Massive Nambu-Goldstone Fermions and Bosons for Non-relativistic Superconformal Symmetry: Jackiw-Pi Vortices in a Trap

We discuss a supersymmetric extension of a non-relativistic Chern-Simons matter theory, known as the SUSY Jackiw-Pi model, in a harmonic trap. We show that the non-relativistic version of the superconformal symmetry, called the super-Schr\"odinger symmetry, is not spoiled by an external field including the harmonic potential. It survives as a modified symmetry whose generators have explicit time dependences determined by the strength of the trap, the rotation velocity of the system and the fermion number chemical potential. We construct 1/3 BPS states of trapped Jackiw-Pi vortices preserving a part of the modified superconformal symmetry and discuss fluctuations around static BPS configurations. In addition to the bosonic massive Nambu-Goldstone modes, we find that there exist massive Nambu-Goldstone fermions associated with broken modified super-Schr\"odinger symmetry generators. Furthermore, we find that eigenmodes form supermultiplets of a modified supersymmetry preserved by the static BPS backgrounds. As a consequence of the modified supersymmetry, infinite towers of explicit spectra can be found for eigenmodes correspond- ing to bosonic and fermionic lowest Landau levels.


Introduction
Non-trivial external background fields are useful tools to study various aspects of field theories.
When a generic background field is turned on in a physical system, it may break a symmetry of the system and drastically change the structure of the model. However it has been known that if an external field can be viewed as a chemical potential term associated with a conserved charge, a version of the Nambu-Goldstone (NG) theorem can still be applied even when a symmetry appears explicitly broken by the external field. The crucial difference from the standard NG theorem is that the corresponding NG mode in this case has a non-vanishing mass precisely determined by the symmetry algebra. Such massive Nambu-Goldstone bosons have been discussed in [1,2,3,4] and scattering amplitudes of massive NG modes was recently studied in Ref. [5].
In Refs. [6,7,8,9,10], various properties of the massive NG modes associated with the nonrelativistic conformal symmetry, called the Schrödinger symmetry [11,12] , have been revealed in the 2 + 1 dimensional non-linear Schrödinger system in a harmonic trap. One of the most important observations is that the Schrödinger symmetry survives even in the presence of external background fields including the harmonic potential. More precisely, a modified Schrödinger symmetry generated by time dependent operators remains in such a background. In general, when a symmetry generated by an operator with an explicit time dependence is spontaneously broken, the associated NG modes has a non-vanishing mass determined by the commutation relation between the corresponding broken generator and the Hamiltonian. As in the case of the Lorentz and Galilean symmetry, a time dependent symmetry can be used to study dynamical properties of the system. For example, in the non-linear Schrödinger system in a harmonic trap, time dependent solutions can be generated from static ones by applying the time dependent modified Schrödinger symmetry.
In this paper, we discuss the supersymmetric Jackiw-Pi model and study vortices and massive NG modes in a non-trivial background. The Jackiw-Pi model is a field theoretic framework describing anyons in terms of the non-linear Schrödinger system coupled with a Chern-Simons gauge field [13]. As with the standard non-linear Schrödinger model, the Jackiw-Pi model has a modified (time dependent) Schrödinger symmetry in various backgrounds. Non-topological vortex solutions, called the Jackiw-Pi vortices [14,15], have been discussed in such backgrounds [16,17,18,19,20,21,22], and in particular time dependent solutions were constructed by making use of maps between the models with and without the external fields.
The Jackiw-Pi model without background field has a supersymmetric extension which possesses a non-relativistic superconformal symmetry, called the super-Schrödinger symmetry [23,24,25]. In this paper, we show that external background fields corresponding to the harmonic potential, the spatial rotation, the flavor and fermion number chemical potentials, do not spoil the superconformal symmetry as well as the Schrödinger symmetry. In the presence of such external fields, the whole super-Schrödinger symmetry becomes a time dependent symmetry of the type which has been discussed in the context of the supersymmetric harmonic oscillator in quantum mechanics [26,27].
