Frame-like gauge invariant Lagrangian formulation of massive fermionic higher spin fields in AdS_3 space

We construct the frame-like gauge-invariant Lagrangian formulation for massive fermionic arbitrary spin fields in three-dimensional $AdS$ space. The Lagrangian and complete set of gauge transformations are obtained. We also develop the formalism of gauge-invariant curvatures for the massive theory under consideration and show how the Lagrangian is formulated in their terms. The massive spin-5/2 field is discussed as an example.


Introduction
Despite the significant progress in the theory of massless higher-spin fields in various dimensions (see reviews [1][2][3], see also [4] for details), the construction of full higher spin field theory is still far from being complete. In particular it remains unclear how the non-linear theory of massive higher spins should look like. Investigations of massive higher spin interactions are very important in-particular for understanding the relation between higher spins and (super)string theories that is assumed to realize a kind of spontaneous symmetry breaking mechanism. Thus a natural framework for such investigations is a gauge invariant formulation for massive higher spin fields (similar to the one appearing in string field theory) that becomes possible due to introduction of appropriate number of Stueckelberg fields. Currently gauge invariant description for the free massive higher-spin fields is rather well studied, moreover various authors have developed different ways of such description [5][6][7][8][9].
At the same time studying the massive higher spin interactions in arbitrary dimensions appears to be even more complicated that the ones for the massless fields. Therefore it would be instructive to investigate a structure of the massive higher spin theories coupled to external fields or among themselves in three-dimensional space where situation becomes simpler than in a space of arbitrary * Corresponding author. dimension. As a result we can get a nice playground to gain useful experience and possibilities for generalizations. In particular it turns out that in d = 3 there exist examples of interacting models with finite number of higher spin fields [10][11][12][13]. Moreover the specific properties of three dimensional space allow us to construct the more exotic higher spin models [14]. Therefore we can expect that the massive higher-spin theory also becomes easier in d = 3.
First Lagrangian formulation for massive higher-spin fields in d = 3 was considered in [15] however in gauge invariant form it has been developed for bosonic fields only [16][17][18]. In this work we fill this gap and give gauge invariant formulation for massive fermionic higher spins in three-dimensional AdS 3 space. In this we think that the most convenient formalism is the frame-like one [19][20][21], which in particular allows one to work with an explicitly invariant objects.
The paper is organized as follows. In Section 2 we collect the basic information about frame-like formulation of free massless higher spins in AdS 3 focusing on the fermionic fields only. In Section 3 we give the frame-like gauge-invariant formulation for free massive fermionic higher spins in AdS 3 . Here at first we discuss a field content which we need to have gauge invariant description of massive fields. Then in terms of these fields we derive the Lagrangian and corresponding gauge transformations. We also construct the full set of gauge invariant linearized curvatures and show how the free Lagrangian can be rewritten in their terms. At the end of Section 3 we consider the concrete example of massive spin-5/2 field. In conclusion we summarize the main points.

Massless fermionic fields
In this section we briefly review a frame-like formulation of massless higher spin fermionic fields in AdS 3 space. The Lagrangian, gauge transformation and gauge invariant curvatures for free fields are given.
Frame-like formulation of massless higher-spin fields can be treated as generalization of the frame formulation of gravity in terms of vielbein field h μ a and Lorentz connection ω μ a,b . Such generalization was successfully developed both for the bosonic fields and for the fermionic ones [19][20][21]. We will focus here only on half-integer spins, the details for integer spins can be found in [19,20].
It is known [19] that in four dimensions the field with halfinteger spin s > 3/2 in the frame-like formulation is described by a total bunch of spin-tensor one-forms (here we omit spinor index) is the absence of these extra fields that greatly simplifies the calculations in construction of consistent higher spin field theories. Moreover the description can be simplified even more by the use of multispinor frame-like formalism where all fields are still oneforms but with all local indices replaced by the spinor ones [11].
Free Lagrangian being three-form in three dimensional AdS 3 space looks as follows [1,11] Using this curvature the free Lagrangian can be rewritten as follows In order to verify gauge invariance of the Lagrangian in such form we should use the Bianchi identity. It is easy to see that the curvature just gives the equations of motion for the Lagrangian (1). Note that the possibility to work in terms of the curvatures is a peculiarity of the frame-like formalism.

Massive fermionic fields
In this section we develop the frame-like gauge invariant description for massive arbitrary half-integer spin s ≥ 3/2 in AdS 3 space. To provide the gauge invariance we introduce the auxiliary Stueckelberg fields and construct the Lagrangian in their terms. Then we generalize the formalism of gauge-invariant curvatures for massive fields. General formulation is illustrated on the example of massive spin-5/2 theory.

