On Nonlinear Higher Spin Curvature

Abstract

We present the first nonlinear term of the higher spin curvature which is
covariant with respect to deformed gauge transformations that are linear in the
field. We consider in detail the case of spin 3 after presenting spin 2 as an
example, and then construct the general spin s quadratic term of the
deWit-Freedman curvature.

Full text

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arXiv:1102.0306v3 [hep-th] 28 Feb 2011
hep-th/November 2010
On Nonlinear Higher Spin Curvature
Ruben Manvelyanab∗, Karapet Mkrtchyanab†,
Werner R¨ uhla‡and Murad Tovmasyanb§
aDepartment of Physics
Erwin Schr¨ odinger Straße
Technical University of Kaiserslautern, Postfach 3049
67653 Kaiserslautern, Germany
bYerevan Physics Institute
Alikhanian Br. Str. 2, 0036 Yerevan, Armenia
Abstract
We present the first nonlinear term of the higher spin curvature which is covariant
with respect to deformed gauge transformations that are linear in the field. We
consider in detail the case of spin 3 after presenting spin 2 as an example, and then
construct the general spin s quadratic term of the deWit-Freedman curvature.
∗manvel@physik.uni-kl.de
†karapet@yerphi.am
‡ruehl@physik.uni-kl.de
§mtovmasyan@ysu.am