We also discuss the Jackiw-Pi vortices in the non-trivial background fields and construct their 1/3 BPS states which is invariant under a part of the time dependent supersymmetry.
The moduli matrix formalism, which has been used to describe the moduli space of non-Abelian vortices [28,29,30,31,32,33], can also be applied to write down a formal solution of the 1/3 BPS equation in this system. For each choice of a holomorphic matrix H 0 (z), we can obtain a BPS configuration of trapped Jackiw-Pi vortices by solving the Gauss law equation. Generic 1/3 BPS solutions turn out to be Q-soliton-like configurations, that is, they are time dependent stationary configurations stabilized by conserved charges. They are new time dependent solutions which are different from the known solutions obtained by using the maps between the models with and without the external fields [16,17,18,19,20,21].
The BPS solutions becomes static configurations if H 0 (z) takes one of special forms corresponding to the fixed points of the spacial and flavor rotation. We discuss fluctuations around them and show that bosonic and fermionic eigenmodes form supermultiplets of the unbroken time dependent supersymmetry. There are two types of supermultiplets: one is a generic supermultiplet composed of a pair of bosonic and fermionic modes and the other is a short supermultiplet consisting only of a bosonic component. In particular, we show that in addition to bosonic massive NG modes associated with spontaneously broken generators of the modified Schrödinger symmetry, there exist massive Goldstino corresponding to spontaneously broken modified supercharges. They consistently form supermultiplets as expected from the super-Schrödinger algebra.
In addition to those massive NG modes, we exactly derive eigenvalue spectra of infinite towers of short and long supermultiplets corresponding to the bosonic and fermionic lowest Landau levels, respectively.
The organization of the paper is as follows. In Sec. 2, we briefly review the super-Schrödinger symmetry in the supersymmetric Jackiw-Pi model and show that there exists a modified super-Schrödinger symmetry even in the presence of generalized chemical potential terms including the harmonic potential. In Sec. 3, we discuss 1/3 BPS solutions of trapped non-Abelian Jackiw-Pi vortices which preserve a part of the modified superconformal symmetry. By applying the moduli matrix formalism, we write down formal solutions and show that static configurations correspond to fixed points of the rotation and flavor symmetry. In Sec. 4, we investigate fluctuations around static BPS backgrounds and elucidate the structure of supermultiplets of eigenmodes including bosonic and fermionic massive NG modes. Sec. 5 is devoted to a summary and discussions. The supersymmetric Jackiw-Pi model consists of a gauge field and pairs of bosonic and fermionic matter fields. For simplicity, we consider the case of a U(N) gauge field A µ with N F matter pairs (φ I , ψ I ) (I = 1, · · · , N F ) in the (N, N F ) representation of the U(N) gauge group and the SU(N F ) flavor symmetry. It would be straightforward to extend the following discussion to more general settings. By using N-by-N F matrix notation for the matter fields the action of the supersymmetric Jackiw-Pi model can be written as where the trace is taken over the flavor indices. Just for notational convenience, we have introduced the N F -by-N F matrix M and the N-by-N matrix Y deffined by 3) The symbol∆ 0 denotes the differential operator which gives the standard non-relativistic kinetic The covariant derivative and the field strength are defined by The parameter k is the Chern-Simons level and S CS is the Chern-Simons term normalized as By rescaling the gauge field as A µ → A µ /k, we can see that in the infinite level limit k → ∞, this model reduces to the free theory whose equations of motion are given by the Schrödinger equation. In addition to the standard Schrödinger symmetry, the action is invariant under the non-relativistic version of superconformal symmetry, namely the super-Schrödinger symmetry.
We can show that this system has the same symmetry as the free supersymmetric Schrödinger system even for finite k.

