移位对称spin-1理论：大重力下（A)dS空间的对称性与解耦新发现

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JHEP09(2019)029

To take the massless limit we introduce a St¨uckelberg vector A

patterned after the linear

diﬀeomorphism symmetry that is restored in the massless limit,

µν

7→ h

µν

(∇

+ ∇

) , (2.6)

where the mass scaling is chosen so that the kinetic term for the vector is canonically nor-

malized up to a constant factor. After the replacement (2.6), the theory has a gauge sym-

metry

δh

µν

= ∇

+ ∇

, δA

= −mξ

, (2.7)

where ξ

(x) is a vector gauge parameter. We can then take the massless (A)dS limit

m → 0, L ﬁxed, and h

µν

, A

, ξ

ﬁxed , (2.8)

after which the Lagrangian (2.5) becomes

L = L

m=0

√

−γ



−

µν

−



, (2.9)

where L

m=0

is the Lagrangian (2.5) evaluated at m = 0. We see the appearance of a

massive vector with the mass (2.2), which carries the three extra degrees of freedom of a

massive graviton compared to a massless graviton.

In the massless limit (2.8), the gauge symmetry (2.7) becomes

δh

µν

= ∇

+ ∇

, δA

= 0 . (2.10)

This is the transformation that survives the decoupling limit for arbitrary ξ

, so the vector

ﬁeld has no gauge symmetry in the limit (2.8). There are, however, special choices of ξ

such that A

does transform after taking the limit. To look for these, we deﬁne

= mξ

, (2.11)

and consider taking the decoupling limit holding

ﬁxed, so that (2.7) becomes

δh

µν



∇

+ ∇



, δA

. (2.12)

In general, the expression for δh

µν

in (2.12) blows up in the massless limit. However, in the

special case where

is a Killing vector, satisfying the Killing equation ∇

+ ∇

= 0,

then δh

µν

vanishes and we are left with a ﬁnite shift symmetry for A

, recovering (2.3).

3 Symmetry algebra

In this section we use the ambient space formalism to describe the algebra formed by the

shift symmetries along with the (A)dS isometries and to classify possible deformations.

This gives us information about the possible invariant interactions for shift-symmetric

spin-1 theories.

– 5 –

JHEP09(2019)029

3.1 Ambient space

To classify possible symmetries, it is convenient to use the (A)dS ambient space formal-

ism [30, 31], where we describe (A)dS

as a surface X

(x) embedded in a 5-dimensional ﬂat

ambient space with Cartesian coordinates X

(see appendix B of ref. [21] for more details

of the ambient space formalism in our conventions). A vector ﬁeld A

(x) with mass (2.2) is

bijectively mapped to a vector ﬁeld, A

(X), in the ambient space satisfying the following

homogeneity and transversality conditions:

∂

= A

and X

= 0, (3.1)

where the (A)dS vector is recovered via the pullback,

(x) = ∂

(x)A

(X(x)) . (3.2)

(A)dS Killing vectors are represented in ambient space by constant antisymmetric

tensors. We call the antisymmetric tensor associated to the (A)dS isometries J

. The

Killing vector 

of (2.4) is then given through the relation



(x) = ∂

(x)J

(x). (3.3)

We call the antisymmetric tensor associated to the shift symmetries S

, so the Killing

vector ξ

of (2.3) is given through the analogous relation

(x) = ∂

(x)S

(x). (3.4)

3.2 Abelian shift symmetry

We now describe the symmetry algebra of the free theory. The isometries acting on the

ambient space vector take the form

≡ J

= (X

∂

− X

∂

) A

+ (J

)

, (3.5)

where (J

)

≡ η

−η

is the Lorentz generator in the vector representation.

These satisfy the commutation relations of the (A)dS

isometry algebra, so(5),

, J

] = η

− η

+ η

− η

. (3.6)

The shift symmetry (2.3) written in ambient space is δA

= S

, so its form is

≡ S

= η

C[A

. (3.7)

Due to the (A)dS covariance of the shift generators S

, their commutators with the

isometries (3.5) are

, S

] = η

− η

+ η

− η

. (3.8)

Since the shifts (3.7) are independent of the ﬁeld, they commute among themselves,

, S

] = 0. (3.9)

The commutators (3.6), (3.8), and (3.9) together close to form the algebra of the free

theory. It is a semi-direct product of so(5) with the abelian algebra of the shifts.

– 6 –

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