The supersymmetric (SUSY) Jackiw–Pi model consists of a gauge field and pairs of bosonic matter fields and fermionic matter fields . For simplicity, we consider the case of a gauge field with matter pairs in the representation of the gauge group and the flavor symmetry. It would be straightforward to extend the following discussion to more general settings. By using matrix notation for the matter fields, (2.1) the action of the supersymmetric Jackiw–Pi model can be written as (2.2) where the trace is taken over the flavor indices. Just for notational convenience, we have introduced the matrix and the matrix defined by (2.3)

The symbol denotes the differential operator which gives the standard non-relativistic kinetic term (2.4) where and are the complex coordinates on the two-dimensional plane. The covariant derivative and the field strength are defined by , and , etc.

The parameter is the Chern–Simons level and is the Chern–Simons term normalized as (2.5)

By rescaling the gauge field as , we can see that in the infinite level limit , this model reduces to the free theory whose equations of motion are given by the Schrödinger equation. In addition to the standard Schrödinger symmetry (see Appendix B for the details of the 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 .

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 (corresponding to the special Schrödinger transformation) to the Hamiltonian. As shown in Appendix A, a Hamiltonian with such generalized chemical potential terms possesses a modified symmetry even though the original symmetry appears explicitly broken. In the present case, the generalized chemical potential terms can be turned on by introducing the following external gauge field as : (2.22) where is the fermion number operator, (2.23) and are the flavor number operators: (2.24)

The parameters correspond to the following generalized chemical potentials:

: the strength of the harmonic trap

In the presence of the external gauge fields, the differential operator in the kinetic terms is replaced by the differential operator obtained by replacing the covariant derivatives with those with the external field (2.25) where denotes the covariant derivative including the external field, (2.26)

Since the differential operator does not commute with some generators of the super-Schrödinger transformation, it appears that the 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. By using the general method explained in Appendix A, we can find the following modified super-Schrödinger symmetry (see Eqs. (B.19)–(B.23) in Appendix B for the explicit forms of the symmetry transformations).

Since the SUSY Jackiw–Pi system has the modified super-Schrödinger symmetry even in the harmonic trap, it is possible to consider BPS states of the Jackiw–Pi vortices [14], which preserve part of the modified super-Schrödinger symmetry. A BPS condition can be obtained by requiring for each choice of the transformation parameters . We can obtain a BPS equation with no explicit time dependence by setting (3.1)

By using the differential operators and defined by (3.2) we can write the BPS equation corresponding to Eq. (3.1) as (3.3)

Any solution of this BPS solution satisfies the full set of equations of motion if the following first-order differential equations are also satisfied: (3.4) where we have defined (3.5)

In the following, we consider field configurations satisfying the set of equations in Eqs. (3.3) and (3.4) with asymptotic behaviors² (3.6)

To write down the general solution of Eqs. (3.3) and (3.4) it is convenient to introduce an arbitrary holomorphic matrix , called the moduli matrix [28,29]. By using , we can formally solve Eqs. (3.3) and (3.4) as (3.7) (3.8) (3.9) where the matrix is an element of the complexified gauge group satisfying (3.10)

This equation ensures that the Gauss law is satisfied. The BPS equation , which can be rewritten as (3.11) is automatically satisfied for an arbitrary choice of the holomorphic matrix . The remaining equation determines the time dependence of the solution as (3.12)

It follows from this equation that and have no explicit dependence if they are written in terms of the coordinates and defined as (3.13)

This implies that the whole system is rotating in the -plane with angular velocity . By solving Eq. (3.10) for , physical quantities such as energy density profiles can be explicitly obtained for an arbitrarily chosen .

Note that for , Eq. (3.10) can be rewritten into the vortex equation classified as follows [34]: 0, ~k < 0 && \mbox{Ambj{\o}rn-Olesen} \quad\quad && \mbox{[35,36]} \notag \\ m\omega &> 0, ~k > 0 && \mbox{Taubes} && \mbox{[37]} \notag \\ m\omega &> 0, ~k = \infty \quad\quad && \mbox{Bradlow} && \mbox{[38]} \notag \\ m\omega &< 0, ~k < 0 && \mbox{Popov} && \mbox{[39]} \notag \end{alignat}$$]]>

Although Eq. (3.10) has an identical form to the vortex equation, the boundary conditions are different and consequently the vortices in our setup have some distinctive physical properties.

