Shift Symmetries for p-Forms and Mixed Symmetry Fields on (A)dS

Massive fields on (anti) de Sitter space realize extended shift symmetries at particular values of their masses. We find these symmetries for all bosonic p-forms and mixed symmetry fields, in arbitrary spacetime dimension. These shift symmetric fields correspond to the missing longitudinal modes of mixed symmetry partially massless fields where the top row of the Young tableau is activated.


Introduction
In [1], shift symmetries acting on massive symmetric tensor fields on (anti)-de Sitter space ((A)dS) were found, generalizing the extended shift symmetries of the galileon [2,3] and special galileon [4][5][6][7] on flat space. We refer to the introduction of [1] for more extensive background and motivation.
A summary of the results of [1] is as follows. A massive spin s field on AdS D of radius L, carried by a symmetric tensor field φ µ 1 ···µs , satisfies on shell the Klein-Gordon equation, The shift symmetry reads δφ µ 1 ···µs = S A 1 ···A s+k ,B 1 ···Bs X A 1 · · · X A s+k ∂X B 1 ∂x µ 1 · · · ∂X Bs ∂x µs , (1.3) where the X A are embedding coordinates of AdS D into a flat spacetime of dimension D + 1, and the tensor S A 1 ···A s+k ,B 1 ···Bs is a fully traceless, constant embedding space tensor with the index symmetries of the following Young tableau, As explained in [1], the k-th shift symmetric spin s field can be thought of as the missing longitudinal mode of a partially massless (PM) spin s + k + 1 field of depth s (i.e. it has a rank s gauge parameter). There is a symmetric traceless 'field strength' with the same indices as this partially massless field, made from k + 1 derivatives of the shift symmetric field in a way that mirrors the PM gauge transformation, F µ 1 ...µ s+k+1 = ∇ (µ s+1 · · · ∇ µ s+k+1 φ µ 1 ...µs) T . Here, we will extend the results of [1] by finding the analogs of the above shift symmetries for all the remaining bosonic fields, the anti-symmetric p-form fields and the mixed symmetry fields. In the process, we will also see that there are no further shift symmetries of this type beyond those we find, and none further for the symmetric tensors beyond those in (1.2).
We will find these shift invariant fields as longitudinal modes of various mixed symmetry fields as they approach PM points. In what follows, we will therefore make frequent use of the classification of PM points for general mixed symmetry fields [8][9][10][11][12][13][14][15]. A short summary is as follows. Consider a general massive field with the symmetries of a p row Young tableau with row lengths [s 1 , s 2 , . . . , s p ], which on-shell is traceless and divergenceless in all indices and is annihilated by the Klein-Gordon operator ∇ 2 −m 2 . This field has dual conformal field theory (CFT) dimensions ∆ ± found from the massm 2 by finding the greater and lesser roots ofm The partially massless points occur when squares in the tableau from a row which is longer than the row below it are 'activated'. If it is the q-th row that is being activated, then the number of squares that can be activated ranges from 1, 2, . . . , s q − s q+1 . We assign a depth t = 0, 1, · · · , s q − s q+1 − 1, which indicates that s q − s q+1 − t of the squares in the q-th row are activated. By removing the activated squares, we get the Young tableau of the gauge parameter, and each activated square becomes a derivative in the gauge transformation law.
These PM points occur at integer values of the dual CFT dimension given by The gauge transformation is generally reducible, with q − 1 levels of gauge-for-gauge parameters. It will be important that the only irreducible gauge symmetries are for q = 1, i.e.
when the first row is activated; it is these whose longitudinal modes will give rise to the shift symmetric fields. Tensors are symmetrized and antisymmetrized with unit weight, e.g. t [µν] = 1 2 (t µν − t νµ ), and (· · · ) T means the symmetric fully traceless part of the enclosed indices. Young tableaux are denoted [s 1 , s 2 , . . . , s p ] where s i is the number of boxes in the i-th row, and are deployed in the manifestly anti-symmetric convention. We sometimes use the shorthand of using an exponent to denote multiple rows of the same length, e.g. [4, 2 3 , 1] ≡ [4, 2, 2, 2, 1]. We denote the corresponding Young projectors by Y [s 1 ,s 2 ,...,sp] . The notation T on a tableau or projector indicates that it is fully traceless.