Lagrangian formulation
Gauge invariant Lagrangian formulation of massive fields is based on introduction of the auxiliary Stueckelberg fields. We follow the procedure proposed in [6] to use the minimal number of such fields. In the case under consideration the full set of field variables includes the following one-forms Ψ α(n) , n = 1, 3, ..., 2(s − 1) (each one with its own gauge transformations) and zeroform φ α . We will look for the free Lagrangian for massive field as the sum of kinetic terms for all these fields as well as the most general mass-like terms gluing them together: and a n , b n are free parameters to be determined. The most general form of the corresponding gauge transformations looks like δ 0 Ψ α(n) = Dξ α(n) + α n e α β ξ α(n−1)β + β n e αα ξ α(n−2) + γ n e ββ ξ α(n)ββ where α n , β n , γ n are also free parameters. Our aim is to find the restrictions on the parameters in the Lagrangian and gauge transformations providing the gauge invariance of the Lagrangian (2) under the transformations (3).
First of all we consider variations of the Lagrangian that are of the first order in derivatives. This allows us to express the parameters a n , b n in the Lagrangian through parameters in the gauge transformations a n = n(n − 1) and also imposes one restriction on the parameters of the gauge transformations Note that this condition is valid because β n is defined for 3 ≤ n ≤ 2(s − 1) and γ n for 1 ≤ n ≤ 2(s − 2). The remaining free parameters α n , γ n are fixed from the invariance conditions under the transformations (3) for the variations without derivatives. Direct calculations yield the following equations 2(n + 2)a n α n − 2b n−2 γ n−2 = 0 n ≥ 3 Last equation allows us to express the only remaining free parameter in the Lagrangian b 0 b 0 = −3α 1 Taking into account the relations (4), (5) we see that first two equations in (6) are identical and lead to a simple recurrent relation Denoting the maximal value of n = 2(s − 1) =ŝ and expressing everything through αˆs we get the general solution for α n in the form α n =ŝ (ŝ + 2) n(n + 2) αˆs (7) Now let us consider the third equation (6) containing two parameters Here we have used the relations (4), (5). First, note that the γ n is absent for n =ŝ. Therefore one can express αˆs through γˆs −2 (as we will see later the parameter γˆs −2 will remain as the only free one and will play the role of a mass parameter) Taking into account (7), we see that the relation (8)  (ŝ − n + 2)(ŝ + n + 2) 4n(n + 1) where we introduced mass parameter m 2 =ŝγ s−2 2 . Finally from the fourth equation of system (6) it follows that Let us also introduce convenient combination so that now Thus all parameters are found and are defined by (4), (5), (10)- (13) with m 2 as the only free one. Then the final expression for the Lagrangian (2) takes the form: where we denote γ n−2 = m n and α 0 = m 0 while the gauge transformations (3) look as follows δ 0 Ψ α(n) = Dξ α(n) + (ŝ + 2)M 2n(n + 2) e α β ξ α(n−1)β + 2 n(n − 1) m n e αα ξ α(n−2) + m n+2 e ββ ξ α(n)ββ Note that such gauge invariant description works not only in AdS (and Minkowski) space but in dS space as well provided m 2 > 1 4ŝ 2 Λ, Λ = −λ 2 so that massless limit is possible in the Minkowski and AdS spaces only. Inside the unitary forbidden region there exists a number of partially massless cases [22][23][24]5,6]. Namely each time when one of the parameters m n becomes zero the whole system decomposes into two disconnected ones. One of them with the fields Ψ α(2s−1) . . . Ψ α(n) describes partially massless field while the remaining fields describe massive field with spin n 2 − 1.

