Extended Jackiw-Pi Model and Its Supersymmetrization

We present an extended version of the so-called Jackiw-Pi (JP) model in three dimensions, and perform its supersymmetrization. Our field content has three multiplets: (i) Yang-Mills vector multiplet $(A_\mu{}^I, \lambda^I)$, (ii) Parity-odd extra vector multiplet $(B_\mu{}^I, \chi^I)$, and (iii) Scalar multiplet $(C^I, \rho^I, f^I)$. The bosonic fields in these multiplets are the same as the original JP-model, except for the auxiliary field $~f^I$ which is new, while the fermions $\lambda^I, \chi^I$ and $\rho^I$ are their super-partners. The basic difference from the original JP-model is the presence of the kinetic term for $C^I$ with its modified field-strength $H_\mu{}^I \equiv D_\mu C^I + m B_\mu{}^I$. The inclusion of the $C^I$-kinetic term is to comply with the recently-developed tensor hierarchy formulation for supersymmetrization.

the clash between two local invari-ances. It is conjectured that such a bifurcation effect could appear in the JP-model, since it conforms two local invariances.
The importance of JP-model can be found in a different context. It has been conjectured that the super-algebra OSp(1|32) is the full symmetry group of M-theory [10] [11]. It was pointed out in [12] that CS theory for the super-algebra OSp(32|1) appears to contain the so-called M-theory matrix models [13]. Therefore the aforementioned advantage of JP-model over CS theory mandates the supersymmetrization of the original JP-model [2].
The original JP-model [2] has the following lagrangian in our notation: where D µ is the usual Yang-Mills (YM) gauge-covariant derivative, while F µν I and G µν I are the field strengths of A µ I and B µ I defined by [2] F µν I ≡ +2∂ ⌊ ⌈µ A ν⌋ ⌉ I + mf IJK A µ J A ν K , (1.2a) (1.2b) The vector B µ I has its proper 'gauge' invariance: The latter transformation combined with the peculiar F ∧ C -term in (1.2b) maintains the invariance δ β G µν After the recent development of non-Abelian tensor formulations [14] [15], the sophisticated structures (1.1) through (1.4) can be now understood as a special case of more general 'tensor hierarchy' whose supersymmetrization has been also accomplished. Therefore it is imperative to encompass the JP-model into this newly developed formulation and also study it's supersymmetrization. In passing, we note that the 4D formulation of non-Abelian tensor multiplet [15] has three multiplets: vector multiplet (A µ I , λ I ), a tensor multiplet (B µν I , χ I , ϕ) and a compensator vector multiplet (C µ I , ρ I ). These are 4D multiplets, and their 3D analogs are respectively our present vector multiplet (VM) (A µ I , λ I ), an extra vector multiplet (EVM) (B µ I , χ I ) and the scalar multiplet (SM) (C I , ρ I ). 3) The fact that the compensator vector multiplet (C µ I , ρ I ) in 4D has its own kinetic term indicates the SM (C I , ρ I ) in 3D should have also its own kinetic terms to accomplish its supersymmetrization, even though the original JP-model had no such a kinetic term for the C I -field [2].
From this viewpoint, we first extend the original JP-model with the kinetic term of the C I -field, and establish its consistency. We refer to this bosonic model as the extended JPmodel. Having accomplished this step, we next perform its N = 1 supersymmetrization.
In the next section, we present the relevant details of the extended JP-model by including the kinetic term of the C I -field. Subsequently, the super-invariant action is presented in section 3. We investigate the consistency of field equations in section 4. In section 5, we perform superspace re-formulation as an addition confirmation on our component formulation. Concluding remarks are given in section 6.
2. Extended JP-Model. As has been alluded to, we comply with the general pattern of tensor-hierarchy formulations [14][15] by introducing the C I -kinetic term 4)

3)
We introduce an auxiliary field f I later for off-shell formulation for the SM.

