s = 2 the above formula gives (4.2) up to a sign andnumerical factors that don’t matter,

Γ(1)ρ;µν = ∂ρφµν − 2∂(µφν)ρ = ∂ρφµν − ∂µφνρ − ∂νφµρ

We can then recursively define the higher-rank christoffel symbols using [5],

Γ(m)ν1...νm,µ1...µs = ∂ν1Γ(m−1)

ν2...νm,µ1...µs −s

m∂(µ1Γ

(m−1)|ν2...νm,ν1|µ2...µs) (4.10)

where (a1...|b1...|c1...) denotes symmetrisation over the indices ai and ci only. At this point wecould go on and, using the symmetries of the symbols obtain the formula for the nth christoffelsymbol. It is then apparent that for m = s the christoffel symbol is gauge invariant, and wedefine this to be our generalised curvature tensor. However for us it is sufficient for us to considerthe second christoffel symbol determined by the recursion relation (4.10) and using (4.9) it is[5],

Γ(2)ν1ν2,µ1...µs = ∂ν1Γ(1)

ν2,µ1...µs −s

2∂(µ1Γ

(1)|ν2,ν1|µ2...µs)

= ∂ν1∂ν2φµ1...µs − s∂ν1∂(µ1φµ2...µs)ν2

− s

2∂(µ1∂|ν2φν1|µ2...µs) +

s2

2∂(µ1∂|(ν1|φµ2...µs))ν2

= ∂ν1∂ν2φµ1...µs − s∂ν1∂(µ1φµ2...µs)ν2 +1

2s(s− 1)∂(µ1∂µ2φµ3...µs)ν1ν2

Now we define the Fronsdal tensor Fµ1...µs = Γ(2)ρ

ρµ1...µs . Setting this to zero the equation

Fµ1...µs = 0 is a rather impressive-looking second-order equation for the spin-s field,

Fµ1...µs = φµ1...µs − s∂(µ1∂ρφµ2...µs)ρ +

1

2s(s− 1)∂(µ1∂µ2φ

ρµ3...µs)ρ = 0 (4.11)

This is exactly the gauge-invariant equation found by Fronsdal [24] that describes a masslessbosons of arbitrary spin! We will refer to (4.11) as the Fronsdal equation. To verify thisresult we can check that upon gauge fixing the equation reduces to the wave equation for amassless particle and describes exactly 2 propagating degrees of freedom. Let’s first check thatthe Fronsdal equation generalises the case of Maxwell’s source-free equations and the linearisedvacuum Einstein equations (4.5). For the spin-1 field we have that,

φµ − ∂µ(∂νφν) = ∂ν(∂νφµ − ∂µφν) = ∂νFµν = 0

which are exactly Maxwell’s source-free equations where Fµν = ∂νφµ − ∂µφν is the familiarfield-strength tensor. In the spin-2 case the Fronsdal equation is,

Fµν = φµν − (∂µ∂ρφρν + ∂ν∂

ρφρµ) + ∂µ∂νφ = 0

32


which agrees exactly with (4.5).Let us check that (4.11) is gauge invariant under the spin-s gauge transformation δφµ1...µs =

s∂(µζµ2...µs). It is first useful to compute the variations of the divergence and trace of the spin-sfield under a gauge transformation:

δ(∂µ1φµ1...µs) = s∂µ1∂(µ1ζµ2...µs) = ∂µ1(∂µ1ζµ2...µs + ∂µ2ζµ1...µs + ...+ ∂µsζµ2...µ1)

= ζµ2...µs +(s− 1)!

(s− 2)!∂ρ∂(µ2ζ

ρµ3...µs) = ζµ2...µs + (s− 1)∂ρ∂(µ2ζ

ρµ3...µs)

δ(ηµ1µ2φµ1µ2...µs) = ηµ1µ2s∂µ1∂(µ1ζµ2...µs) = ηµ1µ2(∂µ1ζµ2...µs + ∂µ2ζµ1...µs + ...+ ∂µsζµ2...µ1)

= 2∂ρζρµ3...µs +(s− 2)!

(s− 3)!∂(µ3ζ

ρµ4...µs)ρ = 2∂ρζρµ3...µs + (s− 2)∂(µ3ζ

ρµ4...µs)ρ

(4.12)Where we used that ζ is totally symmetric. Now using the above results we can determine thevariation of the terms appearing in (4.11),

δ(φµ1...µs) = s∂(µ1ζµ2...µs)

δ(∂(µ1∂ρφµ2...µs)ρ) = ∂(µ1ζµ2...µs) + (s− 1)∂(µ1∂

ρ∂(µ2ζµ3...µs))ρ

= ∂(µ1ζµ2...µs) + (s− 1)∂(µ1∂µ2∂ρζµ3...µs)ρ

δ(∂(µ1∂µ2φρµ3...µs)ρ) = 2∂(µ1∂µ2∂

ρζµ3...µs)ρ + (s− 2)∂(µ1∂µ2∂(µ3ζρµ4...µs))ρ

= 2∂(µ1∂µ2∂ρζµ3...µs)ρ + (s− 2)∂(µ1∂µ2∂µ3ζ

ρµ4...µs)ρ

Now we simply plug these into (4.11), we find that the variation of the Fronsdal tensor is,

δF = δ(φµ1...µs)− sδ(∂(µ1∂ρφµ2...µs)ρ) +

1

2s(s− 1)δ(∂(µ1∂µ2φ

ρµ3...µs)ρ)

= s∂(µ1ζµ2...µs) − s∂(µ1ζµ2...µs) − s(s− 1)∂(µ1∂µ2∂ρζµ3...µs)ρ + s(s− 1)∂(µ1∂µ2∂

ρζµ3...µs)ρ

+1

2s(s− 1)(s− 2)∂(µ1∂µ2∂µ3ζ

ρµ4...µs)ρ

=1

2s(s− 1)(s− 2)∂(µ1∂µ2∂µ3ζ

ρµ4...µs)ρ

But for arbitrary gauge parameters, the result is non-zero for spin > 2. Therefore theFronsdals equation is not gauge invariant under arbitrary gauge transformations as we mighthave hoped, but only those for which the trace of the gauge parameter vanishes, which is clearlynon-trivial only for spin ≥ 3. Let’s move on without worrying about this constraint for themoment.

It is worth noting that the Fronsdal tensor is not equal to the generalised curvature tensor.The generalised curvature Rµ1ν1...µsνs formed by using the recursively defined christoffel symbolsturns out to be related to the antisymmetrisation of the Fronsdal tensor viz [15],

Rµ1ν1...µsνsην1ν2 = −1

2Fµ1µ2[µ3[...[µs,νs]...]ν3]

In the spin 2 case the result is particularly simple and gives,

Rµρνσηρσ = Rµν = −1

2Fµν

So for spin-2 case the Fronsdal tensor is a multiple of the Ricci tensor, which explains why theFronsdal equation F = 0 gave us the Ricci flat condition.

We can write down the generalised Einstein tensor G that generalises (4.4) to the case ofarbitrary spin. This is given in terms of the Fronsdal tensor as follows [25],

Gµ1...µs = Fµ1...µs −1

4s(s− 1)η(µ1µ2Fµ3...µs)ρ

ρ (4.13)

33


The equation Gµ1...µs = 0 then implies that the terms on the right, above vanish. Then con-tracting with ηµ1µ2 gives,

ηµ1µ2Gµ1...µs = ηµ1µ2Fµ1...µs −1

4s(s− 1)ηµ1µ2η(µ1µ2Fµ3...µs)ρ

ρ

= Fµ3...µsρ ρ − 1

4s(s− 1)δµ1

µ1F(µ3...µs)ρρ

= Fµ3...µsρ ρ − 1

4s(s− 1) · 4 · Fµ3...µsρ ρ

= (1− s(s− 1))Fµ3...µsρ ρ = 0

But the equation 1− s(s−1) = 0 has non-integer solutions, so for integer spin the above impliesthat the trace of the Fronsdal tensor vanishes. Plugging this back into (4.13) gives the Fronsdalequation G = F = 0. Now we would like to look for a lagrangian whose equations of motion givesimply G = 0. One can construct the canonical Lagrangian L = φµ1...µsGµ1...µs [26]. Since theEuler-Lagrange equation ∂L

∂φ = 0 simply implies that G = 0 as required. If we had included termslike ∂φ∂G, then the resulting Lagrangian would depend on higher-than first-order derivatives ofthe field and as a result it would be non-canonical. Since for traceless gauge paremeters F isguage invariant, then (4.13) implies that G is also gauge invariant. Therefore the variation ofthe action S =

∫ddxL under gauge transformations is,

δS =

∫ddx δ(φG) =

∫ddx ∂(µ1ζµ2...µs) G

µ1...µs

=

∫ddx ∂(µ1 [ζµ2...µs) G

µ1...µs ]− ζ(µ2...µs∂µ1) Gµ1...µs

The total derivative term vanishes. Now demanding invariance of S under gauge transformationsrequires that ∂ · G = 0. We have all of the pieces we need in order to analyse this constraint. In[25] it was found that ∂ · G ∼ φ′′. Therefore we also require the field to be doubly traceless!

ηµ1µ2ηµ3µ4φµ1µ2µ3µ4...µs = φ′′µ3...µs = 0

Clearly, this condition only becomes relevant for spin s ≥ 4. This arises more generally dueto physical constraints [5], but the above result which requires that we be able to obtain theFronsdal equations from a suitable Lagrangian is also strong motivation for the double-tracelesscondition.

Ideally what we would like is a second order field equation for unconstrained higher-spinfields which is invariant under arbitrary gauge transformations. One suspects that our resultis a partially gauge-fixed relic of an unconstrained system of equations. It is possible to recastthe equations of motion (4.11) in a way which removes both of these constraints by a fieldredefinition and this was done in [27]. The price we have to pay for removing the constraints isthat the equations of motion become explicitly non-loca!! Upon partial gauge fixing this systemindeed reduces to (4.11) with constrained fields and gauge parameters [15]. Non-locality is oneof the exotic features of higher-spin gauge field theories. We will have more to say about thisshortly.

Now we will check that the Fronsdal equation describes massless particles of arbitrary integerspin. We can proceed by analogue with what we did for linearised Einstein gravity. Namely,we simply seek to gauge fix the equations of motion and ensure that the transverse-tracelessgauge can be reached, and that the resulting fields carry exactly 2 degrees of freedom. Wewill proceed by counting degrees of freedom, for which we will need to know that the generalformula for the number Ns of independent components of a totally symmetric rank -s tensor fieldin d = 4 is Ns =

(s+3

3

)[26]. But the trace of a totally symmetric rank-s tensor is a rank-(s-2)

tensor. It then immediately follows that a totally symmetric, traceless rank-s tensor field hasNs −Ns−2 = (s+ 1)2 independent components.

34


Now we proceed to gauge fix (4.11). In the spin-2 case, choosing the De-donder gaugecondition reduced the linearised equations to a wave equation. Clearly by eliminting the lasttwo terms on the right of (4.11) with some gauge choice then it reduces to a wave equation. Thecorrect choice is the generalised De-Donder gauge condition [5],

Dµ2...µs = ∂ρφρµ2...µs −1

2s(s− 1)∂(µ2φ

ρµ3...µs)ρ = 0 (4.14)

indeed using this condition the second term in (4.11) becomes,

s∂(µ1∂ρφµ2...µs) =

1

2s(s− 1)∂(µ1∂(µ2φ

ρµ3...µs))ρ =

1

2s(s− 1)∂(µ1∂µ2φ

ρµ3...µs)ρ

This exactly cancels the last term on the right of (4.11), so indeed this gauge choice gives us thewave equation,

φµ1...µs = 0

Now the condition (4.14) is traceless,

ηµ2µ3Dµ2...µs = ∂ρφρσ µ4...µs −1

2ηµ2µ3(∂µ2φ

ρµ3...µsρ + ∂µ3φ

ρµ2...µsρ + (s− 2)∂(µ4φ

ρµ5...µs)µ2µ3ρ)

= ∂ρφρσ µ4...µs −1

2· 2∂ρφρσ µ4...µs −

1

2(s− 2)∂(µ4φ

′′µ5...µs) ∼ ∂φ

′′ = 0

since the double-trace vanishes. Therefore Dµ2...µs = 0 fixes all of the components of a totallysymmetric, traceless rank-(s-1) tensor; a total of s2 components. which is the wave equation formassless particles. Further gauge transformations are possible because, using the results (4.12)we find that the gauge variation of (4.14) is,

δDµ2...µs = sζµ2...µs + s(s− 1)∂(µ2∂ρζµ3...µs)ρ −

1

2s(s− 1)[2∂(µ2∂

ρζµ3...µs)ρ

+ (s− 2)∂(µ1∂µ2∂(µ3ζρµ4...µs))ρ]

= sζµ2...µs + s(s− 1)∂(µ2∂ρζµ3...µs)ρ − s(s− 1)∂(µ2∂

ρζµ3...µs)ρ

= sζµ2...µs

where, as we recall, the trace of the gauge parameter vanishes. Therefore (4.14) holds for gaugetransformations with parameter ζ satisfying the wave equation ζµ2...µs = 0. As with the spin-2case we may choose the plane wave solution,

ζµ1...µs−1 =

∫dd kRe[−iCµ1...µs−1(k)eik·x]

where k2 = 0. We also consider the plane-wave solutions of the massless field equation,

φµ1...µs =

∫ddk Re[φµ1...µse

ik·x]

Now we will gauge away the trace using the method followed in [15]. Switching to lightconecoordinates for convinience, we can choose the standard null momenta k = (k+, 0, 0, 0) . Thetrace is gauged away via transformation that solves φ′+ δφ′ = 0. Using (4.12) and the fact thatthe gauge parameter is traceless, the gauge variation of the trace of the field is,

δφ′ = δ(s

∫ddk Re[φν νµ1...µse

ik·x])

= 2∂νζνµ2...µs =

∫dd kRe[−iCµ1...µs−1(k)∂νeik·x]

=

∫ddk Re[−2i · ikνC+µ2...µs−1e

ik·x] =

∫ddk Re[2k+C+µ2...µs−1e

ik·x] =

−φ′ =∫ddk Re[−φν νµ2...µse

ik·x]

35

4.3 Philosophy of Higher-Spin Gauge Theories

Therefore the required choice of C is C+µ2...µs−1 = − 12k+

φν νµ2...µs . With the trace gauged awaythe condition (4.14) reduces to the transversality condition,

∂ρφρµ2...µs = 0

So we have shown explicitly that the transverse-traceless gauge can be reached. Having fixedthe trace to be zero, further gauge transformations that are compatible with this must haveδφ′ = 2∂νζνµ2...µs = 0, so the divergence of the regauge parameter vanishes. Now the gaugeparameter is a totally symmetric, traceless rank-(s-1) tensor therefore having s2 independentcomponents. It’s divergence is a totally symmetric, traceless rank-(s-2) tensor therefore having(s − 1)2 independent components. The divergenceless regauge parameter then has a total of(s2 − (s− 1)2) independent components.