Super-Schrödinger algebra
The generators of the super-Schrödinger symmetry is summarized in Table 1. The nonvanishing part of their commutation relation is given by

Bosonic part of super-Schrödinger symmetry
Let ξ µ be the non-relativistic version of the conformal Killing vector where ǫ O are transformation parameters. Then the bosonic part of the super-Schrödinger transformations takes the form where the real functions λ and α are given by

Fermionic part of super-Schrödinger symmetry
To see the invariance of the action under the supersymmetry, let us first note that the following transformation does not change the action: where ζ q and ζ Q are fermionic SUSY transformation parameters corresponding to the supercharges q and Q, respectively. Actually, there exists one more supersymmetry generated by the supercharge S whose transformation law can be obtained from (2.16)-(2.19) by promoting the transformation parameters ζ q and ζ Q into the following functions depending on the coordinates where (ε q , ε Q , ε S ) are transformation parameters corresponding to the supercharges (q, Q, S)

Harmonic trap and modified super-Schrödinger symmetry
Now let us put the SUSY Jackiw-Pi system in a harmonic trap. The harmonic potential term can be introduced by adding the Noether charge C (corresponding to the special Schrödinger transformation) to the Hamiltonian. We can turn on such generalized chemical potential terms by introducing the following external gauge field A ex whereN f is the fermion number operator andN a (a = 1, · · · , N F ) are the flavor number operator The parameters (ω,ω, µ f , µ a ) correspond to the following generalized chemical potentials • ω : the strength of the harmonic trap •ω : the angular velocity of the rotation • µ f : the fermion number chemical potential • µ a : the flavor symmetry chemical potential In the presence of the external gauge fields, the differential operator∆ 0 in the kinetic terms is replaced by∆ obtained by replacing the covariant derivatives with those with the external field whereD µ denotes the covariant derivative including the external field Since the differential operator∆ does not commute with some generators of the super-Schrödinger transformation, it appears that a part of the super-Schrödinger symmetry including the supersymmetry is broken in the Jackiw-Pi system in the harmonic trap. Although the original super-Schrödinger transformation is no longer a symmetry of the action, there exists a modified super-Schrödinger symmetry even in the presence of the external field.

Bosonic part of modified super-Schrödinger symmetry
To write down the bosonic part of the modified super-Schrödinger symmetry it is convenient to introduce η I (t) defined by the following differential equations By using these functions, "the non-relativistic conformal Killing vector ξ µ for the modified Schrödinger symmetry" can be written as Then the bosonic part of the modified super-Schrödinger transformations takes the form with λ = η D and By appropriately identifying the integration constants of the differential equations (2.27)-(2.28) with the transformation parameters ε O , we can confirm that this transformation reduces to the standard Schrödinger symmetry when the chemical potential terms are turned off (ω =ω = 0).
It is worth noting that the SU(N F ) flavor symmetry is also not broken but modified as

Fermionic part of modified super-Schrödinger symmetry
The fermionic part of the modified super-Schrödinger transformation takes the same form as the unmodified one (2.16)- (2.19), if the covariant derivative is promoted as D µ →D µ and ζ q and ζ Q are replaced with the functions satisfying the differential equation The general solution takes the form where the function f (t) is given by and f ′ (t) is the time derivative of f (t). The integration constants (ε q , ε Q , ε S ), which can be interpreted as the transformation parameters of the modified symmetry, are chosen so that ζ q and ζ Q reduce to the original forms (2.20) in the limit ω,ω, µ f → 0.

1/3 BPS condition
Since the SUSY Jackiw-Pi system has the modified super-Schrödinfer symmetry even in the harmonic trap, it is possible to consider BPS states of the Jackiw-Pi vortices [14] which preserve a part of the modified super-Schrödinfer symmetry. A BPS condition can be obtained by requiring δψ = 0 for each choice of the transformation parameters (ε q , ε Q , ε S ). We can obtain a BPS equation with no explicit time dependence by setting By using the differential operators ∇ z and ∇z defined by we can write the BPS equation corresponding to (3.1) as Any solution of this BPS solution satisfies the full set of the equations of motion are satisfied if the following first order differential equations are also satisfied where we have defined In the following, we consider field configurations satisfying the set of equations (3.3) and (3.4) with asymptotic behaviors