4)
We assign the engineering dimension 0 (or 1/2) for fundamental bosons (or fermions), so that our lagrangians have the dimension of (mass) 2 . We can recover the usual (mass) 4 for dimensionless action I JP ≡ κ −2 d 3 x L JP , by using a constant κ with the dimension of length. Accordingly, the gauge-coupling constant m has the dimension of mass.
Here H µ I is the C I -field strength [2] H µ I ≡ D µ C I + mB µ I .

(2.2)
Even though this modified field strength was introduced in the original paper by Jackiw-Pi [2], the kinetic term of the C -field was not explicitly introduced. As has been mentioned, this modification is motivated by the recently-developed 'tensor hierarchy' formulation [14][15], as a special case. Due to the modified field-strength for C I , the original B µ I -field equation in [2] is modified to The important consistency question is Note here that these remaining terms vanish exactly due to the C I -field equation: The second equality here is only an identity, similar to the Bianchi identity.
Eq. (2.6) is also related to the invariance of our action I JP under the vectorial symmetry Because of this property, it is straightforward to confirm δ β I JP = 0. This action invariance leads to re-producing the previous result (2.6).
There is an alternative better method of variations for supersymmetric variations which we present later. We can show that the general variations of G and H -field strengths are According to (2.7), the first three terms of (2.1) are manifestly invariant, while the mB ∧ F -term yields In other words, neither the bare B nor the bare C -field term arise in terms of the modified variation δ B µ I , so that the invariance δ β I JP = 0 becomes manifest.
3. N = 1 Superinvariant Action. As has been mentioned, for supersymmetrization of the extended JP-model, we introduce the three multiplets: where f I is an auxiliary field, such that all of our multiplets are off shell. Our total action I ≡ κ −2 d 3 x L has the lagrangian The definition of the field strengths F and G are exactly the same as (1.2), while that of H is given by (2.2). These field strengths satisfy their proper Bianchi identities: Similar to the previous section, the invariance δ β I = 0 under β -transformation is easily confirmed.
Our action I is also invariant under N = 1 supersymmetry Notice that there is no fermionic-quadratic terms in δ Q λ, while λρ or λχ -terms exist in δχ and δρ, respectively. They are determined by the supersymmetric invariance δ Q I at where the symbol Φ stands for any fundamental field in our system, which may contain derivative(s). Our multiplets VM and EVM are all off-shell, as can readily be established by counting their degrees of freedom (DOF) 1 + 1 (on-shell), and 2 + 2 (on-shell). Our SM has 1 + 1 (on-shell) and 2 + 2 (off-shell) DOF, because the auxiliary field f I carries one off-shell DOF. The C I -field plays the role of Nabmu-Goldstone field that is absorbed into the longitudinal component of B µ I , making the latter massive.
For completeness, the DOF of our fields are listed in Table 1  The invariance confirmation δ Q I = 0 is summarized as follows. They are confirmed order-by-order in terms of the power of fundamental fields, such as Φ 2 , Φ 3 , · · ·. First, at the quadratic order, there are two categories of terms: (I) m 0 Φ 2 -terms and (II) m Φ 2 -terms.
The sector (I) is rather a routine confirmation, while there is one subtlety in sector (II), associated with the variation of the mF ∧ B -term in the lagrangian. This is because δ Q B µ I in the first expression in (3.3c) contains the bare C -field. However, as the arbitrary variation of the mF ∧ B -term shows in (2.10), the bare C -field term does not arise.
Relevantly, the supersymmetry transformation rule δ Q B µ I is the second expression in (3.3c). This is a common feature of a potential field whose field strength is a modified (generalized) CS-term.
After tedious cancellations and by the use of the relationships, such as the Jacobi identity h ⌊ ⌈IJ,K]L ≡ 0, the final form of the lagrangian is obtained, e.g., the absence of the χ 2 ρ 2 -terms in the lagrangian, and the absence of (Fermion) 2 -terms in δ Q λ. We have found that these structures are uniquely determined by the cancellation of these terms at m 0 Φ 4 . The f I -dependent terms cancel each other, justifying the ρλ 2 -term in δ Q f I and f I -linear term in δ Q ρ I . As for all of the auxiliary-field f I -dependent terms in δ Q I, they cancel themselves manifestly, if we use the last expression of (3.3g).
However, we expect that the renormalizability of the supersymmetric JP-model presented here will be much improved from its original form due to supersymmetry, a feature common to all supersymmetric theories. Typical examples are non-linear sigma-models, which are originally not renormalizable, but become even finite by supersymmetrization, such as finite N = 2 supersymmetric sigma-models [16].