Altogether, the number of physical degrees of freedom of φ after the gauge has been fullyfixed is equal to the number of independent degrees of freedom of a totally symmetric andtraceless rank-s tensor, minus the number of independent components fixed by (4.14), minus thenumber of independent components of the remaining regauge parameter [26] 7. This gives,

(s+ 1)2 − s2 − (s2 − (s− 1)2) = s2 + 2s+ 1− s2 − (s2 − s2 + 2s− 1) = 2

Therefore the transverse-traceless gauge can be reached and the physical field propagates exactlytwo degrees of freedom as required!

We have so far completely foregone the discussion of fermions. Essentially the same kind ofmachinery that we have used here can be used to describe the fermionic case. The bosonic casetherefore sufficiently illustrates the main methodology in the construction of the free higher-spinfield equations and highlights mostly the same generic features that we observe in the case offermions.

4.3 Philosophy of Higher-Spin Gauge Theories

A number of exotic features of interacting higher-spin gauge theories have been observed thatare assumed to be generic; no counter-examples are known. We briefly review some of thesenow.

It is an unproven conjecture that all consitent higher-spin vertices are necessarily higher-derivative [1]. Already, the vertex of the form 0-s-0 must contain more than two derivativesto absorb the field indices. If this is the case then the inclusion of higher-spin vertices impliesthat the Lagrangian acquires a dependence on greater than first-order derivatives of the field.Thus the very canonical formulation of field theory is at stake! Indeed, the existence of higher-derivative terms drastically modifies the canonical theory [28]. However, we can reconcile thisresult by insisting that the higher derivative terms are coupled via some parameter l such that atl = 0 we recover the canonical theory. One immedietaly has the problem that higher-derivativeterms are already of the wrong mass-dimensions, so we require additionally that the parameterl is dimensionful. One context in which we naturally have a dimensionful parameter with whichto dress powers of derivatives is in AdS where we have a non-zero cosmological constant.

Another of the generic features of higher-spin gauge theories is that consistency requiresan infinite tower of interacting fields of higher-spins. More specifically, to have a consistentlymassless spin-s field requires the existence of the spin s+ 1 field [29]. It will come as no surprisethen that Vasiliev’s fully interacting higher-spin system describes such an infinite tower of fieldsof increasing spin!

Non-locality also appears to be a generic feature of higher-spin gauge theories [1]. Althoughit is not our task here to discuss the full implications of non-locality in field theory we notethat it is something which is far from disastrous. Non-locality emerges often in the context of

7We did not have to worry about how many degrees of freedom were fixed by transversality, because (4.14)and tracelessness automatically imply the transversality condition.

36

4.4 Free Higher-Spin Fields in Anti-de Sitter Space

effective field theory [28]. However it is an interesting feature because it potentially suggests aninterpretation in terms of extended objects [1].

One particularly novel feature of interacting Higher-Spin gravity is the inability to performa higher-spin gauge transformation without also performing gauge transformations on the cou-pled lower-spin fields. This generally occurs because spin-s gauge transformations contain alllower-spin gauge transformations as subgroups [2]. For example, coupling electromagnetismto gravity, one cannot perform a general coordinate transformation without also performing agauge transformation of the electromagnetic potential! Much more unsettling is the situation inwhich the metric is coupled to a spin-3 gauge field. Then as one might now expect, performingthe spin-3 gauge transformation results in a also performing a diffeomorphism! In this case thetwo different geometries are gauge equivalent. Formally, the causal structure of the spacetimedescribed by the metric is not invariant under higher-spin gauge transformations. One is thenpresented with the seemingly wild possibility that naively very different spacetime geometriesare gauge equivalent! In particular, this strikes at the heart of general relativity, since good“coordinate invariant” statements are no-longer meaningful [30].


Anti De-sitter space is a maximally symmetric spacetime that solves Einstein’s equations witha non-zero cosmological constant Λ [23]:

Gµν = −Λgµν

The cosmological term actually generates a mass term for the higher-spin fields [26]. Conse-quently the Hamiltonian can be positive definite for fields with negative mass, so long as whatwe could call the effective mass which results from the addition of the mass of the field and the”mass” generated by the cosmological constant is positive definite. The result is that the canon-ical mass of a field in AdS is bounded from below by a negative value, this is the well-knownBreitenlohner-Freedman (BF) bound. The simplest way to derive the BF bound [31] is to lookat the action for a massive scalar field φ in AdSd:

S(φ) = −η2

∫dzddx

zd+1z2∂zφ∂zφ+ z2ηµν∂µφ∂νφ+m2R2φ2 (4.15)

where η is a normalisation constant and R is the radius of AdSd+1. Now we perform a rescaling

of the field φ = zd2ψ and a change of coordinates y = − ln z ⇔ z dy = dz. In terms of the

rescaled field and the new coordinates the action above becomes,

S(ψ) = −η2

∫dyddx

zdz2∂z(z

d2ψ)∂z(z

d2ψ) + z2+dηµν∂µψ∂νψ + zdm2R2ψ2

= −η2

∫dyddx

zdd2

4zdψ2 + d zd+1∂zψ + zd+2∂zψ∂zψ + z2+dηµν∂νψ∂µψ + zdm2R2ψ2

= −η2

∫dyddx

d2

4ψ2 + d ∂yψ + ∂yψ∂yψ + e−2yηµν∂νψ∂µψ +m2R2ψ2

The underline term is a total derivative, it gives a contribution at the boundary of AdSd+1,but we can always add boundary terms so that the resulting boundary term is positive. Theresulting action, ignoring the boundary term, is canonical:

S(φ) = −η2

∫dyddx ∂yψ∂yψ + e−2yηµν∂νψ∂µψ +

[d2

4+m2R2

]ψ2

It is now easy to see that the Hamiltonian, derived from the time-time components of the energymomentum tensor, is positive definite for d2

4 + m2R2 ≥ 0. So the mass is bounded from belowby a negative value,

m2R2 ≥ −d2

4

37


This is the BF bound. Therefore we see that the scalar field acquires an additional mass termin AdS. This derives from the coupling of the field to the non-trivial cosmological constant.

Now to construct Fronsdal’s equation in AdS we start by noting that, in general curvedbackgrounds, the spin-s gauge transformation is deformed by the covariantisation of derivatives:

δφµ1...µs = s∇(µ1ζµ2...µs) (4.16)

In particular, because the covariant derivatives do not commute then simply replacing ∂ → ∇in (4.11) is not enough to ensure that the result is invariant under (4.16). One must add furtherterms to compensate for this fact . We will forego the analysis and simply present the end resultthat was found in [26]; Fronsdal’s equations in AdS4 are,

(∇2 −m2)φµ1...µs − s∇(µ1∇ρφµ2...µs)ρ +

s(s− 1)

4sg(µ1µ2∇

ν1∇ν2φµ3...µs)ν1ν2 = 0 (4.17)

with m2 = s2 − 2s− 2 in AdS4 [4]. In particular, we note the addition of the mass term whicharises due to the coupling of the field to the cosmological constant.

Key Points:

• Linearised Einstein gravity is the quantum field theory of the massless, spin-2 particle: thegraviton.

• By generalising the Christoffel symbols of linearised Einstein gravity we are able to con-struct a gauge invariant second order equation for the spin-s field, the Fronsdal equation(with traceless gauge parameters and doubly traceless fields).

• In a fully fixed gauge the Fronsdal equation describes the propagation of massless higher-spin particles.

• One can reformulate the Fronsdal equation in terms of unconstrained fields and gaugeparameters, however the equations of motion become explicitly non-local.

• Higher-spin interactions generically exhibit non-locality and higher derivative terms. Ad-ditonally, the causal structure is not invariant under higher-spin gauge transformations.

• AdS is a solution of Einstein’s equations with a non-zero cosmological constant. Gaugefields in AdS acquire an additional mass-term due to their coupling with the cosmologicalconstant.

• The spin-s gauge transformation is deformed in curved backgrounds so one must modify(4.11) with compensatory terms to ensure deformed gauge invariance.

38

5 Vasiliev Gauge Theory

Vasiliev gauge theory represents what is perhaps currently the most successful realisation of afully interacting gauge theory of higher-spin fields. In a nutshell, Vasiliev gauge theory describesan infinite tower of interacting higher-spin gauge fields and a massive scalar field in AdS4. Toreally understand how this was achieved requires an depth of understanding of the alternativegeometric formulations of gravity, their higher-spin generalisations and the so-called higher-spin algebras. This would take us far of the course that we intend to follow in this report, sowe will not address these issues at length. Nevertheless in order to see that Vasiliev theorysuccessfully realises the aforementioned qualities (the main task of this chapter!) we will notneed to understand these things in any detail.

5.1 Background: Frame Formalism

Vasiliev gauge theory appears immediately very obscure! Indeed, the full dynamical content ofthe theory is vast and complex and so before plowing on we will briefly discuss some of the stepsthat lead to its realisation.

The search initiated by Vasiliev for interacting higher-spin equations of motion was largelyinspired by geometric reformulations of Einstein gravity due to Stelle-West-Macdowell-Mansouri[32]. The fundamental dynamical objects in the alternative formulation of Einstein gravity arenot the metric, as usual, but the vierbein and Lorentz spin-connections which characterise thelocal-Lorentz frames on the manifold. One can then generalise the construction to the caseof higher-spins by utilising the generalised curvatures and spin connections a la de Wit andFreedman. Because one can formulate higher-spin gauge theory in this was by generalisingEinstein gravity, it is often referred to as “higher-spin gravity”. It was found that one couldneatly write down the massless higher-spin equations of motion by packaging the higher-spinfields and the higher-spin connections into a single “frame field” [33]. The result is called theCentral On-Mass-Shell Theorem [29]. This reflects the situation that we will review for Vasilievtheory, where the vierbein and spin connections that characterise the vacuum geometry, as well asthe higher-spin fields that propagate on this background, are all contained in a single dynamicalone-form object called a “master field”. It was found that consistency of the interacting higher-spin equations required the introduction of a spacetime zero-form in addition to the masterone-form [1]. For this reason Vasiliev’s system is also endowed with a massive scalar field.

5.2 Twistors and the Star Product

At the heart of the construction of Vasiliev’s system is the realisation of the higher-spin algebrapreviously mentioned by the introduction of a twistor space with non-commutative star product[29]. This plays a similar role to the supercoordinates in supersymmetry which realise thesupersymmetry algebra.

Vasiliev’s theory comprises a set of non-linear equations of motion for a set of master fieldsliving on a product of spacetime and a non-commutative twistor. The latter is space spannedby a pair of commuting spinors (Y,Z) = (yα, yα, zα, zα) or ”ghosts”. The twistors y, z andy, z transform as left-handed and right-handed spinors of SU(2) respectively. They obey thecommutation relations [34],

[yα, yβ] = [zα, zβ] = [yα, yβ] = [zα, zβ] = 0 (5.1)

As we will shortly see, the dependence on the twistor coordinates allows the master fields toencode a tower of increasing-spin degrees of freedom, whose dynamics are in turn governed bythe master equations.

39

5.3 Master Fields and Equations of Motion

Indices are raised and lowered using yα = εαβyβ, yα = εαβ yβ etc. A useful property of thetwistors obeying these commutation relations is that their contraction vanishes,

yαyα = εαβyβyα = εαβyαyβ = −εβαyαyβ = −yβyβ ⇔yαy

α = 0(5.2)

The inner product on the twistor space is the non-commutative star product “∗”. The starproduct of two two test functions f ,g of the twistors is [2],

f(Y,Z) ∗ g(Y, Z)

= f(Y,Z)exp[εαβ(←−∂ yα +

←−∂ zα)(

−→∂ yβ −

−→∂ zβ ) + εαβ(

←−∂ yα +

←−∂ zα)(

−→∂yβ−−→∂zβ

)]g(Y, Z)

(5.3)

Using (5.3) we find the useful star-commutation relations between the twistors,

[yα, yβ]∗ = yα(1 + εab←−∂ ya−→∂ yb)y

β − yβ(1 + εab←−∂ ya−→∂ yb)y

α = εabδαa δβb − ε

abδαb δβa = 2εαβ (5.4)

[zα, zβ]∗ = zα(1− εab←−∂ za−→∂ zb)z

β − zβ(1− εab←−∂ za−→∂ zb)z

α = −εabδαa δβb + εabδαb δ

βa = −2εαβ

Where [, ]∗ denotes star commutation. The twistors with non-commutative star product aremerely a convinient, albeit powerful, tool for encoding the higher-spin algebra and a tower ofhigher-spin fields in Vasiliev theory. The twistor fields themselves are completely auxilliary andas such the physical fields do not depend on them.

It is useful to define the functions [33],

K(t) = etyαzα, K(t) = etyαz

α(5.5)

The functions K(1) := K and K(1) := K are called Kleinians. Taking the star products ofKleinians with arbitrary twistor functions gives the following identities [4],

f(y, z) ∗K = f(−z,−y)K, K ∗ f(y, z) = Kf(z, y) (5.6)


As we have mentioned, the dynamical objects of Vasiliev theory are bundled into a set ofmaster fields which depends on spacetime and the twistor coordinates. These are a one-formon space time W (x|Y, Z) = Wµ(x|Y, Z)dxµ, a scalar field B(x|Y, Z) and an auxilliary one-formon twistor space S(x|Y,Z) = Sα(x|Y, Z)dzα + Sα(x|Y,Z)dzα. We have denoted the dependenceon spacetime and the twistor space by (x|Y,Z) [34]. The physical fields are contained withinW and B, whereas the twistor one-form S was originally found convinient for writing down theequations of motion [2].