Formulation in terms of curvatures
In this subsection we develop the formalism of gauge invariant curvatures for massive higher spin fermionic field in AdS 3 and rewrite the Lagrangian, found in previous subsection, in terms of these curvatures. The gauge invariant curvatures have been introduced before to construct the Lagrangian formulation for massless higher spin fields in frame-like approach. We will show that the objects with analogous properties can also be constructed for massive higher spin fields. Since the gauge-invariant description of massive fields uses the auxiliary Stueckelberg fields, it is natural to construct the corresponding curvatures for them as well.
First of all we introduce the additional auxiliary zero-forms C α(n) , n = 3, ..., 2(s − 1) 1 which transform under the gauge transformations as follows δ 0 C α(n) = η n ξ α(n) (16) where the parameters η n are yet to be fixed. We combine these fields with our spinor field denoting φ α = C α and similarly for parameter of gauge transformations η 1 = α 0 . The most general ansatz for the full set of curvatures looks like R α(n) = DΨ α(n) + α n e α β Ψ α(n−1)β + β n e αα Ψ α(n−2) where α n , β n , γ n are the same as in previous subsection and guarantee that curvature R is invariant under the part of gauge transformation containing derivatives. Other parameters are arbitrary and will be fixed by the gauge invariance under the transformations (3), (16). Here C α(n) are the curvatures corresponding to the zero-forms C α(n) . Parameters l n , k n , q n , p n are found from gauge invariance for curvatures C. We have δC α(n) = D η n ξ α(n) + l n Dξ α(n) + α n e α β ξ α(n−1)β + β n e αα ξ α(n−2) + γ n e ββ ξ α(n)ββ + k n e α β η n ξ α(n−1)β + q n e αα η n−2 ξ α(n−2) + p n e ββ η n+2 ξ α(n)ββ It leads to the relations l n + η n = 0 l n β n + q n η n−2 = 0 l n α n + k n η n = 0 l n γ n + p n η n+2 = 0 Using the arbitrariness in definition of C α(n) related with multiplication by constant, C α(n) → c n C α(n) (c n are some numerical coefficients) we can set p n = γ n . Then the general solution of the relations above looks like l n = −η n , The last condition is the recurrent relation. To solve it we use as initial data η 1 = α 0 and then obtain η n = η 1 = α 0 .
To find the last parameters f n we consider the invariance of the Invariance condition leads to −λ 2 + 4α n 2 − 2(n − 1)β n γ n−2 + 2(n + 3)γ n β n+2 + f n η n = 0 2(n + 2)α n β n − 2(n − 2)β n α n−2 = 0 Comparing with relations (6), we conclude that As a result we have the final expressions for the curvatures R α(n) = DΨ α(n) + α n e α β Ψ α(n−1)β + β n e αα Ψ α(n−2) + γ n e ββ Ψ α(n)ββ (18) Here the curvatures corresponding to n = 1 are written separately. Now let us rewrite the Lagrangian (2) in terms of curvatures. The most general expression for it has the form where A n and B n are the arbitrary coefficients and are fixed by requirement to reproduce initial Lagrangian (2). Since the fields C α(n) for n ≥ 3 do not enter in (2) we can immediately put B n = 0, n ≥ 3.
Comparing (19) with Lagrangian (2) we obtain As a result we rewrite the Lagrangian (2) in terms of gauge invariant curvatures (19).

Massive spin 5/2 example
Massive field with spin s = 5/2 is described by a set of fields Ψ α(3) , Ψ α , φ α , C α(3) , where the second and third fields are Stueckelberg ones while the latter is an auxiliary field and enter in the expressions for curvatures only. Then in accordance with (14) the Lagrangian looks like According to (18) the expressions for the linearized gauge-invariant curvatures have the form The set of gauge transformations is written as follows Accordingly to (19) the Lagrangian in term of curvatures can be written in the form

Conclusion
In this paper we have formulated a gauge-invariant Lagrangian description for massive fermionic higher spins in three-dimensional AdS 3 space. Using suitable set of Stueckelberg fields we have derived the gauge-invariant Lagrangian and obtained a full set of corresponding gauge transformations. We have also constructed a complete set of linearized gauge-invariant curvatures on AdS 3 and found that the gauge invariance required to introduce an additional set of auxiliary fields which do not enter the free Lagrangian. Massive spin-5/2 example is considered in details. We hope that our results provide a ground for the development of frame-like gaugeinvariant formalism for interacting massive half-integer spins in three dimensional space.
In three dimensional space it is convenient to use twocomponent spinor formalism and, in contrast to the d = 4, only one type of spinor indices is used. For instance AdS 3 background vielbein is described by one-form e α(2) and massless arbitrary halfinteger spin s is described by one-form Ψ α(n) , n = 2(s − 1) is odd (here α = 1, 2 is spinor index) where the argument of index means the number of totally symmetrized indices. Further we present the list of our notations and conventions for two-component spinor formalism and differential form language (sign ∧ of wedge product in the text of the paper is everywhere omitted): • For spinor indices labeled by one letter and standing on the same level we use agreement of total symmetrization without normalization factor for example e α β ∧ Ψ α(n−1)β = e (α 1 β ∧ Ψ α 2 ...α n )β = e α 1 β ∧ Ψ α 2 ...α n β + (n − 1) sym. terms • For anti-symmetric matrices ε αβ , ε αβ we use the following basic relation and rule for lowering and raising spinor indices ε αγ ε γ β = −δ α β ,