Consistency of Field Equations.
We first list up the field equations of all of our fields obtained from our action I of (3.1): δI As has been discussed in the non-supersymmetric case with (2.6), the most crucial consistency question is whether the divergence of the B µ I -field equation vanishes. This is confirmed as the supersymmetric generalization of the purely bosonic case. The result is simply Note that the middle equality here is an identity, and no field equation has been used. This is formally the same as the non-supersymmetric case (2.6), since this is nothing but the δ β -invariance of our action: Note that the second equality in (4.2) can be explicitly confirmed for our field equations (4.1). In particular, when we apply the covariant derivative to (4.1e), all terms cancel themselves, including the quartic-fermion terms. Crucial cancellations occur where identities are needed, such as where k IJ,K,LM ≡ f IJN f N KP f P LM . These identities are confirmed by the relationships We can also confirm similar consistency for the A µ I -field equation: This is nothing but the YM-gauge invariance of our action: By the use of (4.3), the (δ α C)(δI/δC) -term in (4.8a) is replaced by m −1 α CD(δI/δB) -term, which in turn is replaced by (4.9) and the last mB(δI/δB) -term will be cancelled by the like-term in (4.8a). Eventually, we end up with (4.8b).

Superspace
Re-Formulation. We can reconfirm our component-field result in terms of superspace language [17]The basic ingredients are the superfield strengths F AB I , G AB I and H A I , 6) satisfying the Bianchi identities The constraints at engineering dimensions 0 ≤ d ≤ 1 are Other independent components, such as F αβ I are all zero. The constraints at d = 3/2 are The ρ α I -field equation is obtained by the 'on-shell-ness' requirement f I . with (4.1c) in component which is skipped here. As for λ I and χ I -field equations, they can be obtained only by the action invariance. We can confirm their consistency with supersymmetry by taking their spinorial derivative ∇ α , yielding the bosonic field equations (4.1d) through (4.1g).
Note that the off-shell structure of our system is consistent with our own component result. This also provides the supporting evidence of the total consistency of our system. From this viewpoint, we regard our system is the unique supersymmetrization of the original JP-model [2], which necessitates the existence of the physical SM (C I , ρ α I ; f I ).
In this Letter, we have accomplished the N = 1 off-shell supersymmetrization of the extended JP-model [2]. This necessitates the introduction of the kinetic term of the C I -field.
There are two reasons for our introduction of the kinetic term of the C I -field: First, it is motivated by the recent development of tensor hierarchy formulation [14] [15]. The consistency of the B µ I -field equation is associated with the δ β -invariance of our action which is not well stressed in the original JP-model [2]. Second, it excludes the extra constraint f IJK F µν J G µν K .
= 0, because this served as the obstruction to supersymmetrizations.
We have also confirmed the total consistency of our supersymmetric system. We have confirmed the identities (4.2) and (4.6) by using our field equations in (4.1). In particular, these consistencies have been explicitly confirmed even with non-trivial fermionic quartic terms. Involving all field equations, this non-trivial confirmation procedure has established the total consistency of our system. Additional confirmation has been performed also in superspace.
Our supersymmetric system is non-trivial. We can not simply truncate the kinetic term of the SM (C I , ρ I ; f I ), because the action invariance no longer respects invariance for the truncated system. This again justifies the necessity of the kinetic terms for C I and ρ I .
We believe our present result should help in generating other and new consistent topological massive non-Abelian gauge theories and their supersymmetrization.