General BPS solution
To write down the general solution of the equations (3.3) and (3.4) it is convenient to introduce an arbitrary N-by-N F holomorphic matrix H 0 (t, z), called the moduli matrix [28,29]. By using H 0 (t, z), we can formally solve the equations (3.3) and (3.4) as 9) where the N-by-N matrix S(t, z,z) is an element of the complexified gauge group U(N) C ∼ = GL(N, C) satisfying This equation ensures that the Gauss law iF zz + π k φφ † is satisfied. The BPS equation ∇zφ = 0, which can be rewritten as is automatically satisfied for an arbitrary choice of the holomorphic matrix H 0 (t, z). The remaining equation ∇ t φ = 0 determines the time dependence of the solution as These equations imply that S and H 0 have no explicit t-dependence if they are written in terms of the coordinates z * andz * defined as z * ≡ e i(ω−ω)t z,z * ≡ e −i(ω−ω)tz . (3.13) This implies that the whole system is rotating in the z-plane with angular velocityω − ω. By solving the equation (3.10) for Ω, physical quantities such as energy density profiles can be explicitly obtained for an arbitrarily chosen H 0 (z * ). Note that for N = N F = 1, Eq. (3.10) can be rewritten into the vortex equation classified as follows [34] mω = 0, k < 0 Jackiw-Pi [14] mω > 0, k < 0 Ambjørn-Olesen [35,36] mω > 0, k > 0 Taubes [37] mω > 0, k = ∞ Bradlow [38] mω < 0, k < 0 Popov [39] Although Eq. The stability of the solution is guaranteed by the conserved charges associated with the spatial rotation and the internal phase rotation. We can show that for given values of the Noether charges, the energy of the system is bounded from below as where J and N a are the angular momentum and the flavor symmetry Noether charges The BPS solution saturates this lower bound for the energy. This is an example of Q solitons, that is, solitons which are stabilized by Noether charges.
It is worth noting that we can obtain more general solutions of the equations of motion (breathing solutions, etc) by applying the modified Schrödinger transformation to the BPS configurations discussed in this section. Such solutions preserve different combination of the supercharges and satisfy a certain time-dependent BPS equations. By appropriately fixing the gauge, the static solution with e.g. µ aNa φ = µφ ≡ diag(µ 1 , · · · µ N )φ, can be written as (3.19) where L = diag(l 1 , l 2 , · · · , l N ) is an N-by-N diagonal matrix with l i ∈ Z ≥0 . The matrix σ = diag(σ 1 , σ 2 , · · · , σ N ) denotes a set of real profile functions satisfying 20) with asymptotic behavior σ i → mω|z| 2 . We can show that the subleading part of σ i takes the form σ i → mω|z| 2 − ρ i log |z| 2 , (3.21)

Static BPS solution
From this asymptotic behavior, it follows that the real parameters ρ i correspond to the magnetic fluxes and the flavor charges

Spectrum of fluctuation modes in BPS background
In this section we consider fluctuations of the fields (δA µ , δφ, δψ) around a BPS background (A µ , φ) and show that in addition to the so-called massive Nambu-Goldstone modes in the bosonic fluctuations, there exist fermionic massive Nambu-Goldstone modes associated with the broken supercharges of the modified super-Schrödinger symmetry.

Linearized equations for fluctuations
When we discuss fluctuations of the bosonic fields, it is convenient to remove the gauge zero modes by imposing the gauge fixing condition on the fluctuations as This gauge fixing condition does not completely remove unphysical gauge zero modes since there still exist remaining gauge degrees of freedom generated by Λ ∈ u(N) such that This residual gauge degrees of freedom can be fixed by imposing the additional gauge fixing condition as The Gauss law equation iF zz + π k φφ † = 0 reduces to the linearized Gauss law for the fluctuation fields If the gauge fixing condition (4.1) and the linearized Gauss law (4.4) are satisfied, the linearized equations for the fluctuation fields can be written as where the differential operators are given by 1 The operatorsφ andφ † denote the right multiplications of φ and φ † , e.g.φ · δAz = δAzφ a , and the differential operators ∇ z and ∇z are defined by