To write the equations of motion it is convenient for us to package the one-form fields intothe field,

A =W + S +1

2zαdzα +

1

2zαdzα (5.7)

Vasiliev’s gauge invariant, non-linear master equations of motion are [3],

dxA+ A ∗ A = f∗(B ∗K)dz2 + f∗(B ∗ K)dz2 (5.8)

dxB + A ∗B −B ∗ π(A) = 0 (5.9)

where f is an arbitrarys function and f∗ is the function obtained by writing f as a polynomialand replacing all ordinary products with star products. Also we have defined the operation π(·)which flips the signs of (y, z, dz) only.

Roughly, the rationale behind Vasiliev’s master equations is as follows. The master equationscomprise a set of constrained differential equations for the master fields. At the level of the

40


twistor components of the master fields (we will see what this means shortly), the unconstraineddifferential equations equate to an infinite set of equations relating these components. The keypoint is that the equations acquire dynamics only when one imposes non-differential constraintson the master fields [2]. This is essentially a generic feature of the so-called “unfolded formalism”of higher-spin gauge theories [26]. The trick of Vasiliev and others was then to construct asuitable set of differential equations involving the master fields with additional constraints chosenso that the resulting dynamical equations describe the free propagation of higher-spin gaugefields (and a massive scalar) in AdS4 at the linearised level and that they describe interactionsat higher orders. We don’t need to worry about this in what follows, but it is nice to have somejustification for the immediately obscure-seeming formalism!

The master equations (5.8), (5.9) are invariant under the gauge transformations [32],

δA = dε+ [A, ε]∗, δB = −ε ∗B +B ∗ π(ε)

where ε(x|Y,Z) is the gauge parameter which depends on spacetime and twistor space. Wenotice that the above gauge transformation of A takes the form of an ordinary non-abeliandeformation of a 1-form gauge transformation a la Yang-Mills. We impose a truncation conditionon the master fields [3],

[KK,W ]∗ =KK, S

∗ = [KK,B]∗ = 0 (5.10)

Using (5.6) we see that this simply implies that W and B are even functions of Y ,Z,

[KK,W ]∗ = KK ∗W (x|y, y, z, z)−W (x|y, y, z, z) ∗KK= KKW (x|z, z, y, y)−W (x| − z,−z,−y,−y)KK = 0⇔

W (x|y, y, z, z) = W (x| − y,−y,−z,−z)

and similarly for B; with S we would simply get another minus sign above from the anticom-mutator, implying that it S is an odd function of Y ,Z.

Naively the appearance of the function f in (5.8) describes infintely many theories. Howeverone can consitently redefine the fields in such a way that it is equivalent to a redefinition of f .In otherwords f carries physically redundant information. It turns out that the most generalform of f is [3],

f(x) =1

4+ x exp(i

∞∑n=0

θ2nx2n) =

1

4+ xeiθ(x) (5.11)

so that the theory is characterised by the phase θ(x) =∑∞

n=0 θ2nx2n. In relation to this, parity

symmetry on the non-commutative twistor space conjugates the twistors i.e y ↔ y, z ↔ z [4].Under parity, the master field B may be either even or odd. Now the term on the right-handside of (5.8) goes like,

f∗(B ∗K)dz2 + f∗(B ∗ K)dz2 → f∗(±B ∗ K)dz2 + f∗(±B ∗K)dz2

where the top/bottom-sign corresponds to B being parity even/odd. Invariance under parityimplies that either f(x) = f∗(x) or f(x) = f∗(−x). These imply that either,

fA(x) =1

4+ x, fB(x) =

1

4+ ix

when B is parity even or odd respectively [3]. Therefore parity invariance implies that eitherθ0 = 0, π2 , these choices give the type-A and type-B Vasiliev theories respectively. General θ0

therefore characterises Vasiliev theories with broken parity symmetry.

41

5.4 Physical Fields.

5.4 Physical Fields.

The physical fields represent fluctuations around the AdS4 background solution that we willsee is given by W = W0, B = S = 0 with suitable chosen W0 [29]. The physical higher-spinfields and a massive scalar are then identified with the components of the perturbations aroundthe vacuum solution. In particular we set W = W0 + W (There is no point writing similarexpressions for B and S because their vacuum values vanish). It is supposed that all of thephysical degrees of freedom are contained in the master fields at Z = 0 [4], so it is useful todefine the Z-independent components of W and B,

C(x|Y ) = B(x|Y,Z)|Z=0 Ωµ(x|Y ) = Wµ(x|Y,Z)|Z=0 (5.12)

The physical pure-space fields are a massive scalar field which is the bottom component C(0,0)(x)of C(x|Y ) and a tower of higher-spin fields contained within Ωµ(x|Y ) [3]. Now we will how thetwistor dependence of the master fields allows them to encode higher-spin degrees of freedom.We can then explicitly observe the higher-spin fields contained therein. Firsty it is instructiveto consider a test-function f(x|Y ), subject to the truncation conditions (5.10) for B and W .Expanding f in Y we have,

f(x|Y ) = f (0,0)(x) + f(2,0)αβ yαyβ + f

(0,2)

αβyαyβ + f

(1,1)αα (x)yαyα + ...

=

∞∑n,m=0

f (n,m)(x)α1...αnα1...αmyα1 ...yαn yα1 ...y ˙αm ,

(5.13)

where n+m is even and f (n,m)(x) is the pure-space component of f that which multiplies ynym

and which we will call the (n,m) component of f. Now we can use the winding dictionary (3.18)to relate the spinorial components of f to their corresponding vectorial quantities,

(n, n) : f (n,n)(x)α1...αnα1...αn = σµ1α1α1...σµnαnαnf

(n,n)µ1...µn(x), (5.14)

So that the (n, n) component of f corresponds to a metric-like (totally symmetric and traceless)spin-n field. In fact using (3.18) we can define the vector [4],

Y µ = σµααyαyα = yσµy

Using the expansion (5.13) then see that f is a generating function for its bosonic higher-spincomponents because,

δ

δY µ1...

δ

δY µnf(x|Y )|Y=0 = ...+ f (n,n)(x)ν1...νnδ

ν1µ1 ...δ

νnµn + ...|Y=0 = f (n,n)(x)µ1...µn (5.15)

Since n+m is even and since the winding dictionary (3.18) relates the components of f to integerspin degrees of freedom via (5.14), then f encodes bosonic higher-spin degrees of freedom only.Therefore B and W , also obeying (5.10), encode only bosonic degrees of freedom. Since thesecontain the physical fields of the theory then we conclude that Vasiliev theory describes aninfinite tower of bosonic higher-spin gauge fields.

Now the spin-s fields are contained within

Ω(s−1+k,s−1−k)µ (x|Y ) := Ω(k)(x|Y ), C(s+k,s−k)(x|Y )

With k = 0 above, Ω(s−1,s−1)µ (x) contains the spin-s metric-like tensor field as can be seen by

using (5.15),δ

δY µ1...

δ

δY µ(s−1)Ων(x|Y )|Y=0 = Ω(s−1,s−1)(x)νµ1...µs−1

which is a spin-s field as advertised. The other components with k 6= 0 are related to derivativesof the higher-spin fields. For example for k = 1, we have a total of (s − 2) pairs of dotted

42

5.5 AdS4 Background

and undotted indices and one pair of undotted indices. The winding relates a pair of dottedand undotted indices to a vector, but now we need an object that related pairs of the samekind of index to vector indices. The object in question is (σµν)α

β = 14(σµσν − σν σµ)α

β with

σαβ = εβαεαβσµαβ

[35], and then we have schematically that,

Ω(s−1+1,s−1−1)

µ,α1...αs−2β1...βs−2(x) ∼ σµ1

α1β1...σµ1

αs−2βs−2· σρσαβ Γ(1)

ρ,σνµ1...µs−2

Where Γ(1) is the christoffel symbol (4.2)! It is then not hard to imagine that the other com-ponents Ω(k) are related to the kth Christoffel symbol, and therefore these objects encode thehigher-spin curvatures of de Wit and Freedman.

5.5 AdS4 Background

Jusy looking at Vasiliev’s master equations it is not at all obvious that their vacuum solutiondescribes an AdS background! Nonetheless this result is not actually particularly difficult to seeso long as the twistor manipulations are handled carefully. How this works is that the maximallysymmetric solution for the master one form A is given by [3],

A = W0(x|Y ) = (e0)(x|Y ) + (ω0)(x|Y ) = (e0)αβyαyβ + (ω0)αβy

αyβ + (ω0)αβ yαyβ (5.16)

The components e0 and ω0 can be interpreted as a vierbien and spin-connection for a manifold.The master equations then reduce to a set of geometric equations which describe the geometryof AdS4! We will now show this explicitly. We begin by inserting ths solution (5.16) into themaster equation (5.8) and setting B = 0, S = 0 the master equation reduces to [29],

(dxW0)µν +W ν0 ∗W ν

0 = 0 (5.17)

The first term on the left above is simply,

(dxW0)µν = (dx(e0)αβ)µνyαyβ + (dx(ω0)αβ)µνyαyβ + (dx(ω0)αβ)µν yαyβ (5.18)

Now we’ll undergo a long computation of the star product on the left of (5.17), it is

Wµ0 ∗W

ν0 = [(e0)µ

αβyαyβ + (ω0)µαβy

αyβ + (ω0)µαβyαyβ] ∗ [(e0)ν

γδyγ yδ + (ω0)νγδy

γyδ + (ω0)νγδyγ yδ]

= (e0)µαβ

(e0)νγδ

[yαyβ ∗ yγ yδ] + (ω0)µαβ(e0)νγδ

[yαyβ ∗ yγ yδ] + (ω0)µαβ

(e0)νγδ

[yαyβ ∗ yγ yδ]

+ (e0)µαβ

(ω0)νγδ[yαyβ ∗ yγyδ] + (ω0)µαβ(ω0)νγδ[y

αyβ ∗ yγyδ] + (ω0)µαβ

(ω0)νγδ[yαyβ ∗ yγyδ]

+ (e0)µαβ

(ω0)νγδ

[yαyβ ∗ yγ yδ] + (ω0)µαβ(ω0)νγδ

[yαyβ ∗ yγ yδ] + (ω0)µαβ

(ω0)νγδ

[yαyβ ∗ yγ yδ].(5.19)

All of the star products above are computed in the appendices.To get the equations that describe an Anti-de sitter background we only need to consider

the (1,1),(2,0) and (2,0) components of (5.17), so we’ll plug the products (A.1 - A.9) into (5.19)to get:

W0 ∗W0 = (e0)µαβ

(e0)νγδ

[εαγ yβ yδ + εβδyαyγ ] + (ω0)µαβ(e0)νγδ

[εαγyβ yδ + εβγyαyδ]

+ (ω0)µαβ

(e0)νγδ

[εαδyβyγ + εβδyαyγ ] + (e0)µαβ

(ω0)νγδ[εαγ yβyδ + εαδyβyγ ]

+ (ω0)µαβ(ω0)νγδ[εαγyβyδ + εαδyβyγ + εβγyαyδ + εβδyαyγ ]

+ (e0)µαβ

(ω0)νγδ

[εβγyαyδ + εβδyαyγ ]

+ (ω0)µαβ

(ω0)νγδ

[εαγ yβ yδ + εαδyβ yγ + εβγ yαyδ + εβδyαyγ ] + ...

(5.20)

43

5.5 AdS4 Background

Where ”...” denotes all terms of order 0 and 4 in the Y . Next it is helpful to organise terms oforder (2,0), (1,1) and (0,2). Then we swap a whole load of pairs of indices so that the twistorsappearing in the (2,0),(1,1),(0,2) order terms all match and then collect the terms, the end resultis:

Wµ0 ∗W

ν0 = yαyβ

[(e0)µ

αβ(e0)ν

βδεβδ + (ω0)µδα(ω0)νγβε

δγ + (ω0)µγα(ω0)νβδεγδ

+ (ω0)µαδ(ω0)νγβεδγ + (ω0)µαγ(ω0)νβδε

γδ]

+ yαyβ[(e0)µαα(e0)ν

γβεαγ + (ω0)µ

δα(ω0)ν

γβεδγ + (ω0)µγα(ω0)ν

βδεγδ

+ (ω0)µαδ

(ω0)νγβεδγ + (ω0)µαγ(ω0)ν

βδεγδ]

+ yαyβ[(ω0)µβα(e0)ν

γβεβγ + (ω0)µαβ(e0)ν

γβεβγyαyβ + (ω0)µ

αβ(e0)ν

αδεαδyαyβ

+ (ω0)µβα

(e0)ναδεαδyαyβ + (e0)µ

δβ(ω0)νγαε

δγyαyβ + (e0)µγβ

(ω0)ναδεγδyαyβ

+ (e0)µαδ

(ω0)νγβεδγyαyβ + (e0)µαγ(ω0)ν

βδεγδyαyβ

]+ ...

Now we notice that the last four terms on the top fours rows above all differ by trading δ ↔ γand or swapping α↔ β which, because Y s commute, doesn’t change their sign and hence theyare all equal. The same is true of the last four terms in the middle rows. Similarly the first andsecond, third and fourth, fifth and sixth, seventh and eigth terms on the bottom two rows differby trading the indices appearing in ε and swapping the indices of e0 and ω0, hence for the samereasons as before these pairs are also all equal and we have,

Wµ0 ∗W

ν0 = yαyβ

[(e0)µδα(e0)ν

γβεδγ + 4(ω0)µ

δα(ω0)ν

γβεδγ]

+ yαyβ[(e0)µαγ(e0)ν

βδεγδ + 4(ω0)µδα(ω0)νγβε

δγ]

+ yαyβ[2(ω0)µαδ(e0)ν

γβεδγ + 2(ω0)µ

αβ(e0)ν

αδεαδ + 2(e0)µ

δβ(ω0)νγαε

δγ + 2(e0)µαδ

(ω0)νγβεδγ]

+ ...

(5.21)Now we can use antisymmetry of ε to re-write the right-hand terms on the first two lines above,and again trading pairs of indices appearing in εs and swapping indices in e0 and ω0 we obtain,

Wµ0 ∗W

ν0 = yαyβ

[(e0)

[µγα(e0)

ν]

δβεγδ + 4(ω0)

[µγα(ω0)

ν]

δβεγδ]

+ yαyβ[(e0)

[µαγ(e0)

ν]

βδεγδ + 4(ω0)

[µδα(ω0)

ν]γβε

δγ]

+ yαyβ[8(e0)

[µ

δβ(ω0)ν]

γαεδγ − 8(e0)

[µ

αδ(ω0)

ν]

γβεδγ]

+ ...