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 (3.14) where and are the angular momentum and the flavor symmetry Noether charges, (3.15)

The BPS solution saturates this lower bound for the energy. This is an example of 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.

Although a generic BPS configuration is a stationary solution which depends on time , the solution in Eqs. (3.7)–(3.9) becomes static if is chosen so that the resulting scalar field is invariant under the rotation and the flavor transformations, that is, (3.16)

By appropriately fixing the gauge, the static solution with, e.g., , can be written as (3.17) (3.18) (3.19) where is an diagonal matrix with . The matrix denotes a set of real profile functions satisfying (3.20) with asymptotic behavior . We can show that the subleading part of takes the form (3.21)

From this asymptotic behavior, it follows that the real parameters correspond to the magnetic fluxes and the flavor charges (3.22)

See Figs. 2 and 3 for the profiles of the charge density.

As in Ref. [20], time-dependent solutions can be obtained from these static solutions by applying the modified Schrödinger symmetry. Although such solutions do not satisfy the BPS equation in Eq. (3.3), they preserve a certain time-dependent linear combination of the supercharges.

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 (4.1)

This gauge-fixing condition does not completely remove unphysical gauge zero modes since there remain gauge degrees of freedom generated by such that (4.2)

This residual gauge degrees of freedom can be fixed by imposing the additional gauge-fixing condition as (4.3)

The Gauss law equation reduces to the linearized Gauss law for the fluctuation fields: (4.4)

If the gauge-fixing condition in Eq. (4.1) and the linearized Gauss law in Eq. (4.4) are satisfied, the linearized equations for the fluctuation fields can be written as (4.5) where the differential operators are given by³ (4.6) and (4.7)

The operators and denote the right multiplications of and , e.g. , and the differential operators and are defined by (4.8)

Here we consider fluctuations around the static BPS background in Eqs. (3.17)–(3.19). Let us consider the eigenmode expansion of the fluctuations (4.9) where and are bosonic and fermionic mode functions satisfying the eigenmode equations (4.10) where is the operator which can be obtained from in Eq. (4.6) by replacing the time derivative with an eigenvalue . Then the linearized equations reduce to the following equations for the bosonic and fermionic degrees of freedom and : (4.11)

Note that since commutes with and , (4.12) the solution to the bosonic (fermionic) linearized equation can be decomposed into simultaneous eigenmodes of and ().

Since the BPS background configuration preserves a linear combination of three complex supercharges , eigenmodes of the bosonic and fermionic fluctuations are paired so that they form supermultiplets of the unbroken supersymmetry. Let be a bosonic eigenmode with eigenvalue . The partner fermionic eigenmode can be obtained as (4.13)

On the other hand, any fermionic eigenmode with eigenvalue can be mapped to its partner bosonic eigenmode as (4.14)

Since and are eigenmodes of and respectively, the sequential mappings () and () do not give new eigenmodes, (4.15)

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, . This is due to the unbroken supersymmetry being part of the modified supersymmetry which explicitly depends on the time . One can check that the unbroken supersymmetry becomes independent of when .

Since the static BPS configuration in Eqs. (3.17)–(3.19) breaks part of the super-Schrödinger symmetry, there exist NG modes in the fluctuations of the fields.

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 is satisfied by Eqs. (3.7)–(3.9) for an arbitrary matrix , the linearized BPS equation can be solved by using the linearized version of Eqs. (3.7)–(3.9), which takes the form (4.25) where is the matrix determined by the constraint⁵ (4.26)

This solution of the linearized BPS equation satisfies the eigenmode equation with eigenvalue if the matrix is chosen so that . This condition is satisfied when has one non-zero component given by a monomial of . For example, if has in its -component, (4.27) the eigenfrequency of the corresponding short multiplet is given by (4.28)

The NG mode generated by corresponds to a linear combination of the modes with . 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, is an eigenmode with frequency (4.29) if has a non-zero monomial in the -component (4.30)

The fermionic massive NG modes in Eqs. (4.23) and (4.24) correspond to the linear combinations of the eigenmodes with and , respectively. The corresponding bosonic mode can be obtained by applying the map in Eq. (4.14), (4.31) where . As we have seen above, this bosonic mode has eigenfrequency related to that of the fermionic mode as in Eq. (4.13), (4.32)

The towers of the eigenmodes in Eqs. (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.