Eigenmode expansion and supermultiplets
Here where (u g, n , u s, n ) and u f, n are bosonic and fermionic mode functions satisfying the eigenmode where ∇(ǫ) is the operators which can be obtained from ∇ t in (4.6) by replacing the time derivative i∂ t with an eigenvalue ǫ. Then the linearized equations reduce to the following equations for the bosonic and fermionic degrees of freedom ϕ n (t) and χ n (t) i∂ t ϕ n (t) = ǫ b, n ϕ n (t), i∂ t χ n (t) = ǫ f, n χ n (t). (4.11) Note that since ∇(ǫ) commutes with ∇∇ and ∇∇ ∇(ǫ b, n ), ∇∇ u g, n u s, n = 0, ∇(ǫ f, n ),∇∇ u f, n = 0, (4.12) the solution to the bosonic (fermionic) linearized equation can be decomposed into simultaneous eigenmodes of ∇(ǫ) and ∇∇ (∇∇).
Since the BPS background configuration preserves a linear combination of three complex supercharges (q, Q, S), eigenmodes of the bosonic and fermionic fluctuations are paired so that they form supermultiplets of the unbroken supersymmetry. Let (u g , u s ) be a bosonic eigenmode with eigenvalue ǫ b . The partner fermionic eigenmode u f can be obtained as On the other hand, any fermionic eigenmode u f with eigenvalue ǫ f can be mapped to its partner bosonic eigenmode as (4.14) Since (u g , u s ) and u f are eigenmodes of ∇∇ and∇∇ respectively, the sequential mappings (boson → f ermion → boson) and (f ermion → boson → f ermion) do not give new eigenmodes Therefore, a generic supermultiplet consists of a pair of bosonic and fermionic eigenmodes.
It is worth noting that unlike the case of ordinary supersymmetry, bosonic and fermionic eigenmodes in a supermultiplet have different eigenfrequencies ǫ b − ǫ f = µ f + ω +ω. This is due to the unbroken supersymmetry is a part of the modified supersymmetry which explicitly depends on time t. One can check that the unbroken supersymmetry becomes independent of t when ǫ b − ǫ f = µ f + ω +ω = 0.  where Λ ∈ gl(N) is an N-by-N matrix satisfying ∇(ǫ b ) Λ = 0 and

Short supermultiplets
In this way, we can find physical short multiplets satisfying the linearized Gauss law equation and the gauge fixing condition.
2∇ ∇ = ∇ z ∇z − mω is a negative definite operator, since for any function f ,

Bosonic and Fermionic massive Nambu-Goldstone modes
Since the static BPS configuration (3.17)-(3.19) breaks a part of the super-Schrödinger symmetry, there exist Nambu-Goldstone (NG) modes in the fluctuations of the fields.

Bosonic massive Nambu-Goldstone modes
In the presence of the external fields, the super-Schrödinger symmetry is modified in such a way that the generators explicitly depend on time t. Consequently, the corresponding NG modes become massive. The bosonic NG modes satisfying the constraints (4.4) and (4.3) takes the form where the first NG mode generated by P z − iωBz is in a short multiplet and we have used the symmetry (4.18) so that it satisfies the constraint (4.4) and (4.3). These three complex modes (and their complex conjugate) correspond to the broken modified symmetry generated by six real operators (translation, Galilean, dilatation and special conformal symmetry). There also exist massive NG modes corresponding to the broken modified flavor symmetry (2.34).

Fermionic massive Nambu-Goldstone modes
Since the BPS configuration breaks a part of the supersymmetry, there also exist fermionic NG modes. As in the bosonic case, the modified supersymmetry transformations explicitly depend on time and hence the corresponding fermionic NG modes are massive. There are two such fermionic massive NG modes corresponding to the broken fermionic generators q and Q + iωS We can check that the NG modes generated by (P z + iωBz, q) and (D + 2iωC, Q + iωS) are the pairs of supermultiplets related by the boson-fermion mapping discussed above.