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1Introduction
Geometry plays a very important role in field theory. General relativity is the most
beautiful and obvious example to prove this. In higher spin gauge field theory the
elements of geometry were studied first in the classical paper by Bernard de Witt and
Daniel Freedman [1]. Geometrical interpretations of Yang-Mills theory and general
relativity hint on possible geometric structures of higher spin gauge field theories in
general [1], [2], [3], [4], [5]. For the well analyzed existing theories gauge symmetry
itself has a hidden geometrical origin. Since the gauge symmetry is a redundancy in
the (non-observable) variables of the physical system, the observables of any gauge
theory are gauge invariant. For electrodynamics the observables are the components
of the Maxwell tensor Fµν (curvature of the electromagnetic field Aµ), for gravity
these are the components of the Riemann curvature tensor Rµνλρ(constructed from
the dynamical field, the metric tensor gµν). In analogy with these cases, in the higher
spin gauge field theory the curvature of the higher spin field should be observable.
The linear curvature as well as linearized “Cristoffel symbols” and connections for
any higher spin field are known from [1]. Taking traces of the linearized curvature
one can get geometrical free equations of motion for the unconstrained higher spin
gauge field [2], [6], which is nonlocal, but can be localized with the help of partial
gauge fixing after which the geometrical equation coincides with Fronsdal’s equation
[7].
There are two most common covariant approaches to the task under consider-
ation. The first one is frame-like formalism, which is developed by Vasiliev and
coauthors (see [8] for a review). The second approach is generalization of linearized
gravity - the metric-like theory of higher spin fields [7] (for recent development see
[9] and references therein). We use the second one.
In this letter we construct a higher spin curvature that is of second order in
the field, which can be used to find that part of the geometric equation of motion
that is quadratic in the field, which, after the same partial gauge fixing (ruling out
the trace of the gauge parameter and the double trace of the field), may or may
not coincide with the corresponding second order of Fronsdal’s equation of motion
coming from the cubic Lagrangian [10]. The cubic (self)interaction Lagrangian and
the corresponding gauge transformation laws for Fronsdal’s fields that are of first
order on field, are known [10] - [18]. The connection between these two independent
constructions is a subject for future investigations. Now we are going to construct
the first nonlinear term in the higher spin curvature independently, and with no use
of Fronsdal’s constraints on the fields involved. In order to get a gauge covariant
curvature, we don’t need to impose the constraint of tracelessness on the gauge
parameter respectively the constraint of double tracelessness on the higher spin
field. We don’t make use of Fronsdal’s theory in this letter, because the nonlinear
curvature that we construct here, describes not only Fronsdal’s (constrained) fields,
but, if it exists to all orders in the field, gives us a possibility for the construction
of theories of Higher Spin fields alternative to Fronsdal’s (consider the nonlinear
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continuation of conformal higher spin theory [19, 20] as an example).
In Section 2 we discuss as a simple example linearized gravity and construct
the second order of the Riemann curvature using the gauge covariance condition.
In Section 3 we construct the curvature of second order in the field for the spin
3 gauge field. In the final Section 4 we derive that part of the nonlinear gauge
covariant curvature that is of second order in the field for any higher spin field
which reproduces all lower spin cases and has the same symmetry properties as the
linear curvature.
The most remarkable property of the curvature that is of second order in the field
is that it is a sum of terms, which are all squares of the linear connection Γ(s−1), the
s minus first member in the hierarchy of de Witt - Freedman - Christoffel symbols
[1]. Using shorthands explained later, we can summarize the curvature up to second
order in the field in the following compact way
R = dΓ + Γ ⋆ Γ(1.1)
for any higher spin gauge field.
2Quadratic term of linearized Riemann curva-
ture
