BPS Equations and the Stress Tensor

We exploit the relationship between the space components of the energy-momentum tensor and the supercurrent to discuss the connection between the BPS equations and the vanishing of the components of the stress tensor in various supersymmetric theories with solitons. Using the fact that certain combination of supercharges annihilate BPS states, we show that $T_{ij}=0$ for kinks, vortices and dyons, displaying the connection between supersymmetry and non-interacting BPS solitons.


Introduction
First order BPS equations were originally obtained either by looking for a bound of the soliton mass [1] or by imposing the stress tensor to vanish [2]. Already in this last work the relation between supersymmetry and the possibility of reducing the second order equations of motion to BPS equations at certain critical values of the coupling constants was stressed and afterwards exploited in the search of classical solutions to two dimensional supersymmetric models [3].
The origin of such a connection was finally clarified by Witten and Olive [4] by considering the supersymmetric extension of bosonic models exhibiting topological soliton solutions. Studying the supersymmetry algebra, it was shown in this work that the soliton topological charge can be identified with the central charge of the supercharge algebra and gives a lower bound for the soliton mass. This was done for the supersymmetric version of a scalar field theory in 1 + 1 dimensions with kink solutions and a N = 2 Yang-Mills theory in 3 + 1 dimensions with dyon solutions. Afterwards, the case of vortices in N = 2 supersymmetric gauge theories in 2 + 1 dimensions and instantons in 4 dimensional Euclidean space was discussed along the same lines [5]- [7] and the extension to the case of supergravity models was also studied [8]. The question on how supersymmetry protects the Bogomol'nyi bound at the quantum level also deserved a lot of attention [9]- [10].
We show in the present note how the alternative derivation of BPS equations from the vanishing of the soliton stress-tensor T ij (i, j = 1, 2, 3) can be also understood supersymmetry point of view studying the supercurrent-supercharge algebra. As it is well known, the algebra of supersymmetry itself already imposes a very intimate relationship between the supercurrent and the stress tensor. This relationship stems from the connection between the energy-momentum vector and the supercharge [11]. In fact both the supercurrent and T µν must belong to the same supermultiplet and then it is not difficult to understand how an identity between the stress tensor and an appropriate trace containing the supersymmetric transform of the supercurrent connects BPS states and the condition BP S|T ij |BP S = 0 .
In order to construct the supercurrent and show how its connection with the energy-momentum tensor leads to eq.(1) we will work with specific models having BPS 1 + 1 dimensional kinks and 2 + 1 vortices and also explain how the results can be easily extended to the case to BPS dyons in 3 + 1 dimensions. In fact our derivation indicates that the same result should be also valid for any other model with BPS soliton solutions.
It should be mentioned that our work was prompted by a recent work of Manton [12] where new scaling identities for solitons are derived in terms of the stress tensor, showing the relevance of T ij in connection with the study of soliton solutions. As mentioned above, already in the case of vortices it was recognized [2] that the critical point at which the topological bound for the energy of the Abelian Higgs vortices is saturated corresponds to the limiting value between type-I and type-II superconductivity, precisely where forces between vortices (and hence the surface integral of T ij ) vanish [13]. We shall see below that supersymmetry provides a way to construct models where general noninteracting solitons equations can be studied by analyzing the Noether supercurrent.

Scalar field theory in two dimensions
The action for the simplest two-dimensional supersymmetric model admitting solitons in its bosonic sector reads, in component fields [14], with φ a real scalar field, ψ a 2-component Majorana spinor, F an auxiliary field and V [φ] an arbitrary function. We take the metric g µν with signature (+, −) and the Dirac matrices as With this conventions the charge conjugation matrix satisfying C · γ µ · C −1 = −γ µ is given by C = −γ 0 . Given a spinor the charge conjugate ψ c is then The energy momentum tensor associated with action (2) takes the form and its symmetric on-shell components The (off-shell) supersymmetric transformations leaving action(2) invariant are δφ =ǭψ and the associated conserved supercurrent is where the auxiliary field has been eliminated using its equation of motion. More explicitly The chiral components Q ± of the supersymmetry charge take then the form From this, one finds for the supercharge algebra (in the rest frame) where Concerning Z, it is given by and coincides with the topological charge which is non-trivial for soliton states.
