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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 925 (2017) 431–454

www.elsevier.com/locate/nuclphysb

On the phase space of free higher-spin theories

and conformal transformations

Henrique Flores

Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II –

Barra Funda, CEP: 01140-070, São Paulo, Brazil

Received 9

March 2017; received in revised form 5 October 2017; accepted 26 October 2017

Available

online 31 October 2017

Editor: Leonardo

Rastelli

Abstract

write a ﬁrst order action for higher-spin ﬁelds and construct a canonical map to Fronsdal theory.

The ﬁrst-order description is deﬁned over complex ﬁeld conﬁgurations and has conformal invariance. We

show that it is possible to push forward these transformations to a set of symmetries in Fronsdal theory that

satisﬁes the conformal algebra but is not given by standard conformal change of coordinates.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Higher-spin theories have an important application in the AdS/CFT context. When the AdS

radius is small, it is conjectured that a subset of large string excitations decouples from the

remaining degrees of freedom and is described by an interacting higher-spin theory [1]. Unfortu-

nately, interactions are subtle to construct, but there is a comparati

vely easier case which we can

study: in the N →∞limit, we have free massless higher-spin theories.

There are tw

o known descriptions of free theories, which are referred to as Fronsdal and Pen-

rose formulation. In Fronsdal theory [3,4], we have constrained spacetime tensors that form an

irreducible representation of the little group on-shell, while in Penrose theory one uses twistor

geometry to construct irreducible representations of the little group [7]. Both theories are well

described by an action which is in

variant under higher-spin gauge symmetries. It is interesting,

E-mail address: henrique@ift.unesp.br.

https://doi.org/10.1016/j.nuclphysb.2017.10.020

The Author. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

432 H. Flores / Nuclear Physics B 925 (2017) 431–454

however, that Penrose formulation is invariant under conformal symmetries while Fronsdal for-

mulation is not [10].

At ﬁrst glance, it may seem strange that tw

o descriptions of the same theory have different

symmetries. The present paper aims to solve this conﬂict. After a brief review of the two formu-

lations, we construct a ﬁrst order action for Penrose theory and show that both theories describe

the same classical phase space via a canonical transformation. Using the canonical map, we can

push forw

ard the conformal transformations of Penrose theory to a set of non-local conformal

symmetries in Fronsdal description.

Plan of this paper We or

ganize our presentation as follows. Section 2 is a brief review, where

we explain the two approaches for free massless higher-spin theories.

In section 3 we write an action for Penrose higher

-spin theory. To our knowledge, such action

for general higher-spins has never appeared before in the literature. First-order formulations,

however, were used by Fradkin and Vasiliev in [14] for AdS space, where they were also ex-

tended to interactions. More recently, Kirill Krasnov described full self-dual gra

vity in [2] using

an action that resembles ours; but, in our case, this action is deﬁned over complex ﬁeld conﬁgura-

tions, and it describes off-shell a doubled set of the higher-spin modes. In phase space, however,

there is a well-deﬁned notion of reality, and it is where we obtain a single cop

y of the spectrum.

At this point, we look at some e

xamples, so the spins 1, 3/2 and 2 cases are discussed in

detail, each of which highlights a particular feature of our construction outlining our strategy for

dealing with general spins. The spin s case is done in section 4; our construction is a particular

instance of the prescription gi

ven in [13], where a set of equations of motion and a presymplectic

structure are shown to lift to a well-deﬁned Lagrangian.

With this map, we can in

vestigate conformal invariance. In section 5 we show that Penrose

action does have conformal symmetry for every spin s. Therefore one is able to push forward

these transformations to the Fronsdal case. For spins lower than 2, these new transformations

agree with usual conformal change of coordinates. The ﬁrst non-tri

vial case is linearized gravity.

We write explicitly the resulting transformation, where one is able to see the difference from

standard Lie derivatives.

On notation Our con

ventions follow those of [8]; we are concerned with 4-dimensional Min-

skowski space; so, through out the paper, the various indices will always be running over ﬁxed

intervals. Small Latin letters, for example, are spacetime indices running from 0to 3, so that

is a spacetime covector. Capital Latin letters, in turn, are spinor indices in Va n der Warden

notation, that is, dotted and undotted running from 0to 1. In particular, a Dirac spinor is a two

component Weyl and anti-Weyl spinor written like

 =





(1.1)

for some chiral spinor ψ

and anti-chiral χ

Such notation is designed so that there is a correspondence between spacetime and spinor

indices where, for instance, m will correspond to the pair M

M. The explicit realization is given

by the Pauli matrices with index structure σ

, where

=−1 and σ = (σ

,σ

The epsilon symbol satisﬁes 



= δ

for undotted and dotted indices. This enables one

to raise the indices of σ

to obtain

H. Flores / Nuclear Physics B 925 (2017) 431–454 435





···m

= 

∂

···m

, (2.20)

so that the remaining gauge symmetry obeys



∂

···m

= 0 and 

···m

= 0. (2.21)

The gauge ﬁxing (2.20) ensures that 

···m

is an irreducible representation of the little group.

The on-shell degrees of freedom are then described by a ﬁeld  which satisﬁes



∂



···m

= 0 and 



···m

= 0 (2.22)

thus proving 

···m

describes an spin s = h + 1/2 representation.

2.2. Penrose theory of free massless higher-spin ﬁelds

Penrose’s description of massless higher-spin ﬁelds is obtained from the Penrose transform. It

relates homogeneous functions of deﬁnite degree in twistor space to massless higher-spin ﬁelds in

Minkowski space. For an introduction to twistors, see reference [5] as well as references therein.

Here we describe the inte

gral expressions obtained by Penrose in [6] only to give some con-

text. These integral formulas are not necessary for the rest of this paper. We are only interested

in the spacetime ﬁelds they deﬁne.

Let Z = (ω

, π

) be the coordinates of a twistor inside the complex projective line P

. These

are constrained by the twistor equation:

= x

, (2.23)

where x

parametrizes the Minkowski space. Consider also a point Z = (λ

, μ

) in the dual

twistor space and ﬁx two closed cycles of integration: γ inside P

and γ

∗

inside the dual line P

∗

Deﬁne the following spacetime spinors

B···

(x) =

2πi



...π

  

f(Z)π

dπ

(2.24a)

and

AB··· D

(x) =

2πi



∗

...λ

  

f(Z)λ

dλ

(2.24b)

for some semi-integer number s.

Remark. These integrals are well deﬁned over P

if the integrands are homogeneous functions

of degree 0. Hence, the complex functions f(Z) and

f(Z) must have homogeneity −2s − 2in

and λ

respectively.

These spinors form an irreducible representation of the Lorentz group SL(2, C) and satisfy,

by consequence of their deﬁnitions, the differential equations

∂

B···

(x) = 0 (2.25)

and

∂

AB··· D

(x) = 0. (2.26)

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