(5.22)

= yαyβ(W0 ∗W0)µναβ + yαyβ(W0 ∗W0)µναβ

+ yαyβ(W0 ∗W0)µναβ

Where we used the fact that,

yαyβ(e0)µαα(e0)νββεαβ =

1

2yαyβ[(e0)µαα(e0)ν

ββεαβ − (e0)µαα(e0)ν

ββεβα]

=1

2yαyβ[(e0)µαα(e0)ν

ββ− (e0)µ

ββ(e0)ναα]εαβ = yαyβ(e0)

[µαα(e0)

ν]

ββεαβ

because, since ys commute, we can swap α and β indices of the (e0)s freely and then we used

antisymmetry of εαβ and then traded α ↔ β. Similar identities hold for all of the terms in(5.21). Now we simply read off the components from (5.22),

(W0 ∗W0)αβ = (e0)γα ∧ (e0)δβεγδ + 4(ω0)αγ ∧ (ω0)βδε

γδ

(W0 ∗W0)αβ = (e0)αγ ∧ (e0)βδεγδ + 4(ω0)αγ ∧ (ω0)βδε

γδ

(W0 ∗W0)αβ = 4(e0)γβ ∧ (ω0)αδεγδ − 4(e0)αγ ∧ (ω0)δβε

γδ

44

5.6 Linearised Equations of Motion

Where we used the fact that for two spacetime one-forms a and b that (a ∧ b)µν = 2a[µbν].Combining the above results with (5.18), the (2,0), (1,1) and (0,2) components of (5.17) givethe following equations which relate the vierbein to the spin-connection,

dx(e0)αβ + 4(e0)γβ ∧ (ω0)αδεγδ − 4(e0)αγ ∧ (ω0)δβε

γδ = 0

dx(ω0)αβ + (e0)αγ ∧ (e0)βδεγδ + 4(ω0)αγ ∧ (ω0)βδε

γδ = 0

dx(ω0)αβ + (e0)γα ∧ (e0)δβεγδ + 4(ω0)αγ ∧ (ω0)βδε

γδ = 0

(5.23)

Now using the relation between the spinor notation and SO(4) vector notations [3],

(e0)αβ =1

4eaσa

αβ, (ω0)αβ =

1

16ωabσabαβ, (ω0)αβ = − 1

16ωabσab

αβ

We eventually find that equations (5.23) imply,

dxea + ωab ∧ eb = 0, dxwab + ωac ∧ ωcb = 6eb ∧ ea

where the first equation sets the torsion tensor to zero and the second equation expresses theRicci tensor in terms of the vierbein [3]. These equations precisely identify the geometry ofAdS4!

5.6 Linearised Equations of Motion

The free higher-spin fields and massive scalar arise in Vasiliev theory as small fluctuations aroundthe AdS background. We can illustrate this by setting the dynamical field equal to their vacuumvalue plus a small perturbation. To extract the scalar and higher-spin equations of motion from(5.8), (5.9) it is useful for us to re-write them in terms of the fields W,S,B and then seperatethem into several more manageable equations. We expand A using the definition (5.7) so thatthe first master equation (5.8) becomes,

dxW+dxS+dZW+dZS+W∗W+S∗S+W,S∗ =[1

4+eiθ0(B∗K)

]dz2+

[1

4+e−iθ0(B∗K)

]dz2+o(B2)

Where we used the general form of f given by (5.11). Equating coefficients in dz above givesthe three equations,

dxW +W ∗W = 0

dZW + dxS + W,S∗ = 0

dZS + S ∗ S =[1

4+ eiθ0(B ∗K)

]+[1

4+ e−iθ0(B ∗ K)

]+ o(B2)

(5.24)

In terms of the compoment fields, the second master field equation (5.9) becomes,

dxB + dZB +W ∗B +1

2dzαz

α ∗B + S ∗B +1

2dzαz

α ∗B

−B ∗ π(W )−B ∗ π(S)−B ∗K ∗ 1

2dzαz

α ∗K −B ∗K ∗ 1

2dzαzα ∗K

= dxB + dZB +W ∗B + S ∗B −B ∗ π(W )−B ∗ π(S) = 0

Equating coefficients in dz then gives the two equations,

dxB +W ∗B −B ∗ π(W ) = 0

dZB + S ∗B −B ∗ π(S) = 0(5.25)

45

5.7 The Scalar Field

Now we set up the perturbation theory. Recall that in the vacuum the solutions are W = W0,S = B = 0. Now we set the fields equal to a small perturbation of order λ around this solution.The master fields are

W = W0 + λW , S = λS, B = λB

to linear order in λ, where hatted fields are the field perturbations. Plugging this into theequations of motion (5.24),(5.25) we get,

dx(W0 + λW ) + (W0 + λW ) ∗ (W0 + λW ) = λ(dxW +

W0, W

∗

)+ ... = 0

dZ(W0 + λW ) + dxλS +

(W0 + λW ), λS∗

= λ(dZW + dxS +

W0, S

∗

)+ ... = 0

dZλS + λS ∗ λS −[1

4+ eiθ0(λB ∗K)

]−[1

4e−iθ0(λB ∗ K)

]+ ... = 0

dxλB + (W0 + λW ) ∗ λB − λB ∗ π(W0 + λW ) = λ(dxB +W0 ∗ B − B ∗ π(W0)) + ... = 0

dZλB + λS ∗ λB − λB ∗ π(λS) = λ(dZB) + ... = 0

where ”...” corresponds to other terms not linear in λ. Now simply equating coefficients in λ wefind the linearised equations of motion,

D0W = 0,

dZW +D0S = 0,

dZ S = eiθ0(B ∗K) + e−iθ0(B ∗ K),

D0B = 0,

dZB = 0.

(5.26)

where we have defined D0 = dx + [W0, ·]∗ and D0 = dx + W0 ∗ · − · ∗ π(W0). Now we can usethese equations to solve for the components of the master fields and demonstrate the we obtainthe equations describing a massive scalar and a tower of higher-spin fields in AdS4. We knowthat the components of the master fields are related to these fields, so the general strategy willbe to analyse the appropriate linearised equations of (5.26) to tease out the equations for thecomponent fields.


The Scalar field is, as we have described, given by the pure-space component of the master fieldB at Z = 0. To extract the equation for the scalar field it then seems appropriate to look at thefourth line of (5.26) and set Z = 0 [3]. Doing this we obtain,

D0B(x|Y,Z)|Z=0 = D0C(x|Y ) = dxC(x|Y ) + [ω0, C] + eo, C∗

which in spacetime components reads,

(D0C)µ = ∂µ + [(ω0)µ, C]∗ +

(e0)µαβyαyβ, C

∗

= ∇µLC +

(e0)µαβyαyβ, C

∗

(5.27)

where we have expanded both the vierbein and C(x|Y ) in Y . The general form of the above staranticommutator is worked out in the appendix, (A.12). Using this equation (5.27) becomes,

∇µLC(x|Y ) + 2(e0)µαβyαyβC(x|Y ) + 2(e0)µ

αβ∂α∂βC(x|Y ) = 0 (5.28)

Equating coefficients in Y above, we get an infinite tower of equations relating the components ofC. Now multiplying by yαyα an arbitrary function increases the degree in y, y by one. Similarlyacting on an arbitrary function with ∂α∂α reduces the degree in y, y by one. We can also show

46


that ∇L does not change the degree in y or y because using the identities (A.15), (A.16) provedin the appendix we find that,

[(ω0), A(Y )]∗ = 2[(ω0)αβ(yα∂β + yβ∂α) + (ω0)αβ(yα∂β + yβ∂α)]A(Y )

for some A(Y ) so that for each degree in y or y killed by a derivative, another factor is insertedand so the overall degree in y and y is not changed as claimed. Therefore by equating coefficientsin ymym (5.28) reads, in components,

∇µLC(n,m)(x|Y ) + 2(e0)µ

αβyαyβC(n−1,m−1)(x|Y ) + 2(e0)µ

αβ∂α∂βC(n+1,m+1)(x|Y ) = 0 (5.29)

So (5.28) relates components with n−m fixed. To find the second order equation for C(0,0) it issufficient to consider the (0,0) and (1,1) components of (5.29) which using (5.29) are given by,

∂µC(0,0)(x) + 2(e0)µαβ∂α∂βC(1,1)(x|Y ) = 0 (5.30)

∇µLC(1,1)(x|Y ) + 2(e0)µ

αβyαyβC(0,0)(x) + 2(e0)µ

αβ∂α∂βC(2,2)(x|Y ) = 0 (5.31)

where we used the fact that ∇LC(0,0) = ∂C(0,0) since the star commutator [(ω0), C(0,0)(x)]vanishes. Now we have two equations for C(0,0). We start by using (5.30) to relate C(0,0) toC(1,1). Expanding equation (5.30) in Y we find,

∂µC(0,0)(x) + 2(e0)µαβC

(1,1)γγ (x)∂α∂βyγ yγ = ∂µC

(0,0)(x) + 2(e0)µαβC

(1,1)γγ (x)εαbεβb∂b∂by

γ yγ

= ∂µC(0,0)(x) + 2(e0)µ

αβεαbεβbC

(1,1)γγ (x)δγb δ

γ

b= ∂µC

(0,0)(x) + 2(e0)µαβεαbεβbC

(1,1)

bb(x) = 0⇔

∂µC(0,0)(x) = −2(e0)µ,αβC(1,1)

αβ(x)

We can invert the above relation by contracting with (e0)µ,γδ. We obtain,

(e0)µ,γδ∂µC(0,0)(x) = −2(e0)µ,γδ(e0)µ,αβC

(1,1)

αβ(x) = −2(−1

8δαγ δ

β

δ)C

(1,1)

αβ(x)⇔

C(1,1)

αβ(x) = 4(e0)µ

αβ∂µC

(0,0)(x) (5.32)

We will substitute this relation into (5.31) to eliminate C(1,1). We then get an equation for

C(0,0) only by eliminating C(2,2). In particular, contracting (5.31) with (e0)αβµ we get,

(e0)αβµ ∇µLC

(1,1)(x|Y ) + 2(e0)αβµ (e0)µαβyαyβC(0,0)(x) + 2(e0)αβµ (e0)µ

αβ∂α∂βC(2,2)(x|Y )

=(e0)αβµ ∇µLC

(1,1)(x|Y )− 1

4yαyβC(0,0)(x)− 1

4C

(2,2)

γδγδ(x)εαbεβb∂b∂b(y

γyδyγ yδ)

=(e0)αβµ ∇µLC

(1,1)(x|Y )− 1

4yαyβC(0,0)(x)− yγ yδC

(2,2)αγβδ(x)

Now we can kill C(2,2) by acting with ∂α∂β on the above, we get,

(e0)αβµ ∂α∂β∇µLC

(1,1)(x|Y )− 1

4δααδ

β

βC(0,0)(x)− C(2,2)αγβδ(x)∂α∂βyγ yδ


(1,1)(x|Y )− C(0,0)(x)− C(2,2)αγβδ(x)εαγεβδ


(1,1)(x|Y )− C(0,0)(x)− C(2,2)αγββ(x)


(1,1)

γδ(x)yγ yδ − C(0,0)(x) = 0

47

5.8 The Higher-Spin Fields

Finally plugging in the relation (5.32) into the above we obtain an equation for C(0,0) only [3]:

4(e0)αβµ ∂α∂β∇µL[(e0)ν

γδyγ yδ∂νC

(0,0)(x)]− C(0,0)(x)

= 4(e0)αβµ (e0)νγδ∂α∂β(yγ yδ)∂µ∂νC

(0,0)(x)

+ 4(e0)αβµ ∂α∂β[(ω0)µαβyαyβ + (ω0)µ

αβyαyβ, (e0)ν

γδyγ yδ∂νC

(0,0)(x)]∗ − C(0,0)(x)

= 4(e0)αβµ (e0)νγδ

(δγαδδβ)∂µ∂νC

(0,0)(x)

+ 4(e0)αβµ (ω0)µαβ∂α∂β(e0)νγδ

[yαyβ, yγ yδ]∗∂νC(0,0)(x)

+ 4(e0)αβµ (ω0)µαβ

(e0)νγδ∂α∂β[yαyβ, yγ yδ]∗∂νC

(0,0)(x)− C(0,0)(x)

The above star-commutators are evaluated in the appendix (A.13), (A.14). Continuing we get[3],

= 4(e0)αβµ (e0)ναβ∂µ∂νC

(0,0)(x)

+ 4(e0)αβµ (ω0)µαβ(e0)νγδ∂α∂β[2εαγyβ yδ + 2εβγyαyδ]∂νC

(0,0)(x)

+ 4(e0)αβµ (ω0)µαβ

(e0)νγδ∂α∂β[2εαδyγ yβ + 2εβδyγ yα]∂νC

(0,0)(x)− C(0,0)(x)

= −1

2∂ν∂νC

(0,0)(x)

+ 8(e0)αβµ [(ω0)µαβ(e0)νγβεβγ + (ω0)µ

αβ(e0)ν

αδεαδ]∂νC

(0,0)(x)− C(0,0)(x)

= −1

2[(∂ν − 16(e0)αβµ [(ω0)µαβ(e0)ν

γβεβγ + (ω0)µ

αβ(e0)ν

αδ])∂νC

(0,0)(x) + 2C(0,0)(x)] ⇔

= −1

2(∇µ∂µ + 2)C(0,0)(x) = 0 (5.33)

But this is exactly the Klein-Gordon equation for a scalar field of negative mass-squaredm2 = −2in AdS4!


Finding the spin-s equations of motion requires a lot more work than was needed to extract thescalar field equation! Unfortunately we will not arrive at the final result, but we will at leastdemonstrate that one recovers the equations for the spin-1 field in AdS4 and we will describehow, in principle, one can recover Fronsdal’s equations in AdS4 from the master equations, asit outlined in [3].

The starting point is to recursively solve for the linearised fields in terms of one another. Wecan solve for S from the 3rd line of (5.26). We plug the result into the 2nd line to solve for W .We can then obtain an equation relating the components of Ω, from which one can extract theFronsdal equations in AdS4.