In order to find the Bogomol'nyi bound, Witten and Olive considered [4] the combinations Then, writing one finds that the soliton mass M is bounded by the topological charge, M ≥ |Z| 2 (18) and that the bound is attained for those states |BP S ± such that In view of the explicit form of charges these states correspond to kink solutions satisfying the first order BPS equations which can be written in the form Each of the BPS kink solutions break half of the supersymmetry of the theory according to the choice among eqs. (19) or (20).
Let us now study the supercurrent-supercharge anticommutators. In particular, from the canonical commutation relations (11) one has with J µ the supercurrent and ξ µ the topological current, related to the central charge through the identity Writing one easily finds Explicitly, the l.h.s. takes the form and then From these two equations, one can write two identities for the stress-tensor Then, in view of (49)-(50) and being the currents J 1± given by we conclude that either That is, BPS saturated states preserving half of the supersymmetry correspond to states with vanishing stress tensor.
The N = 2 supersymmetric action associated with the Abelian Higgs model is Concerning boson fields, A µ is an Abelian gauge field, F µν its curvature, φ a complex scalar and N a real scalar field. The covariant derivative is defined as Note that the coupling constant in the gauge symmetry breaking scalar potential is taken as λ = e 2 /8, the condition required in order to have N = 2 supersymmetry. Fermion fields ψ and Σ are Dirac fermions and The energy-momentum tensor components of the bosonic sector are where B = F 12 . Action (41) is invariant under the following N = 2 supersymmetry transformations with η c a complex (Dirac spinor) parameter.

The Noether supercurrent associated with invariance of action (41) under transformations (45) is
so that the conserved charge Q can be defined as Writing and One can now compute the supersymmetry algebra among supercharges Q and Q. Since this will be connected with the Bogomol'nyi bound for the Abelian Higgs model, we shall put N = 0 and, after using fermion anticommutator relations we shall also put all fermions to zero. As one is interested in static configurations with finite energy one should also impose A 0 . The answer is where and the central charge Z is given by Here i, j = 1, 2.
One can see that the central charge (53) coincides with the topological charge (the quantized magnetic flux) of the vortex configuration. Indeed, Z can be rewritten in the form where V i is given by so that, after using Stokes' theorem (and taking into account that D i φ → 0 at infinity) with n ∈ Z an integer characterizing the homotopy class to which A i belongs.