Conventions
Masses for mixed symmetry fields are denoted bym 2 , which are the "bare masses" that appear in the Klein-Gordon equation satisfied by the transverse and traceless field: For symmetric tensors and p-forms, there is a traditional notion of "massless," which is the partially massless point with the largest depth (the only partially massless point in the p-form case), so in these cases we use this more traditional m 2 which is shifted relative tom 2 so that m 2 = 0 corresponds to this massless point.

Shift symmetries for 2-forms
We start by illustrating the general pattern with the simplest case not covered by [1], the massive 2-form field. We will do the simplest instance of this case fully off-shell at the Lagrangian level, then proceed to more on-shell methods as we go on to more general cases.
Consider the Lagrangian for a 2-form field B µν of mass m on AdS D , is the field strength. The equations of motion can be cast in the form The mass is defined such that when m 2 = 0 we get the usual massless 2-form gauge symmetry.
There are no other points of enhanced gauge symmetry besides this.
As we will see, the massive 2-form theory (2.1) gets an enhanced shift symmetry when m 2 is set to the following values, The form of the k-th shift symmetry is where S B 1 ...B k+1 ,A 1 ,A 2 is a constant, fully traceless mixed symmetry embedding space tensor of type and there is a k + 1 derivative shift invariant field strength which is a traceless [k + 2, 1] We will find these shift symmetries by considering partially massless limits of appropriate mixed symmetry fields. For k = 0 the appropriate field is a massive [2,1] hook field (also known as a Curtright field [16,17]). The Lagrangian for a massive [2,1] hook field on AdS D is [18,19], (2.7) The equations of motion can be cast in the form The theory (2.7) has two mass values at which partially massless gauge symmetries arise [20]: • The first is where the top block is activated, giving an antisymmetric tensor gauge symmetry, This gauge symmetry has no gauge-for-gauge reducibilities. This mass value is unitary on AdS and non-unitary on dS.
• The second is where the bottom block is activated, giving a symmetric tensor gauge symmetry, This gauge symmetry has a gauge-for-gauge reducibility where with scalar gauge-for-gauge parameter χ. This mass value is unitary on dS and nonunitary on AdS.
Consider first the decoupling limit where we approach the first partially massless value (2.9) from above, To preserve the degrees of freedom in this limit we introduce a 2-form Stückelberg field B µν and make the Stückelberg replacement where we have inserted the factor of 1/ so that B µν will come out canonically normalized up to numerical factors. This replacement introduces a Stückelberg gauge symmetry under which the Stückelberg field shifts, (2.14) In the limit (2.12), the Lagrangian (2.7) splits up into a partially massless hook (2.9) and a correct-sign massive 2-form with mass m 2 L 2 = 3 (D − 2), This mass is precisely the k = 0 value of (2.3).
As discussed in [1], the shift symmetry arises from reducibility parameters of the Stückelberg gauge symmetry, i.e. values of Λ µν for which the gauge transformation of the hook field vanishes, ∇ [µ Λ ν]ρ − ∇ ρ Λ µν = 0. In terms of the embedding space field Λ AB corresponding to the gauge parameter Λ µν , this equation says that the mixed symmetry part of ∂ A Λ BC should vanish. In addition, Λ AB should be transverse to X A , should satisfy the higher dimensional Klein-Gordon equation (D+1) = 0, and should be of weight 1 in X A (so that the Klein-Gordon equation pulls back to the correct mass). The solution for all this is