The third line of (5.26) reads,

dZS = ∂[αSβ]dzαdzβ + ∂[αSβ]dz

αdzβ = eiθ0(B ∗K)dzαdzα + e−iθ0(B ∗ K)dzαdzα

Equating coefficients in dzα gives,

1

2[∂αSβ − ∂βSα]dzβ = eiθ0(B ∗K)dzα

Next using dzβ

dzα= εβγ

dzγdzα

= εβα the above result implies that,

1

2[∂αSβ − ∂βSα]

dzβ

dzα=

1

2[∂αSβ − ∂βSα]εβα = ∂αS

α = eiθ0(B ∗K)

48


Now the general solution of the spinor equation ∂αfα(z) = g(z) is given by [29],

fα(z) = ∂αε(z) +

∫ 1

0dt tzα g(tz)

This gives the following solution for Sα,

Sα = zαeiθ0∫ 1

0dt t(C(x|y, y) ∗K)|z→tz = zαeiθ0

∫ 1

0dt t C(x| − tz, y)K(t)

where we ignored the total derivative that appears in the identity because it can be set to zeroby a gauge transformatin [29] and we also used the Kleinian identity (5.6) to see that the ydependence of C is traded for z dependence. In particular note that C(x| − tz, y) does notdepend on y. Using the above result we find that,

S = Sαdzα + c.c. = dzαzαeiθ0

∫ 1

0dt t C(x| − tz, y)K(t) + c.c. (5.34)

Now we can use the 2nd line of (5.26) that dZW = dZ(W0 +W ′) = dZW′ = −D0S to solve

for W ′ by integrating in z,z and plugging in (5.34) for S. We begin with,

dZW′ = (dzα∂

α + dzα∂α)W ′ = −D0S = −D0S

αdzα +−D0Sαdzα ⇔

∂αW′ = −D0Sα (5.35)

Now we require the identity that the solution to the spinor equation ∂αf(z) = gα(z) is,

f(z) = C +

∫ 1

0dt zαgα(tz)

for some constant C [29]. Applying this to (5.35) gives the solution for W ′,

W ′(x|Y,Z) = −zα∫ 1

0dt (D0Sα)|z→tz + c.c.

Now plugging in (5.34) and using the definition of D0 we get,

W ′ = −eiθ0zα∫ 1

0dt

∫ 1

0dt′ t′ [W0, zαC(x| − t′z, y)K(t′)]∗|z→tz + c.c. (5.36)

Now we use the identity [A, zαB]∗ = [∂αA,B]∗ [4] so that the above star-commutator becomes,

[∂αW0, C(x| − t′z, y)K(t′)]∗ = [(e0)αβ yβ + (ω0)αβy

β, C(x| − t′z, y)K(t′)]∗

where we recalled the definition (5.16) of W0. Using this result (5.36) becomes,

W ′ = −eiθ0zα∫ 1

0dt

∫ 1

0dt′ t′[(e0)αβ y

β + (ω0)αβyβ, C(x| − t′z, y)K(t′)]∗|z→tz + c.c. (5.37)

Now we need to compute the two star commutators that appear above. We can use theidentities (A.10 - A.11) proved in the appendix that,

[yα, A(y, z)]∗ = 2∂αA(y, z), [yα, A(y, z)]∗ = 2∂αA(y, z)

Using the above results, (5.37) gives,

W ′ = −2eiθ0zα[(e0)αβ∂β + (ω0)αβ∂

β]

∫ 1

0dt

∫ 1

0dt′ t′C(x| − tt′z, y)K(tt′) + c.c.

49


where we took the t,t′ independent quantities out of the integrals and we also let z → tz. Theintegral above is computed in [4] and we get,

W ′ = 2eiθ0zα[(e0)αβ∂β + (ω0)αβ∂

β]

∫ 1

0dt (1− t) C(x| − tz, y)K(t) + c.c.

The quantities in front can go back under the integral, the ∂β is non-zero only when acting onC, whereas the ∂β is non-zero only when acting on K (since C(−tz, y) does not depend on y)where is brings down a factor of tzβ so we get,

W ′ = 2eiθ0zα∫ 1

0dt (1− t) [(e0)αβ∂

β + (ω0)αβtzβ]C(x| − tz, y)K(t) + c.c.

Now that we have solved for W ′ we can plug this into the first equation of (5.26), which gives[3],

D0Ω = −W0,W

′∗|Z=0

= 2eiθ0∫ 1

0dt (1− t)

W0, z

α[(e0)αβ∂β + (ω0)αβtz

β]C(x| − tz, y)K(t)∗|Z=0 + c.c.

= 2eiθ0∫ 1

0dt (1− t)

[∂αW0, [(e0)αβ∂

β + (ω0)αβtzβ]C(x| − tz, y)K(t)

]|Z=0 + c.c.

= 4eiθ0∫ 1

0dt (1− t)εγα[(e0)γδ∂

δ + (ω0)γδtzδ] ∧ [(e0)αβ∂

β + (ω0)αβtzβ]C(x| − tz, y)K(t)|Z=0 + c.c.

= 4eiθ0∫ 1

0dt (1− t)εαγ(e0)γδ∂

δ ∧ (e0)αβ∂βC(x|0, y) + c.c.

where in the fourth line we evaluates at Z = 0. Setting Z = 0, the last line above no longerdepends on t, so the integral just equals 1

2 . The end result is that,

D0Ω = 2eiθ0εαγ(e0)αβ ∧ (e0)γ

δ∂β∂δC(x|0, y) + 2e−iθ0εαγ(e0)β α ∧ (e0)δ γ∂β∂δC(x|y, 0) (5.38)

Now (5.38) agrees exactly with the Central On-Mass Theorem [29] and implies that these are ofthe correct form of the free higher-spin equations in AdS4, written in terms of the generalisedhigher-spin curvatures. This verifies that at the linearised level, Vasiliev theory describes aninfinite tower of higher-spin gauge fields propagating in AdS4! Now we will look at the simplecase of the spin-1 equation as a basic proof of principle. The left-hand side of (5.38) can beexpanded,

D0Ω = dLΩ + (e0),Ω∗ = dLΩ + 2(e0)αβ ∧ [∂αyβΩ + yα∂βΩ]

We may extract the equation for the spin-1 field via the equation,

dxΩ(0,0) = 2eiθ0εαγ(e0)αβ∧(e0)γ

δ∂β∂δC(0,2)(x|0, y)+2e−iθ0εαγ(e0)β α∧(e0)δ γ∂β∂δC

(2,0)(x|y, 0)

Where we used the fact that Ω(0,0)(x) does not depend on y, so its star commutator with (ω0),appearing in dL, vanishes. In component the above result reads,

(∂µΩν)dxµ ∧ dxν = [2eiθ0εαγ(e0)µ,αβ(e0)ν,γ

δ∂β∂δC(0,2)(x|0, y)

+ 2e−iθ0εαγ(e0)β µ,α(e0)δ ν,γ∂β∂δC(2,0)(x|y, 0)]dxµ ∧ dxν

From which it follows that,

∂µΩν − ∂νΩµ = 2eiθ0(e0)[µ,αβ(e0)αδν] ∂β∂δC

(0,2)(x|0, y) + 2e−iθ0(e0)β [µ,α(e0)δαν] ∂β∂δC(2,0)(x|y, 0)

(5.39)

50


Where the antisymmetrisation is over spacetime indices only. To proceed we need to go back to(5.29) which we can use to solve for C(2,0). We have that,

∇µLC(2,0)(x|Y ) + 2(e0)µ

αβ∂α∂βC(3,1)(x|Y ) = ∇µLC

(2,0)(x|Y ) + 6(e0)µαβC(3,1)(x)abαβyayb = 0

(5.40)where we expanded C(3,1)(x|Y ) in Y and evaluated the Y derivatives. Acting on the last term

on the right appearing of the middle expression above with (e0)γδµ ∂γ gives,

6(e0)γδµ (e0)µαβ∂γC

(3,1)(x)abαβyayb = 6(−1

8δγαδ

δβ)[εγaC

(3,1)(x)abαβyb + εγbC(3,1)(x)aγαβya]

=3

2C(3,1)(x)α

bαδyb = 0

The last expression vanishes because C is traceless, therefore acting on (5.40) with (e0)γδµ ∂γgives,

(e0)αβµ ∂α∇µLC(2,0)(x|Y ) = 0

Using this result and its complex conjugate, if we now act on (5.39) with ∇µL we get that,

∇µL(∂µΩν − ∂νΩµ) = ∂2Ων − ∂ν∂µΩµ = 0

which gives (4.17) for s = 1, i.e it is exactly the equation for a free spin-1 field in AdS4! Thespin-1 case is the simplest case of a field equation contained within Vasiliev’s system, but wehave an explicit proof of principle!

Now we will outline how, in general, one can go on and derive the full Fronsdal equation inAdS4 from Vasiliev’s system. However the computation gets extremely messy and unfortunatelywe did not have time to make contact with the final result in these proceedings. Let’s look atthe (s − 1, s − 1) term above, we that this component of the right hand-side of (5.38) vanishesfor s > 1 in general because C(x|0, y)(s−1,s−1) = C(x|y, 0)(s−1,s−1) = 0, since they respectivelydepend only on y and y. With the help of the above result then, the (s − 1, s − 1) componentof (5.38) reads,

dLΩ(s−1,s−1) + 2(e0)αβ ∧ [∂αyβΩ(s,s−2) + yα∂βΩ(s−2,s)] = 0

In fact we have,

dLΩ(s,s−2) + 2(e0)αβ ∧ [∂αyβΩ(s+1,s−3) + yα∂βΩ(s−1,s−1)] = 0

dLΩ(s−2,s) + 2(e0)αβ ∧ [∂αyβΩ(s−1,s−1) + yα∂βΩ(s−3,s+1)] = 0

It is easy to check that the above relations follow generally from [4],

dLΩn + 2(e0)αβ ∧ [∂αyβΩn−1 + yα∂βΩn+1] = 0 (5.41)

with n = 0,±1 only. Now introducing the following definitions,

Ωn++ = yαyβΩαβ, Ωn

−+ = ∂αyβΩαβ, Ωn+− = yα∂βΩαβ, Ωn

−− = ∂α∂βΩαβ

where Ωnαα = (e0)µααΩn

µ. These allow us to decompose Ωnαβ

as follows [4],

(s2 − n2)Ωnαβ

= (∂α∂βΩn++ − yα∂βΩn

+− − ∂αyβΩn+− − yαyβΩn

−−) (5.42)

In [3] it is seen that we may use (5.41) and (5.42) to deduce the following relations,

Ω1++ =

4

syαyβ(∇L)α

γΩ0βγ , Ω1

−− = −4

s∂α∂β(∇L)γ αΩ0

γβ,

Ω1−+ = −2

s[(s− 1)yα∂β(∇L)α

γΩ0βγ + (s+ 1)yα∂β(∇L)γ αΩ0

γβ]

51


for Ω1±±, and the further relations,

Ω0++ =

4

s− 1yαyβ(∇L)α αΩ1

αβ, Ω0

−− = − 4

s+ 1∂α∂β(∇L)β

βΩ1αβ

for Ω0±±, where we have defined (∇L)αα = (e0)µαα(∇L)µ. It is found in [3] that we can gauge

away Ω0±∓ and Ω1

+− with a Z-independent gauge transformation. Now using (5.42) and theabove relations we find that,

s2Ω0AB

= (∂A∂BΩ0++ − yAyBΩ0

−−)

=(∂A∂B

4

s− 1yαyβ(∇L)α α + yAyB

4

s+ 1∂α∂β(∇L)β

β)

Ω1αβ

=(∂A∂B

4


4

s+ 1∂α∂β(∇L)β

β)

· 1

(s2 − 1)(∂α∂βΩ1

++ − yα∂βΩ1−+ − yαyβΩ1

−−)

=1

(s2 − 1)

(∂A∂B

4


4

s+ 1∂α∂β(∇L)β

β)

·(∂α∂β

4

syayb(∇L)b

b + ∂αyβ2

s

[(s− 1)yb∂a(∇L)b

b + (s+ 1)ya∂ b(∇L)a a

]+ yαyβ

4

s∂a∂ b(∇L)a a

)Ω0ab

⇔1

16s3(s2 − 1)Ω0

AB=(∂A∂B

1

(s− 1)yαyβ(∇L)α α + yAyB

1

(s+ 1)∂α∂β(∇L)β

β)

·(∂α∂βy

ayb(∇L)bb + ∂αyβ

1

2

[(s− 1)yb∂a(∇L)b

b + (s+ 1)ya∂ b(∇L)a a

]+ yαyβ∂

a∂ b(∇L)a a

)Ω0ab

The end result is a very messy looking second order equation for Ω0 only! Unfortunately itcould not be seen how to cleanly simplify this result and there was not time to perform the longcomputation that we appear to face. Suffice it to say that extracting the corresponding equationfor the spin-s field contained within Ω0 gives Fronsdal’s equations in AdS4 [3]

Key Points:

• Vasiliev theory comprises a set of non-linear equations of motion for a set of master fieldson spacetime and twistor space.

• The noncommutative twistors allow the master fields to encode an infinite tower of higher-spin degree of freedom.

• The spacetime one-form contains the higher-spin fields and higher-spin connections. Thescalar contains a massive scalar and the higher-spin curvatures. The auxilliary twistorone-form is convenient for writing the equations of motion.

• The maximally symmetric solution for Vasilievs equations involves only the componentsof the spacetime one-form and describes an AdS4 background.

• From the linearised Vasiliev equations one can extract the equations for the free propaga-tion of a massive scalar of mass m2 = −2 and an infinite tower of higher-spin gauge fieldsin AdS4.

52

6 Interactions in Vasiliev Theory: The Holographic 3-point Function

Our discussion now turns to the machinery of the AdS/CFT correspondence and in particularhow to compute holographic correlation functions. We start with a general case of the com-putation of the scalar three-point function via the classical action. We will then demonstratehow one can obtain the same result up to normalization factors by using only the equations ofmotion. The procedure is generic and we will apply this to the computation of the higher-spinthree point function in Vasiliev theory, where we do not have an action but we do have theequations of motion.

6.1 Overview of AdS/CFT and Holographic Three-Point Correlators

The AdS/CFT correspondence is a remarkably beautiful result which, to date, is the mostpoignant realisation of the holographic principle: that any theory of quantum gravity in a givenambient spacetime, referred to as the “bulk” by convention, can be described entirely in termsof the degrees of freedom at the boundary, what are called the “boundary data” [31]. TheAdS/CFT duality implies more generally that a theory of gravity in the bulk AdS space isholographically dual to a CFT living on the boundary [36]. Vasiliev theory is a theory ofclassical gravity in AdS4, so we anticipate that the holographic dual is some 3D-CFT.