Let us now introduce the projector and define Then, we project eq.(51) with P and take the trace getting Taking the expectation value of (59) in an arbitrary state and since the anticommutator of an operator with its adjoint is a positive definite operator we conclude that M ≥ |Z| which are the BPS equations for the Abelian Higgs model. Due to (61), their solution also solves the static Euler-Lagrange equations of motion. As in the kink case, according to the choice of sign in the BPS equations, the corresponding solution will break half of the supersymmetries. Let us finally insist that the condition λe 2 = λ/8 necessary for this last fact, arises in the present approach from the requirement of N = 2 supersymmetry. In order to connect supersymmetry with conditions on the stress tensor, we will analyze the supercurrent-supercharge algebra in the bosonic sector of the model. The relevant terms in the superchargeQ and the spatial components J i of the supercurrent leading to (static) bosonic contributions arē where we have written the supercurrent J i in the form and ellipsis . . . indicate irrelevant terms which will be ignored from here on. From eqs.(65)and (66) we find that and hence Now, the r.h.s. is nothing but the symmetric stress tensor as defined in (44), so that In particular we have But Then, analogously to the kink case, either one has Q + + iQ − |BP S ± = 0, or j 1 + + ij 1 − |BP S ± = 0 and j 2 + + ij 2 − |BP S ± = 0 (a similar statement is valid for Q + − iQ − , j 1 + − ij 1 − , and j 2 + − i j 2 − ). So we can write for BPS vortex states At this point, it should be stressed that eq.(72) from which the vanishing of the stress tensor components was inferred is in general valid for other supersymmetric models in which one can write where T ij is the symmetric stress-tensor and N d a constant depending on the parameters of the specific model. This is valid for the kink (eq. (25)), for the vortex (eq. (72)) but also for the dyon, the instanton taken as a soliton in 4 + 1 dimensions, etc (see also [9]- [15]). In particular, consider the 3 + 1 case, where the supercharge algebra for the N = 2 Yang-Mills theory takes the form where α, β = 1, . . . , 4 and the central charges U and V are surface integrals. If one takes as gauge group O(3) and breaks this symmetry to U (1) by giving a non-zero vacuum expectation value to the scalar field taken in the adjoint, U corresponds to the U (1) magnetic charge and V to the electric charge. A Bogomol'nyi bound can be then derived from (83), and is saturated when the Bogomol'nyi-Prassad-Sommerfield equations are satisfied. Now, one can see that eq.(82) holds in this case with the spatial components of the supercurrent taking the form This formula corresponds to a bosonic sector containing a gauge field A µ in the Lie algebra of O(3) coupled to a Higgs scalar φ in the adjoint (There is an additional pseudoscalar field that should be put to zero to make contact with the Georgi-Glashow model). Concerning the fermion sector, Ψ a (a = 1, 2) are two Majorana fermions. Then, using eq.(82) and proceeding as for the kink and the vortex, one can see that eq.(81) also holds for the Prasad-Sommerfield dyon. That is, the stress-tensor vanishes for BPS dyons, a fact that can be trivially confirmed by explicit computation of T ij .
We have discussed in this note the relation between supersymmetry and the vanishing of the stress tensor for topological solitons in a variety of field theories in different space-time dimensions. Each one of the elements in this relation was already understood but our point was to show how they could be put together, by exploiting the relation that exists in supersymmetric theories between the supercurrent and the energy-momentum tensor. In fact, this relation was already underlying the analysis in ref. [4] where BPS equations were derived from the relation between the supercharge algebra and the energy momentum vector P µ = d 3 xT 0µ which in the rest frame reduces to P 0 = M .
Here, we have instead used the fact that, since the supercurrent and the energy-momentum tensor belong to the same multiplet, we can extend the analysis of the relation between BPS states and supersymmetry to the spatial components of J µ and T µν . If we consider for example the d = 3 + 1 case in the superfield framework, the linear θ component of the multiplet is the supercurrent and the θθ component corresponds the energy-momentum tensor and they should then necessarily transform under supersymmetry one into the other, Similar identities hold in other d + 1 dimensional models. As signaled above, eq.(82) leading to the connection between supersymmetric BPS states with the condition T ij = 0 can be inferred from this formula. Now, as it is well known, T ij gives the force f i acting in a unit volume of the system. This, together with our result means that, in general, supersymmetry can guide the construction of non-interacting solitons bosonic models of interest just by considering the supersymmetric extension as a tool for identifying BPS states. There are also possible applications of our observation in supergravity models, in connection with stability of cosmic strings [16]- [17] and with the cosmological constant problem [18]- [19]. In particular, the so-called dominant energy condition, T 00 ≥ |T ij |, valid for static spacetime, plays a centra role to establish a connection between stability and the sign of the deficit angle [17]. In this context it is natural to study supergravity models with string-like BPS solutions in their bosonic sector. An analysis based on the supercharge algebra has been already presented [8] and it should be worthwhile to study the problem from the point of view of supercurrents presented here. We hope to report on these issues in a future work.