which when pulled back to AdS
gives the k = 0 case of (2.4) 1 This implies that there is a one-derivative 'field strength' with the same symmetries as the parent PM field f µνρ , which is invariant under the k = 0 shift symmetry (2.4). The trace of this field strength is proportional to ∇ ν B µν and so vanishes on-shell. The traceless part is the basic on-shell non-trivial shift invariant operator in the theory, and is the k = 0 case of (2.6).
We can also consider a decoupling limit where we approach the second partially massless In this case we introduce a symmetric Stückelberg field In the limit (2.18), the Lagrangian (2.7) splits up into a partially massless hook and a massive spin-2, where L [2],m 2 is the Fierz-Pauli action for a massive spin-2 field on AdS D (as written in e.g. (5.2) of [35]). This massive spin-2 that we get is not a new shift-symmetric field, rather its mass m 2 L 2 = −(D − 2) is that of the partially massless graviton [36,37]. This is because of the gauge-for-gauge reducibility (2.11), which shows up in the decoupling limit as a partially massless gauge symmetry for the longitudinal field H µν . We can see all of the above from the dual CFT perspective [38,39]. The massive [2,1] field has a dual [2,1] traceless primary state where the mass and conformal scaling dimension are related bỹ Denoting the larger and smaller roots of this as ∆ ± , the PM point of interest (2.9) gives The submodule spanned by the null states is (d + 1, [1,1]).
The mass of a 2-form is related to its conformal dimension by which is also fully traceless and fully divergenceless on shell. We consider the partially massless point where all the possible top blocks are activated, so that the gauge parameter (2.28) This gauge symmetry is irreducible, so the resulting longitudinal mode in the partially massless limit will be a shift field rather than a gauge field.
Using reasoning similar to the k = 0 case we can find the reducibility parameters for the gauge symmetries (2.28) and they are the shifts (2.4). There is a k + 1 derivative field strength with the same symmetries as the PM field f µνρ 1 ...ρ k+1 , made out of k + 1 derivatives of the shift field patterned after the gauge transformation (2.28), which is invariant under these shifts and gives (2.6).
A massive [k + 2, 1] field has a dual [k + 2, 1] primary state where the mass and conformal scaling dimension are related bỹ At the PM point of interest (2.28), the conformal dimension is Since the gauge transformation (2.28) has k + 1 derivatives, the dual state at the value (2.31) is a kind of multiply-conserved current [40] which gets a shortening condition at level k + 1 in the Verma module, P l 1 · · · P l k+1 |f ij l 1 ...l k+1 ∆ = 0. This is a null state of spin [ has a gauge-for-gauge reducibility, and the longitudinal mode will be a depth t = 0 partially massless spin k + 2 field, so this gives no new shift symmetric fields.
The PM points where fewer than the maximal number of top blocks are activated will have gauge parameters which are again mixed symmetry tensors, and since these gauge transformations are irreducible, these will give rise to shift-symmetric points for these mixed symmetry tensors. We will return to this more general case in section 4, after discussing the higher p-forms in the next section.
We can ask if there are other possible shift-symmetric mass values for the 2-form besides those in (2.3), perhaps coming from PM limits of more complicated mixed symmetry tensors.
The answer is no for the following reason. As we have seen, to get a shift-symmetric field for the longitudinal mode the PM gauge symmetry must be irreducible, otherwise the gauge-forgauge reducibility parameters will become gauge symmetries of the longitudinal mode and we will get other PM fields rather than shift fields. The depth of gauge-for-gauge redundancy in a PM field is equal to the row number which is activated [13]. The PM symmetry must therefore come from a PM tableau where only the top row is activated, so that the gauge symmetry is irreducible. To get a shift-symmetric 2-form, we need a PM tableau whose first row is activated and whose gauge parameter is a 2-form. The only such tableaux are those in (2.28). The same reasoning shows that the shift symmetric points (1.2) for symmetric tensor fields, and the shift symmetric points for more general fields that we find below, are the only ones.

Shift symmetries for p-forms
We now move on to a massive p-form field B µ 1 ...µp on AdS D , p ≥ 1. The equations of motion can be cast in the form The mass is defined such that at m 2 = 0 we get the usual massless p-form gauge symmetry.
There are no other points of enhanced gauge symmetry besides this.
The massive p-forms (3.1) get an enhanced shift symmetry at the following mass values (3.2) and the form of the shift symmetry is There is an invariant field strength of type [k + 2, 1 p−1 ], The shift symmetries (3.3) are the reducibility parameters of this PM transformation.
A massive [k + 2, 1 p−1 ] field has a dual primary operator with conformal scaling dimension related to the mass bym At the PM point of interest (3.7), the conformal dimension is The relation between the mass and dual CFT scaling dimension of a p-form field is given by and in addition is also fully traceless and fully divergenceless. This field has a dual conformal dimension related to the massm 2 bỹ At these values, the dual CFT conformal dimension is Since the gauge symmetry has k + 1 derivatives the dual operator gets a conservation-type shortening condition at level k + 1, giving a null state of spin [s 1 , s 2 , . . . , s p ] and dimension Using w = s 1 + k this reproduces the masses (4.8). We can think of the right hand side of (4.9) as the most general kind of Killing-Yanolike object, or spherical harmonic, on AdS D .
There is a k + 1 derivative 'field strength' with the symmetries [s 1 + k + 1, s 2 , · · · , s p ] of the parent PM field which is invariant under the shifts (4.9),