The duality is clearly much more than a pretty face! indeed in its original form, the conjecturerelates a strongly coupled SYM theory with a weakly coupled string theory [37]. Assuming thatthe conjecture holds for all physical regimes, an as-yet unproved result, then this potentially offersus a means to elucidate the dynamics of one or the other theory in a regime where perturbativemethods fail. What is important to us is that the duality allows us to compute correlationfunctions of fields in the bulk by evaluating correlation functions of their dual operators in theCFT , ”holographic” correlation functions. We will exploit this in order to compute a higher-spincorrelation function in Vasiliev theory. This, and the next section will introduce the machinerywe need to do this.

The dynamical objects related by the AdS/CFT correspondence are the fields in the bulkand primary operators in the boundary CFT . The objects that ”bridge the gap” betweenthe bulk and boundary theories are the boundary fields. These are the the non-renomalizablesolutions of the bulk equations of motion that do not decay at the boundary. Indeed, as wewill shortly see, one finds that fields in the bulk are solved completely in terms of the boundaryfields. Conversely the presence of the bulk field φ(~x, z) with boundary field φ0, corresponds tothe source term

∫dd ~xOφ0 in the CFT action. In other words, the boundary field φ0 of the

bulk field φ is a source for the primary operator O in the CFT. Therefore we have a one-to-onecorrespondence between fields in the bulk and primary operators in the boundary CFT [38].

The dynamical statement of the AdS/CFT correspondence is that the bulk action of thefields evaluated on-shell generates correlation functions in the boundary CFT .

Schematically we express this via the relation,

ZCFT [O] = SBulk(φ) (6.1)

where ZCFT is the generating functional for correlation functions in the boundary CFT andwhere SBulk is the bulk action evaluated on-shell [39]. This means that by functionally differen-tiating (6.1) with respect to the boundary sources and setting to zero one obtains the correlationfunctions. In this sense the on-shell bulk action comprises the “boundary-data” for the CFT.

Let us consider a bulk scalar field φ∆(~x, z) with conformal dimension ∆. Here we point outthat we will work with Poincare coordinates (~x, z) in AdS, for which the boundary lives at z = 0.The free equations of motion are (∇2 −m2)φ∆ = 0. Recall that the Euclidean AdS metric isgiven in Poincare coordinates by,

ds2 =1

z2(dz2 + d~x2)

53


where we have set the AdS radius to unity [31]. Now evaluating the scalar Laplacian in Poincarecoordinates, the free equations read [39],

(1√−g

∂µ[√−g∂µ]−m2)φ∆ = (z2∂2

z − (d− 1)z∂z + z2∂2i −m2)φ∆ = 0 (6.2)

The plane wave Anzatz φ(~x, z) = φ(z)ei~k·~x then implies that,

(z2∂2z − (d− 1)z∂z − z2k2 −m2)φ(z) = 0

Since, as we shall see, the boundary value of φ is the vev of the dual operator of conformaldimension ∆, then near z = 0 we may assume that the solution behaves like φ(z) = z∆ andnoting that z2k2 is negligible this ansatz gives [31],

z2∆(∆− 1)z∆−2 − (d− 1)z∆z∆−1 −m2z∆ = 0 ⇔ ∆(∆− d) = m2 (6.3)

This illustrates the general fact that the equations of motion imply a relation between the mass ofthe bulk field and conformal dimension of the CFT dual operator [39]. The conformal dimensiontakes on one of two possible values ∆± and so the bulk field has the following behaviour nearz = 0:

φ∆(~x, z → 0) = φ1(~x)z∆+ + φ0(~x)z∆− , ∆± =d

2±√d2

4+m2R2 (6.4)

A solution of (6.2) is renormalisable if the action is finite as z → 0 [37]. Simply plugging in the

ansatz φ(~x, z) = z∆ei~k·~x into the action (4.15) we eventually get,

S = −η2

∫dd~xdz e2i~x·~k(∆2 + k2z2 +m2R2)z2∆−d−1

In the limit z → 0 the resulting action converges for 2∆− d− 1 > −1 or equivalenty if ∆ > d2 .

We have that ∆+ > d2 , so the solution corresponding to φ1(~x) is renormalisable and decays at

the boundary. However ∆− <d2 , so the solution corresponding to φ0(~x) is non-renormalisable;

it is non-trivial at the boundary, so we define it to be the boundary field:

φ0(~x) = limz→0

z−∆−φ∆(z, ~x)

and conversely we define the boundary value of φ∆(z, ~x) to be the renormalisable solution [31],

φ1(~x) = limz→0

z−∆+φ∆(z, ~x) = 0 (6.5)

As we will soon see, this is the vev (or one-point function) of the dual operator which of coursemust vanish by conformal invariance. One should not confuse the boundary field with theboundary value of the bulk field. A general solution of (6.2) with the correct behaviour nearz = 0 is given by [31],

φ∆(~x, z) =

∫dd~y K∆(~x, z; ~y)φ0(~y) (6.6)

where φ0(~y) is the boundary field that sources φ∆ in the bulk and K∆ is given by,

K∆(~x, z; ~y) = C∆

( z

z2 + (~x− ~y)2

)∆, C∆ =

Γ(∆)

πd2 Γ(∆− d

2)(6.7)

The object K∆ satisfies the bulk equations of motion for φ [38]. Therefore via (6.6), K projectsboundary fields to solutions of the bulk equations of motion, i.e to fields in the bulk [40]. Itis called the bulk-boundary propagator for φ. In particular we observe that the bulk field is

54


Figure 4: Illustration of a bulk field φ(~x, z) which is sourced at the boundary by φ0(~y) [31]. Theboundary of the disc represents the boundary of AdS at z = 0 and the interior represents thebulk AdS space.

determined entirely by the boundary field! We can schematically represent the propagation ofthe bulk field from the boundary via a Witten type diagram (see figure 4).

Using (6.6) and the definition (6.7) we see that near z = 0, the bulk field behaves like [31],

φ∆(~x, z → 0) ∼ zd−∆φ0(~x) + z∆C∆

∫dd~y

φ0(~y)

(~x− ~y)2∆= zd−∆φ0(~x) + z∆φ1(~x) (6.8)

so as required it behaves like (6.4) with,

φ1(~x) = C∆

∫dd~y

φ0(~y)

(~x− ~y)2∆, φ0(~x) = lim

z→0z∆−dφ∆(z, ~x) (6.9)

Let’s look again at the action (4.15) evaluated on-shell, we can integrate by parts to get,

S = −η2

∫√gdd~xdz − (∇2 −m2)φ+ z2∂µ(φ∂µφ) + z2∂z(φ∂zφ) = −η

2

∫√gdd~x z2(φ∂zφ)

In the middle above, the first term vanishes on-shell, the second is a total derivative in thedirections orthogonal to the boundary and these vanish as usual, however the last term gives anintegral over the boundary and gives the term shown on the right above. This illustrates nicelyhow the on-shell bulk action encodes information at the boundary of AdS. Plugging in (6.8)into the above action and using (6.9) we find that is infinite! This is clearly not a good actionprinciple, so we seek to regularise the result and add counter terms to give the finite part. Thefinite part of the above on-shell action turns out to be [31],

S =1

2η(d− 2∆)C∆

∫dd~x φ0(~x)φ1(~x) (6.10)

According to (6.1) the one-point function, which is the vev of the operator O dual to the bulkfield φ∆(~x, z) is,

〈O(~x)〉 =δS

δφ0(~x)|φ0=0 =

1

2η(d− 2∆)C∆

∫dd~x φ1(~y)δd(~x− ~y) ∼ φ1(~x) = 0.

Therefore φ1(~x) is the vev of the dual operator as we claimed! Substituting (6.9) into (6.10) weget,

S =1

2η(d− 2∆)C∆

∫dd~xdd~y

φ0(~x)φ0(~y)

(~x− ~y)2∆

55

6.2 The Scalar 3-point Function.

So for example the two point function is,

〈O(~x)O(~y)〉 =δ

δφ0(~x)

δ

δφ0(~y)S|φ0=0 ∼

1

(~x− ~y)2∆

as is consistent with the requirement of conformal invariance! At this point we have not men-tioned anything about conformal symmetry, so this result is a nice check that the AdS/CFTmachinery is working! If we try to compute the three point function, there aren’t enough powersof φ0 in the action to give a non-trivial result. This is simply because we have so far consideredonly the action for a free scalar, where there are no interactions!


Now we will illustrate the computation of the holographic 3-point function using a simple toymodel for a massive scalar field of conformal dimension ∆ in AdSd+1 with a non-trivial 3 vertexand coupling constant λ [39]. The bulk action for this theory is,

S =

∫√gdd~xdz

1

2∇φ∇φ− 1

2m2φ2 +

λ

3!φ3 (6.11)

Integrating by parts we get,

S =

∫√gdd~xdz φ(∇2 −m2)φ− ∂(φ∂φ) +

λ

3!φ3

As usual we are treating the interaction term as a small perturbation around the free system,for which the free equations are (∇2 −m2)φ = 0. On-shell the first term vanishes, as we sawabove the second term is the boundary term that contributes to the one and two point functionsonly, therefore the non-trivial contribution to the three-point function comes from the last term.Plugging in the form of the solution (6.6) to the free equations, the on-shell action is,

S = ...+ λ

∫√gdd~x′dz dd~x dd~y dd~z K∆(~x′, z; ~x)K∆(~x′, z; ~y)K∆(~x′, z; ~y) φ(~x)φ(~y)φ(~z)

where ”...” denotes the boundary term and from here onwards we denote the boundary field byφ(~x). Therefore the three-point function for the scalar field is,

〈O(~x)O(~y)O(~z)〉 = − δ

δφ(~x)

δ

δφ(~y)

δ

δφ(~z)S|φ=0

= λ

∫√gdd~x′dz K∆(~x′, z; ~x)K∆(~x′, z; ~y)K∆(~x′, z; ~y)

(6.12)

It is not hard to convince oneself that the general result is that the contribution to the N -point function is an integral in the bulk of the N -vertex evaluated on-shell. Therefore it followsthat the N -point function is an integral in the bulk of the bulk-boundary propagators of thefields appearing in the N -vertex. It is instructive to represent holographic correlation functionspictorially with the modified Feynman diagrams due to Witten, called Witten diagrams [37]. Thebulk is represented by a disk, and the boundary is represented by its edge. Lines etirely withinin the disc represent bulk-propagators (the usual propagator factors) whereas lines connectedto the boundary represent bulk-boundary propagators. Vertices in the disk carry a couplingconstant and a bulk integral as is usual for Feynmann diagrams. Figure 5 illustrates the Wittendiagram for the tree-level scalar 3-point function.

This also illustrates that correlation functions of primary operators in the CFT are dual tothe correlation functions of fields in the bulk! This justifies the name “holographic correlationfunction”.

56


Figure 5: The 3-Point Witten Diagram for the scalar cubic vertex

We now have a general procedure for computing three-point functions. We can simplyidentify the interaction vertices in the action and integrate the appropriate bulk-to-boundarypropagators in the bulk. However, the above method appear to be of limited value to us, sincein Vasiliev theory we do not have an action! What we want is a method to generate (6.12) usingthe equations of motion only.

Using only the equations of motion we can solve for perturbative corrections to the bulk field.However we can then use these to solve for perturbative corrections to the boundary value ofthe field, but these are just corrections to the one-point function that are sourced by increasingorders of boundary fields. Schematically we have,

〈O〉 = 〈O〉0(φ) + ε〈O〉1(φ2) + ...+ +εN−2〈O〉N−2(φN−1) + ...

where ε is small, O is the dual operator to φ, 〈O〉n is the nth correction to the one-pointfunction 〈O〉. Of course, if there are no-interactions then the higher-order corrections to the vevall vanish! So, for instance with a non-trivial three-vertex, the 1st order correction to the one-point function is quadratic in the boundary fields, so twice functionally differentiating 〈O〉 andsetting the boundary fields to zero gives a non-trivial three point function! This also illustratesthat in general if we want to compute the N -point function via the equations of motion weshould solve for the correction to 〈O〉 that gives a non-trivial result when we differentiate N − 1times. As can be seen above, this is the (N − 2)th correction [4].

To see how this works for the three-point function in our toy model, let suppose that wehave only the equations of motion,

(∇2 −m2)φ =λ

2φ2

Now we set about trying to solve this equation perturbatively. Defining the perturbed fieldφ = φ0 + εφ1 + ε2φ2 + ... for ε << 1 and plugging this into the equations of motion this gives,

(∂2 −m2)(φ0 + εφ1 + ε2φ2 + ...) =λ(ε)

2(φ0 + εφ1 + ε2φ2 + ...)2 (6.13)

Where we have additionally assumed that λ ∼ ε. Now matching coefficients in ε0 we find thefree-equations of motion for the field φ0,

(∇2 −m2)φ0 = 0

The solution takes the form

φ0(~x, z) =

∫dd~y K∆(~x, z; ~y)φ(~y) (6.14)

where K∆ is given by (6.7) and φ(~x) is the boundary field which sources φ0(~x, z). To linearorder in λ (6.13) implies that,

(∇2 −m2)φ1 = λφ20

57

6.3 Setting Up The Computation.

So, as we claimed, the next-to-leading order field is sourced by terms quadratic in the free field.This tells us that there is a non-trivial three-point vertex. We solve the above using the standardmethod of integrating with the bulk Green’s function,

φ1(~x, z) = λ

∫√gdd~ydw G∆(~x, z; ~y, w)φ2

0(~y, w) (6.15)

with (∂2−m2)G∆(z, ~z;w, ~w) = δd(~z− ~w)δ(z−w). Now we are interested in the boundary valueof φ1(~x, z). To compute this we need the relation between the value of the bulk-propagator whenone of its arguments tends to the boundary and the boundary-to-bulk propagator [4],

K∆(~x, z; ~y) = limw→0

w−∆+G∆(~x, z; ~y, w)

Using this relation, (6.15) and the definition (6.5), we find the boundary value of the φ1 is,

φ1(~x) := limz→0

z−∆+φ1(~x, z) = λ

∫√gdd~ydz K∆(~y, z; ~x)φ2

0(~y, z) (6.16)

where we defined φ1(~x) as the boundary value of φ1(~x, z) according to (6.5). Now plugging inthe solution (6.14) this becomes,

φ1(~x) = λ

∫√gdd~x′dz dd~udd~v K∆(~x′, z; ~x)K∆(~x′, z; ~u)K∆(~x′, z;~v) φ(~u)φ(~v) (6.17)

We can now get the three vertex by functional differentiation with respect to the boundarysource φ and then setting this to zero at the end. It is easy to see that we must get,

[δ

δφ(~x)

δ

δφ(~y)φ1(~z)]|φ=0 = 2λ

∫dd~x′dz K∆(~x′, z; ~x)K∆(~x′, z; ~y)K∆(~x′, z;~z) ∝ 〈O(~x)O(~y)O(~z)〉

We get the three-point function (6.12) up to the overall normalisation and coupling constant!We see that this result is the same as if we had simply obtained the boundary value of the φ1

and then substituted the boundary-bulk propagators for φ0 into (6.16), because we functionallydifferentiate with respect to the boundary sources at the end, which kills them and the associatedboundary integrals anyway.