Infinite towers of eigenmodes in static BPS background
As we have seen in the previous section, the bosonic and fermionic massive NG modes have eigenfrequencies given by the chemical potentials with the integer coefficients determined by the charges of the corresponding generators. Here we show that there are infinite towers of eigenmodes with such eigenvalues.
Let us first consider the case of short multiplets. Since the BPS equation ∇zφ is satisfied by (3.7)-(3.9) for an arbitrary matrix H 0 (t, z), the linearized BPS equation can be solved by using the linearized version of (3.7)-(3.9), which takes the form where Λ is the N-by-N matrix determined by the constraint 3 the eigenfrequency of the corresponding short multiplet is given by The NG mode generated by P z − iωBz corresponds to the linear combination of the modes with (l, J) = (l j − 1, j). The NG modes corresponding to the broken (modified) flavor symmetry are also contained in these towers of eigenmodes.
In addition to the short multiplets, we can also find exact spectra of a class of ordinary long supermultiplets. Such supermultiplets can be obtained from the fermionic eigenmode corresponding to the lowest Landau level. For example, u f is an eigenmode with frequency if u f has a non-zero monomial in the (j, J)-component The fermionic massive NG modes (4.23) and (4.24) corresponds to the linear combinations of the eigenmodes with (l, J) = (l j , j) and (l, J) = (l j + 1, j), respectively. The corresponding bosonic mode can be obtained by applying the map (4.14) where r = |z|. As we have seen above, this bosonic mode has eigenfrequency related to that of the fermionic mode as (4.13) The towers of eigenmodes (4.27) and (4.30) can be interpreted as the lowest Landau levels in the bosonic and fermionic sectors, respectively. As was done in the non-linear Schrödinger system [40], it would be interesting to discuss the low energy dynamics of such degrees of freedom with physically distinctive properties.

Summary and Discussion
In this paper, we discussed the supersymmetric Jackiw-Pi model in the harmonic trap. The super-Schrödinger symmetry of the original SUSY Jackiw-Pi model is modified in the presence of the external background fields which correspond to the generalized chemical potential terms including the harmonic potential. We have seen that the 1/3 BPS states of Jackiw-Pi vortices, which preserve a part of the modified supersymmetry, are stationary configurations rotating around the origin. They become static when the moduli matrix is at the fixed points of the spacial rotation and the flavor symmetry. We have investigated fluctuations around the static BPS backgrounds and revealed the structure of supermultiplets of eigenmodes. In addition to the bosonic massive NG modes, we identified the fermionic massive NG modes associated with the broken modified superconformal symmetry. We have also found the eigenmode spectra of the infinite towers of supermultiplets corresponding to the bosonic and fermionic lowest Landau levels.
While we have discussed one of the simplest examples of (modified) non-relativistic supersymmetry in the Jackiw-Pi model, it has been known that there exist Chern-Simons matter systems with extended non-relativistic supersymmetries [41,42,43]. It would be interesting to investigate bosonic and fermionic massive NG modes in the extended models such as the non-relativistic ABJM model. Another direction to be explored is to clarify the relation between the quantum states of the Jackiw-Pi vortices in the harmonic potential and the spectrum of the chiral primary operators [44,45] from the viewpoint of the non-relativistic version of the sate-operator mapping [46]. If we set µ f + ω +ω = 0 in our model, the explicit time dependence of the supersymmetry preserved by the 1/3 BPS states disappears and hence we can compactify the time direction without breaking the supersymmetry. Such a situation is quite similar to the Ω-background [47,48] and it would be possible to compute certain types of superconformal indices by using the supersymmetric localization method [49]. As in the case of the vortex partition functions in 2d N = (2, 2) theories [50,51], the moduli matrix method, which was used to describe the BPS vortex solution, would play a crucial role in the localization computation and hence it is an important future work to investigate the structure of the space of the BPS solutions from the viewpoint of the moduli matrix formalism and its relation to the ADHM like construction discussed in Ref. [22].