We start from a consideration of the linearized Riemann curvature (spin 2 case).
We use the following self consistency definitions for covariant derivatives, Christoffel
symbols and curvature:
∇µVν= ∂µVν+ Γρ
[∇µ,∇ν]Vλ= R
µνVρ
ρ
µν,λVρ
µν=1
(2.1)
(2.2)
Γρ,µν= gρσΓσ
2(∂ρgµν− ∂µgνρ− ∂νgµρ)
νλ− ∂νΓρ
(2.3)
R
ρ
µν,λ= Rµν,λδgδρ= ∂µΓρ
µλ+ Γσ
µλΓρ
νσ− Γσ
νλΓρ
µσ
(2.4)
Then we note that expression (2.3) is very convenient for a linearization and we can
obtain a linearized Christoffel symbol just replacing the metric gµνby the linearized
field hµν= gµν−ηµν, where ηµνis the Minkowski metric. To obtain the corresponding
expression for the curvature ready for linearization we can use (2.3) and after some
algebra write the full covariant curvature Rµν,λρin the form
Rµν,λρ= ∂µΓρ,νλ− ∂νΓρ,µλ− gσδ(Γσ,µλΓδ,νρ− Γσ,νλΓδ,µρ)(2.5)
Then substituting in (2.5)
gµν= ηµν− hµν+ hµ
2Γρ,µν= ∂ρhµν− ∂µhνρ− ∂νhµρ
σhνσ− ...
(2.6)
(2.7)
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we arrive at the following expansion of the curvature up to the third order in the
field:
Rh
µν,λρ= R(1)
2R(1)
R(2)
µν,λρ+ R(2)
µν,λρ+ R(3)
µν,λρ+ ...
(2.8)
µν,λρ= ∂µ∂ρhνλ− ∂ν∂ρhµλ− ∂µ∂λhνρ+ ∂ν∂λhµρ
(2.9)
µν,λρ= −ησδ(Γσ,µλΓδ,νρ− Γσ,νλΓδ,µρ)
R(3)
(2.10)
µν,λρ= hσδ(Γσ,µλΓδ,νρ− Γσ,νλΓδ,µρ)(2.11)
Finalizing this section we note that the same expansion could be recovered from
the initial linearized curvature (2.9) and gauge invariance of order zero in the field
δ(0)hµν= ∂µǫν+ ∂νǫµusing the ”covariant” (not invariant) Noether’s equation:
δ(1)R(1)
µν,λρ+ δ(0)R(2)
µν,λρ= LǫR(1)
µν,λρ
(2.12)
where LǫR(1)
µν,λρis the Lie derivative of the first order curvature
LǫR(1)
µν,λρ=
ερ∂ρR(1)
+∂µερR(1)
αβ,µν+ ∂αερR(1)
αβ,ρν+ ∂νερR(1)
ρβ,µν+ ∂βερR(1)
αρ,µν
αβ,µρ
(2.13)
It is easy to see that as a solution of the equation (2.12) we will find (2.10) and the
following gauge transformation of first order in the field:
δ(1)hµν= ǫλΓλ,µν+ ∂µfν(h,ǫ) + ∂νfµ(h,ǫ)(2.14)
where fµ(h,ǫ) is at this stage an arbitrary vector function linear in the field h and
the parameter ǫ. The linearized curvature (2.9) is invariant with respect to any
gradient transformation δhµν = ∂µfν+ ∂νfµ with the arbitrary vector parameter
fµ. Nevertheless the next order of Noether’s procedure fixes this ambiguity, and we
obtain the well known gravitational gauge transformation δhµν= ǫρ∂ρhµν+∂µǫρhνρ+
∂νǫρhµρ(fµ= hµρǫρ) in the first order on the field.
In the next sections we generalize this covariant Noether’s procedure for the spin
3 and the general spin s case constructing the unknown quadratic part of the higher
spin curvature.
3The Case of Spin 3
To handle the spin 3 case we should introduce an additional nonabelian charge
to avoid trivialization of the theory [21]. Then our spin three field ha
symmetry parameter εa
the corresponding zero order gauge transformation in the field
αβγand the
βγcarry an additional Lie algebra basis index. Introducing
δ0
εha
αβγ= 3∂(αεa
βγ)= ∂αεa
βγ+ ∂βεa
αγ+ ∂γεa
αβ
(3.1)
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we define the first order curvature
Ra(1)
αα′,ββ′,γγ′ =
∂α∂β∂γha
−∂α′∂β∂γha
+∂α′∂β′∂γha
−∂α′∂β′∂γ′ha
α′β′γ′
αβ′γ′ − ∂α∂β′∂γha
αβγ′ + ∂α′∂β∂γ′ha
α′βγ′ − ∂α∂β∂γ′ha
αβ′γ+ ∂α∂β′∂γ′ha
α′β′γ
α′βγ
αβγ
(3.2)
from the standard condition of gauge invariance
δ0
εRa(1)
αα′,ββ′,γγ′ = 0(3.3)
Turning to the next step we should solve the following Noether’s equation
δ0
εRa(2)
αα′,ββ′,γγ′ + δ1
εRa(1)
αα′,ββ′,γγ′ = 0 + O(Ra(1)
αα′,ββ′,γγ′,εbµν)(3.4)
where O(Ra(1)
curvature and the gauge parameter and has two derivatives, which plays the role of
the spin 3 generalization of the Lie derivative in (2.12).
To present a solution of (3.4) we should first introduce the spin 3 generalization
of the Christoffel symbol (2.7):
αα′,ββ′,γγ′,εbµν) represents some expression that is linear in the first order
Γa
µν,αβγ
= ∂µ∂νha
−1
2∂α∂µha
−1
2∂β∂νha
+∂α∂βha
αβγ
νβγ−1
2∂α∂νha
µβγ−1
2∂β∂µha
ανγ
αµγ−1
2∂γ∂µha
µβν+ ∂β∂γha
αβν−1
2∂γ∂νha
αβµ
µνγ+ ∂α∂γha
αµν
(3.5)
This expression differs from the second generalized Christoffel symbol of deWit-
Freedman [1] only by an additional Lie algebra index. Then we can present the
following expressions for the second order curvature
Ra(2)
αα′,ββ′,γγ′
= fabc(Γbµν
−Γbµν
+Γbµν
−Γbµν
αβγΓc
µν,αβ′γ′ − Γbµν
µν,α′β′γ′
α′βγΓc
αβ′γΓc
µν,α′βγ′ − Γbµν
αβγ′Γc
µν,α′β′γ
α′β′γΓc
µν,αβγ′ + Γbµν
α′βγ′Γc
µν,αβ′γ+ Γbµν
αβ′γ′Γc
µν,α′βγ
α′β′γ′Γc
µν,αβγ)(3.6)
and the first order gauge transformation
δ1
εha
αβγ=
fabc(εbµν∂µ∂νhc
+∂αεbµν∂µhc
+∂α∂βεbµνhc
αβγ
νβγ+ ∂βεbµν∂µhc
µνγ+ ∂α∂γεbµνhc
ανγ+ ∂γεbµν∂µhc
µβν+ ∂β∂γεbµνhc
αβν
αµν)(3.7)
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