This procedure to determine the correlation functions from the equations of motion is generic.The required data specific to a given theory are the next-to-leading order equations of motionfor the outcoming field and the boundary-bulk propagators for the fields.

Now that we know how to discern the interaction vertices from the equations of motion only,one could apply this to Vasiliev theory to construct a perturbatively defined action. We wouldsimply amend the vertices we find to the free action in AdS. This may be interesting but forus is not practically necessary, because once we have found the vertices using the equationsof motion, and solved for the free-fields, we already know how to go and compute correlationfunctions


So far we have said nothing about the dual CFT to Vasiliev theory. For the moment we willsimply work under the proviso that the dual theory exists and we will say a bit more about thislater. The dual CFT is different depending on the boundary conditions that we choose for thescalar field C(0,0) := φ of mass m2 = −2. Using (6.3) we find that ∆ = 1, 2 for the scalar fieldso that near the boundary it behaves as,

φm2=−2(~x, z → 0) ∼ a(~x)z + b(~x)z2

58


clearly a and b characterise two different behaviours near the boundary z = 0. Conformalinvariance implies that we must choose the boundary condition that either a or b vanish [3].Respectively, setting a/b to zero implies that φ has conformal dimension ∆ = 2/1 and b/a is thevev of the dual operator. For simplicity we will choose the ∆ = 1 boundary condition in whatfollows.

Now we are going to use the machinery we have set up in order to compute, in Vasilievtheory, the tree-level three point function for two higher-spin particles interacting and spittingout a scalar particle (see Figure 6). Therefore we need to solve for the next-to-leading correctionto the boundary value of the scalar field which is sourced by two fields of spin s and s at theboundary.

Figure 6: The 3-Point Witten Diagram for s-s-0 cubic vertex.

To do this we need to solve for the analog of the fourth linearised equation of (5.26) but witha source [4],

D0C = J(x|Y )

We essentially use the same analysis which gave us the Klein-Gordon equation for the freemassive scalar because we want to extract the same equation only with sources. The onlydifference will be that solving for the spin-1 component of C will involve the source J , whichwill in turn generate source terms in the resulting Klein-Gordon equation. By performing exactlythe same manipulations which lead up to equation (5.28) we find that the analogous result givesthe simple modification,

∇µLC(x|Y ) + 2(e0)µαβyαyβC(x|Y ) + 2(e0)µ

αβ∂α∂βC(x|Y ) = Jµ(x|Y )

Equating coefficients in Y implies that,

∇µLC(n,n)(x|Y ) + 2(e0)µ

αβyαyβC(n−1,n−1)(x|Y ) + 2(e0)µ

αβ∂α∂βC(n+1,n+1)(x|Y ) = Jµ(x|Y )(n,n)

(6.18)The equations for the (0, 0) and (1, 1) components of J read,

∂µC(0,0)(x) + 2(e0)µαβ∂α∂βC(1,1)(x) = Jµ(x)(0,0)

∇µLC(1,1)(x|Y ) + 2(e0)µ

αβyαyβC(0,0)(x) + 2(e0)µ

αβ∂α∂βC(2,2)(x|Y ) = Jµ(x|Y )(1,1)

(6.19)

Contracting the first line above with (e0)αβµ gives,

(e0)αβµ ∂µC(0,0)(x)− 1

4∂α∂β[C(1,1)(x)γδy

γ yδ] = (e0)αβµ Jµ(x)(0,0) := Jαβ(x)(0,0) ⇔

C(1,1)(x)αβ = 4(e0)αβµ ∂µC(0,0)(x)− 4Jαβ(x)(0,0) (6.20)

59


Where we have defined Jαβ = (e0)µαβJµ. Now contracting the second line of (6.19) with (e0)αβµ

gives,

(e0)αβµ ∇µLC

(1,1)(x|Y )−1

4yαyβC(0,0)(x)−1

4∂α∂βC(2,2)(x|Y ) = (e0)αβµ Jµ(x|Y )(1,1) := Jαβ(x|Y )(1,1)

Then acting with ∂α∂β on the above kills C(2,2)(x)ααββ and implies that,

(e0)αβµ ∂α∂β∇µL[C(1,1)(x)γδy

γ yδ]− C(0,0)(x) = ∂α∂βJαβ(x|Y )(1,1)

finally plugging in the result (6.20) to eliminate C(1,1)(x)αβ we get,

4(e0)αβµ ∂α∂β∇µL[(e0)ν

γδ∂νC

(0,0)(x)yγ yδ]−C(0,0)(x) = ∂α∂βJαβ(x|Y )(1,1)+4(e0)αβµ ∂α∂β∇

µL[Jγδ(x)(0,0)yγ yδ]

The term on the left is none other than the free Klein-Gordon term −12(∇µ∂µ + 2)C(0,0)(x) as

we showed earlier in the analysis that preceeded (5.33). Therefore the above result gives,

(∇µ∂µ + 2)C(0,0)(x) = −2J (x) (6.21)

where we have defined the scalar source,

J (x) = ∂α∂βJαβ|Y=0 + 4∂α∂β∇αβL J

(0,0)

γδyγ yδ (6.22)

Where we used the fact that ∂α∂βJ(1,1)

αβ= ∂α∂βJαβ|Y=0. In the Toy model case we found

that solving for the boundary value of the outcoming field is equivalent to integrating, in thebulk, the source with the scalar bulk-to-boundary propagator. In exactly the same way, theboundary value of the outcoming scalar C(0,0) is given by the integral [4],

limz→0

z−1C(0,0)(~x, z) =

∫d3~ydz−2

√gK∆(~x, z; ~y)J (~y, z) =

∫d3~ydz

π2z3

1

z2 + (~y − ~x)2J (~y, z) (6.23)

where in the last step we used√g = z−D and used the explicit form of K∆ given by (6.7)

with ∆ = 1 and that C∆=1 = −(2π2)−1. Now we have all of the machinery we need, becausewe can get J from the second order equations of motion, plug this into the definition (6.22)and then insert the bulk-boundary propagators for the master fields in place of the master fieldsthat appear in the source J (since this last step is equivalent to functionally differentiating theboundary value with respect to the boundary sources and then setting them to zero). The lastdata we need are the bulk-boundary propagators for the master fields, which we discuss in thenext section.

On the other hand we know that conformal invariance contrains the s-s-0 three-point func-tions to take the form [39],

〈Os( ~x1)Os( ~x2)O0( ~x3)〉 = N · C(s, s; 0)

|~x1 − ~x2|∆0−∆s−∆s |~x1 − ~x3|∆s−∆s−∆0 |~x2 − ~x3|∆s−∆s−∆0

where ∆i is the conformal dimension of the operator dual to the bulk spin-i field, where N is anormalisation factor and the coefficient C(s, s; 0) encodes the dependence of the amplitude onthe spins s and s. The essentially non-trivial part is to determine C(s, s; 0). We could then go onand consider a different scattering channel to determine the overall normalisation. By comparingthe normalised results for different spins we would then obtain a factor of proportionality, thisis the coupling constant of Vasiliev theory, which we may subsequently put in by hand whencomputing further correlators.

60

6.4 Explicit Vertices and Computation of C(s, s; 0)

6.4 Explicit Vertices and Computation of C(s, s; 0)

We have seen by analog with the scalar case that one can determine the bulk cubic vertices byperturbatively solving the equations of motions for the next-to-leading order fields. Vasiliev’slinearised equations describe free higher-spin fields and so we need to solve for the second-orderfields to extract the cubic vertices. It is sufficient to look at the equation of motion for the scalarfield B given by the fourth line of (5.25) [4],

dxB = −W ∗B +B ∗ π(W )

Recall that the perturbed fields were defined as B = λB1+λ2B2+..., and W = W0+λW1+λ2W2.Plugging these into the above we get,

dx(λB1+λ2B2+...) = −(W0+λW1+λ2W2)∗(λB1+λ2B2+...)+(λB1+λ2B2+...)∗π(W0+λW1+λ2W2)

Now equating coefficients in λ2 we obtain an equation for the second order field B2,

dxB2 +W0 ∗B2 −B2 ∗ π(W0) = D0B2 = −W1 ∗B1 +B1 ∗ π(W1) (6.24)

Now we will denote B2 = B for brevity. Recall that we have defined the Z independent partsof the master fields B and W ,

B1(x|Y, Z) = C1(x|Y ) + B′(x|Y, Z) W1(x|Y,Z) = Ω1(x|Y ) +W ′1(x|Y,Z)

Equation (6.24) then reads,

D0C = −[Ω1 +W ′1] ∗ [C1 + B′] + [C1 + B′] ∗ π(Ω1 +W ′1)

= −Ω1 ∗ C1 + C1 ∗ π(Ω1) + ...

Where ”...” denotes other terms. In fact it is shown in [3] that all of these other terms do notcontribute to the scalar terms. This is a long and tedious affair which essentially requires thatwe inspect all of the various powers of y appearing on the right-hand side above, by looking atthe explicit solutions we have found for the linearised fields. Either the contributions from eachof the terms conspire to cancel or simple power counting arguments in y imply that they givezero contribution in the scalar sector, or in other words that they do not source the scalar field!Thus we have identified the source,

Jαβ = −Ωαβ1 ∗ C1 + C1 ∗ π(Ωαβ

1 )

for the scalar field in the bulk. Expanding in Y this gives (dropping the subscript 1s for brevity),

Jαβ = −Ωsαβ∗ C(s−1+s,s−1−s) + C(s−1+s,s−1−s) ∗ π(Ωs

αβ)

+−Ω−sαβ∗ C(s−1−s,s−1+s) + C(s−1−s,s−1+s) ∗ π(Ω−s

αβ) + ...

= −

Ωsαβ

+ Ω−sαβ, C∗

+ ...

Star products of terms that are not of the of the same degree contain terms with deg(y) 6= deg(y)hence only the terms shown above contribute to the (0, 0) and (1, 1) components of J [4]. Nowwe use the explicit expressions for the vierbein and spin connection in Poincare coordinates [3],

(e0)µ = − 1

4zyασµ

αβyβ, (ω0)i = − 1

8z(yασizαβy

β + yασizαβyβ)

61

6.5 The Dual CFT

Now recalling the definition (6.22) and using the above results, we find the following form forthe current in Poincare coordinates [4],

J = −z2∂y(6 ∂ −

2

zσz) 6 Jy − 1

2∂α∂βJαβ|Y=0

=z

2∂y(6 ∂ −

2

zσz)αβ

Ωsαβ

+ Ω−sαβ, C∗y − 1

2∂α∂β

Ωsαβ

+ Ω−sαβ, C∗|Y=0

=z

2(s2 − s2)(6 ∂ − 2

zσz)αβ

∂α∂βΩs

++, C∗

+1

2

Ωs

++, C∗|Y=0 + (s↔ −s, y ↔ y)

(6.25)

where we have defined the objects 6 ∂ = σµ∂µ and similarly 6 J = σµJµ. What we now need arethe bulk-boundary propagators for the master field components that appear above. These arethe kernels of the kinetic operators appearing in their free equations of motion. Recall that thefree equation for C and Ω are the differential equations,

D0C = 0

D0Ω = 2eiθ0εαγ(e0)αβ ∧ (e0)γ

δ∂β∂δC(x|0, y) + 2e−iθ0εαγ(e0)β α ∧ (e0)δ γ∂β∂δC(x|y, 0)

Solving these for the propagators is very messy. In particular the equations incriminate thetwistors, so the propagators are functions of spacetime and the twistors. The bulk-boundarypropagators for the component fields we eventually find are [4],

Ωs++ =

2−z−2

(2s− 1)!

zs

(x−)s+s(yxσ−zx)s∂2s

+

(yxy)s−s

x2, C =

1

2Ke−yσy(T (y)s + T (y)s)

Where the following quantities are defined,

x = xµσµ = xiσ

i + zσz, T (y) =K2

zyxεiσ

iσxy, εi∂i = ∂+

where K = K(~x) is the ordinary scalar propagator with ∆ = 1, K = K(x) and ε is thepolarisation spinor corresponding to the higher-spin source at the boundary [4]. Now all of thepieces are there! We can insert the above propagators into (6.25) and finally plug this into(6.23) to compute the s-s-0 three-point function. However the resulting integral is very difficultto compute and unfortunately there was not time to do justice to the ensuing analysis of [4]here. The result derived therein gives us the explicit form of the spin-coefficient,

C(s, s; 0) = −√π

2Γ(s+ s+

1

2) (6.26)

Since the result is non-trivial, this establishes that, at least at tree-level, Vasiliev theorycircumvents the no-go results and solves the problem of interacting higher-spin fields. At longlast we appear to have overcome the barrier that loomed over us at beginning! The strengthof this conclusion at the quantum level remains an open question. It is certainly plausible thatloop-level corrections to the three-point function could imply that the resulting amplitude isfree! This reflects the general problem that the lack of an action principle for Vasiliev theoryseverely inhibits in our capacity to discern the properties of the quantum theory.