Conclusions and Discussion
We have found the points of enhanced shift symmetry for arbitrary bosonic p-forms and mixed symmetry tensors on (A)dS D space. This extends [1], which covered only the symmetric tensors. The results for the masses are summarized in (4.8), and the form of the shift symmetries is (4.9). These fields capture the longitudinal mode that decouples when a massive field approaches a PM point where the top row is activated, so that the gauge  On dS, the shift symmetric fields correspond to shortened irreducible representations of the de Sitter algebra (of type V, in the notation of [42]) but they generally lie beyond the complementary series and do not correspond to any discrete series points [15,43], so they are non-unitary. A notable exception is the shift symmetric scalars [44][45][46][47][48], which are unitary in dS D and correspond to scalar exceptional series representations [42,[49][50][51]. There are some other lower dimensional exceptions as well, related to the scalars by duality. For example in D = 3 the level k shift symmetric 2 form is dual to the level k + 1 shift symmetric scalar (as can be seen from the fact that their masses are equal) so the 2-form shift fields are unitary in dS 3 .
In fact, all the discussions in this paper must be understood modulo these massive dualities. For example, the construction in Section 2 of the shift symmetric 2-forms as longitudinal modes of a hook field fails in D = 3 since hook fields are non-dynamical for D < 4. But nothing is missed in this case because the shift symmetric 2-form fields are dual to shift symmetric scalars, and these can be constructed as longitudinal modes of massive symmetric tensor fields which are dynamical in D = 3. More generally, for low enough dimension where the parent PM field is non-dynamical but the shift symmetric field is, the construction of the shift field as a longitudinal mode fails and the shift field will typically be equivalent by duality to a different shift field whose parent PM field is dynamical. But this is not always the case, for example in D = 2 the shift symmetric scalars for k ≥ 1 cannot be constructed as longitudinal modes of any dynamical massive field.
A natural question is whether non-trivial shift symmetric interactions can be found for the more general representations studied here. Interactions can always be written using powers of the shift invariant field strength (4.11), so here 'non-trivial' means that the interactions are not simply powers of the field strength. The k = 1, 2 scalars can be given non-trivial self-interactions [1,[52][53][54], as can the k = 0 vector [26,55], but no other examples are currently known.
Whether interactions are non-trivial in the sense mentioned above is also tied to whether there are there are non-trivial algebras that could underlie the symmetries [56,57].
A trivial interaction will not deform the abelian algebra of shift symmetries present in the linear theory, whereas a non-trivial interaction should deform the algebra into a non-abelian algebra. It would be interesting to know if there are finite algebras of the type studied in [58] (which are finite subalgebras of higher spin algebras underlying PM Vasiliev theories [58][59][60][61][62]), that could be candidates to underly non-trivially interacting shift symmetric p-form or mixed symmetry theories.
Non-trivial theories would also presumably have a flat space limit which gives interacting p-form or mixed symmetry theories in flat space. From the point of view of the S-matrix, the non-trivial effective field theories on flat space should be theories with enhanced soft limits that allow for a recursive reconstruction of the amplitudes [4,6,63,64], so it would be interesting to study if there are such possibilities for fields in these other representations. In flat space, interactions for p-form galileons are known [65][66][67], though these are presumably Wess-Zumino-like [68] interactions that do not deform the basic underlying shift symmetries. It would be interesting to see whether these interactions could be extended to (A)dS and/or deformed into non-trivial interactions, or whether they can harbor hidden special galileon-like non-trivial enhancements.