6.5 The Dual CFT

Finally, we will talk about the CFT s that are conjectured duals of the known types of Vasilievtheories. Firstly suppose that we have some higher-spin field φµ1...µs of spin-s in the bulk AdSd+1.We know that this corresponds to the term

∫ddx J i1...isφi1...is in the CFT action, where J i1...is

is the operator dual to φ and φi1...is is the boundary value of the spin-s field with i = 1, ..., d.Clearly J must be a spin-s object. Now it is easy to see that gauge invariance implies that

62

6.5 The Dual CFT

∂iJii2...is = 0. Therefore the dual operator of a higher-spin field in the bulk is a higher-spinconserved current. Now since Vasiliev theory contains an infinite tower of higher-spin fields thisresult implies that the dual theory has an infinite number of higher-spin conserved currents!

It was shown by Maldacena and Zhiboedov in [41] that CFT with exactly conserved higher-spin currents have the correlation functions of a free CFT ! This implies that Vasiliev theory,with unbroken higher-spin gauge symmetry is holographically dual to a free CFT ! Klebanovand Polyakov conjectured that the dual to the type-A, ∆ = 1 Vasiliev theory is the free O(N)vector model [42].

One can try to verify a holographic duality by comparing correlation functions in the pro-posed dual CFT with those in the bulk. One can determine the higher-spin conserved currentsin the O(N)-vector model and compute the correlation function 〈J0JsJs〉 and, rather beautifully,the result agrees exactly with (6.26)! [4]. Perhaps more remarkably, the ∆ = 2 Vasiliev theoryappears to be dual to the critial O(N) vector model! (with a non-trivial 4-point self interac-tion). The correlation functions are computed in [3] and compared with the dual correlators andagain, exact equivalence has been found. These so-called higher-spin/vector model dualities aregorgeous results and it is a shame we could not do the subject more justice here, since they areone of the most captivating aspects of higher-spin field theory. It will be interesting to see whatrole the higher-spin/vector model dualities might have to play in the elucidation of quantumgravity as may be conferred by the AdS/CFT correspondence more generally, but it is clearthat they represent a distinct and highly non-trivial realisation of the correspondence.

Key Points:

• TheAdS/CFT correspondence relates a bulk theory of gravity inAdS to a non-gravitationalCFT living at the boundary.

• The boundary fields source primary operators in the CFT and their dual fields in the bulkvia the bulk-boundary propagator.

• The boundary value of a bulk field is the vev or one-point function of the correspondingdual operator.

• The tree-level three-point function is an integral in the bulk of the bulk-boundary propa-gators that furnish the vertex.

• One can compute the corrections due to interactions to the boundary value of the bulkfield using the equations of motion only. Functionally differentiating these and setting theboundary fields to zero generates correlation functions up to overall normalisation and thecoupling constant.

• Solving for the second order master scalar field in Vasiliev theory implies there are non-trivial higher-spin interactions.

• In order to compute the 0-s-s correlation function at tree-level, one solves for the boundaryvalue of the outcoming scalar field and inserts the bulk-boundary propagators for themaster fields.

• Our result for C(s, s; 0) agrees with the corresponding correlation function of dual currentsin the free O(N) vector model.

63

7 Discussion

We began with a review of the powerful no-go theorems for interacting higher-spin gauge fieldsin flat-space and the means to circumvent them. We proceeded to discuss the basic features thatcharacterise massless particles and higher-spin fields in four-dimensional flat space. The physi-cality constraints that are required of the free-theory were then established. The constructionof the free higher-spin gauge theory in flat-space was presented before we reviewed some of thegeneric features of higher-spin gauge theories. We then discussed and explained the salient fea-tures of Vasiliev gauge theory, verifying that at the linearised level it describes a massive scalarand an infinite tower of higher-spin gauge fields in AdS4. The machinery of the AdS/CFT cor-respondence was introduced and the computation of the 0-s-s higher-spin three-point functionat tree-level in type-A Vasiliev theory with ∆ = 1 boundary conditions was explained. Theresult illustrates that the higher-spin interactions in Vasiliev theory are non-trivial at tree-level.Furthermore, in determining the spin dependence of the three-point function we have obtaineda highly non-trivial piece of evidence that verifies the higher-spin vector model dualities!

The no-go theorems, formidable results in flat-space, have no power in curved backgrounds,where we have seen that we can realise a potentially fully interacting gauge theory in AdS,namely Vasiliev theory! The construction of free higher-spin gauge theories is quite simple inflat space and in four dimensions. This is because, as we have seen, the representation theory isparticularly straightforward. We have forgone a discussion of higher-spin fields in D > 4. TheD = 4 case accomodates a phenomenological discussion but moreover it illustrates most of thesalient features of the subject. Among the nascent features we observe in higher dimensions isthe fact that the representation theory becomes less trivial and in particular symmetric tensorsno longer exhaust all forms of the higher-spin fields [26]. Presently, some of the formalisms thatwe have discussed have yet to be generalised in this direction.

Vasiliev’s theory, as we have often remarked, is certainly not simple! Ultimately it remainsto be seen whether or not the theory represents more than a toy-model for interacting higher-spin gauge fields, or even if the theory is a genuinely fully interacting and consistent theory atthe quantum level. Indeed the action for the theory, something we must take for granted, issomething that presently remains elusive, at least in a form which gives rise to the Fronsdalequations at the linearised level [3]. A particularly interesting open question is whether or notthe massive scalar field induces explicit higher-spin symmetry breaking that could lead to thespectrum of higher-spin states in string theory, as we mentioned in the opening. Vasiliev theoryis extremely fascinating on account of the possible connections to string theory and especiallydue to the remarkable higher-spin/vector model dualities.

Although we briefly reviewed the exotic properties that appear to be generic for interactinghigher-spin gauge theories, there was not time to discuss these issues more formally. An excellentreview of these features, and indeed the whole topic, can be found in [1].

The geometric formulation of gravity due to Stelle-West and their higher-spin generalisationsusing the formalism due to de Wit and Freedman that we mentioned comprise much of theformalism that originally lead to the construction of Vasiliev gauge theory. As we said, a reviewof these subjects would not have been feasible within the scope of this report and moreover theliterature is fairly opaque on this matter!

For brevity, we were rather vague about some of the details of the computation of the higher-spin three-point function. In particular, solving the master field equations for the propagatorsand computing the final integral is a very long and cumbersome task! Nevertheless we haveillustrated the general idea by justifying all of the conceptual steps involved in setting up thecomputation. Interestingly, the most general three-point function for a 3-vertex of all arbitrary-spin fields has yet to be found! However, it is expected that the computation can be carried outalong the lines of the 0-s-s three-point function as we have seen [4].

The subject of interactions higher-spin gauge theories is very rich and diverse and there area number of formalisms that it was not practical to discuss at any length. We have not discussed

64

the so-called BRST formalism [43] of higher-spin gauge fields, in which the higher-spin fieldsand equations of motion and vertices are constructed via the higher-spin BRST invariant statesthat appear in string theory. It was felt that the more historical approach that we have followedwould be more informative.

Higher-spin gauge theories in their various guises continue to represent fascinating and activeareas of research. It is clear that with a deeper understanding of their formulation we potentiallystand to gain much insight into many important areas such as string theory, holography andeven the possibility of realising a deeper framework for quantum field theory.

8 Acknowledgements

I would like to thank Prof. Mukund Rangamani for his patient supervision during this project.

65

9 Appendix

A Star Products

Here we compute a set of star products. These truncate because the are finitely many powersof Y in the quantities that we are computing star products of.

yαyβ ∗ yγ yδ = yαyβ(1 + εab←−∂ a−→∂ b + εab

←−∂ a−→∂ b +

1

2· 2εab

←−∂ a−→∂ bε

ab←−∂ a−→∂ b)y

γ yδ

= yαyβyγ yδ + δαa δγb εabyβ yδ + δβa δ

δbεabyαyγ + εabεabδαa δ

βa δ

γb δδb

= ...+ εαγ yβ yδ + εβδyαyγ + εαγεβδ.

(A.1)

yαyβ ∗ yγyδ = yαyβ(1 + εab←−∂ a−→∂ b)y

γyδ = yαyβyγyδ + εabδαa yβ∂b(y

γyδ)

= yαyβyγyδ + εαbyβ(δγb yδ + δδby

γ) = ...+ εαγ yβyδ + εαδyβyγ(A.2)

yαyβ ∗ yγ yδ = yαyβ(1 + εab←−∂ a−→∂ b)y

γ yδ = yαyβ yγ yδ + εabyαδβa∂b(yγ yδ)

= yαyβ yγ yδ + εβbyα(δγbyδ + δδ

byγ) = ...+ εβγyαyδ + εβδyαyγ

(A.3)


γ yδ = yαyβyγ yδ + εab∂a(yαyβ)βγb y

δ

= yαyβyγ yδ + εaγ(βαa yβ + ββa y

α)yδ = ...+ εαγyβ yδ + εβγyαyδ(A.4)

yαyβ ∗ yγyδ = yαyβ(1 + εab←−∂ a−→∂ b)y

γyδ = yαyβyγyδ + εab∂a(yαyβ)∂b(y

γyδ)

= yαyβyγyδ + εab(δαa yβ + δβay

α)(δγb yδ + δδby

γ)

= ...+ εαγyβyδ + εαδyβyγ + εβγyαyδ + εβδyαyγ

(A.5)

yαyβ ∗ yγ yδ = yαyβ yγ yδ (A.6)


γ yδ = yαyβyγ yδ + εab∂a(yαyβ)δδ

byγ

= yαyβyγ yδ + εaδ(δαa yβ + δβa y

α)yγ = ...+ εαδyβyγ + εβδyαyγ(A.7)

yαyβ ∗ yγyδ = yαyβyγyδ (A.8)

yαyβ ∗ yγ yδ = ...+ εαγ yβ yδ + εαδyβ yγ + εβγ yαyδ + εβδyαyγ (A.9)

where ”...” represents the result of the ordinary product between the twistors.

[yβ, A(Y )]∗ = yβ ∗A−A ∗ yβ = yβ(1 + εab←−∂ a−→∂ b − ε

ab←−∂ a−→∂zb

)A−A(1 + εab←−∂ a−→∂ b + εab

←−∂ za−→∂ b)y

β

= εβb∂bA− εβb∂

zbA− εaβ∂aA− εaβ∂zaA = ∂βA− ∂

zβA+ ∂βA+ ∂

zβA = 2∂βA

(A.10)Similarly we find that,

[yα, A]∗ = yα ∗A−A ∗ yα = yα(1 + εab←−∂ a−→∂ b − εab

←−∂ a−→∂ zb)A−A(1 + εab

←−∂ a−→∂ b + εab

←−∂ za−→∂ b)y

α

= εαb∂bA− εαb∂zbA− εaα∂aA− εaα∂zaA = ∂αA− ∂zαA+ ∂αA+ ∂zαA = 2∂αA(A.11)

Here we computeyαyβ, A

∗, we start with,

yαyβ ∗A = yαyβ(1 + εab←−∂ a−→∂ b + εab

←−∂ a−→∂ b + εab

←−∂ a−→∂ bε

ab←−∂ a−→∂ b)A

= yαyβA+ yβδαa εab∂bA+ yαδβa ε

ab∂bA+ δαa δβb εabεab∂b∂bA

= yαyβA+ yβ∂αA+ yα∂βA+ ∂α∂βA

66

and similarly,

A ∗ yαyβ = A(1 + εab←−∂ a−→∂ b + εab

←−∂ a−→∂ b + εab

←−∂ a−→∂ bε

ab←−∂ a−→∂ b)(e0)µ

αβyαyβ

= yαyβA+ (∂aA)εab(e0)µαβδαb y

β + (∂aA)εab(e0)µαβyαδβ

b+ (∂a∂aA)εabεab(e0)µ

αβδαb δ

β

b

= yαyβA+ (−εαb)(∂bA)yβ + (−εβb)(∂bA)yα + (−εαb)(−εβb)(∂b∂bA))

= yαyβA− yβ∂αA− yα∂βA+ ∂α∂βA

recalling our convention for raising and lowering spinor indices that ya = εabyb etc. Puttingthese results together we find,

yαyβ, A(Y )∗

= (e0)µαβyαyβ ∗A+A ∗ (e0)µ

αβyαyβ

= (2yαyβ + 2∂α∂β)A(Y )(A.12)

Now we evaluate the star commutator,

[yαyβ, yγ yδ]∗ = yαyβ ∗ yγ yδ − yγ yδ ∗ yαyβ

Using the identities (A.2) and (A.1) we get,

[yαyβ, yγ yδ]∗ = εαγyβ yδ + εβγyαyδ − εγαyβ yδ − εγβyαyδ = 2εαγyβ yδ + 2εβγyαyδ (A.13)

Now we evaluate the star commutator,

[yαyβ, yγ yδ]∗ = yαyβ ∗ yγ yδ − yγ yδ ∗ yαyβ

Using the identities (A.3) and (A.7) we get,

[yαyβ, yγ yδ]∗ = εαδyβyγ + εβδyαyγ − εδαyγ yβ − εδβyγ yα = 2εαδyβyγ + 2εβδyαyγ (A.14)

Now we compute the star commutator [yαyβ, A(Y )]∗. We start by computing,

yαyβ ∗A(Y ) = yαyβ(1 + εab←−∂ a−→∂ b +

1

2εab←−∂ a−→∂ bε

cd←−∂ c−→∂ d)A(Y )

= yαyβA(Y ) + εab(δαa yβ + δβay

α)∂bA(Y ) +1

2εabεcd(δαa δ

βc + δβa δ

αc )∂b∂dA(Y )

= (yαyβ + yα∂β + yβ∂α + ∂α∂β)A(y)

similarly we find that,

A(Y ) ∗ yαyβ = A(Y )(1 + εab←−∂ a−→∂ b +

1

2εab←−∂ a−→∂ bε

cd←−∂ c−→∂ d)y

αyβ

= A(Y )yαyβ − εba[∂aA(Y )](δαb yβ + δβb y

α) +1

2εbaεdc[∂a∂cA(Y )](δαb δ

βd + δβb δ

αd )

= (yαyβ − yβ∂α − yα∂β + ∂α∂β)A(Y )

Combining the above two results we find that,

[yαyβ, A(Y )]∗ = 2(yα∂β + yβ∂α)A(y) (A.15)

Now by inspecting the star product (5.3) we see that the computation of [yαyβ, A(Y )]∗ proceedsexactly as above just with y ↔ y and the indices a, b, α, β swapped with abαβ respectively andso we simply get,

[yαyβ, A(Y )]∗ = 2(yα∂β + yβ∂α)A(y) (A.16)

67

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