Component $d=6$ Born-Infeld theory with $N=(2,0)\rightarrow N=(1,0)$ supersymmetry breaking

The formalism of nonlinear realizations is used to construct a theory with $1/2$ partial breaking of global supersymmetry with the $N=(1,0)$, $d=6$ abelian vector multiplet as a Goldstone superfield. Much like the case of the $N=2$, $d=4$ Born-Infeld theory, proper irreducibility conditions of the multiplet are selected by the invariance with respect to the external automorphisms of the Poincar\'e superalgebra. They are found in the lowest nontrivial order in the auxiliary field. The fermionic contributions to the Bianchi identity are restored by assuming its covariance with respect to broken supersymmetry. The invariance of the action with respect to unbroken supersymmetry is checked in the lowest order in the fermionic fields.

Introduction easier to construct. Indeed, one of the simplest theories with scalar and electromagnetic fields was analyzed in [17], where conclusion was reached that it would be highly desirable to formulate the irreducibility conditions of the multiplet in terms of the fermionic superfields. This would eliminate the necessity to solve nonlinear algebraic relations between derivatives of scalar fields and bosonic components of fermionic superfields, which appear in all theories with scalars and can be very complicated in the cases of high supersymmetry (examples can be found in [16]). Also, the components of the vector multiplets, which correspond to the electromagnetic field, satisfy the differential identity (called the Bianchi identity). It should be derived as a consequence of the irreducibility conditions, and this is much easier to do if the conditions are formulated in terms of fermionic superfields. Also, in the theories with spontaneous breaking of supersymmetry this condition is typically highly nonlinear and should be proven equivalent to the usual ∂ [A F BC] = 0, which would also relate the true physical field strength F AB to the components of the multiplet. This is much simpler to do if the identity does not involve scalar fields. As the only physical boson of the N = (1, 0), d = 6 multiplet is the electromagnetic field strength tensor F AB , while N = 2, d = 4 supermultiplet has two additional scalars, the six dimensional case is preferable.
Therefore, our approach to construct the action is the following one.
• At first, we should derive proper irreducibility conditions of the N = (1, 0), d = 6 vector multiplet from the assumption of covariance with respect to broken supersymmetry and the SO(4) group (subgroup of the SO(5) automorphisms of the N = (2, 0), d = 6 superalgebra).
• Secondly, as the consequence of the irreducibility conditions, the nonlinear Bianchi identities should be derived. Let us note that it is sufficient to find them in the bosonic limit and with the auxiliary field removed by its equation of motion. This is acceptable as we are going to construct the action without the auxiliary field, and the fermionic terms in the identity can be restored from the assumption of its covariance with respect to the broken supersymmetry. Then it should be shown that the found nonlinear identities are equivalent to the usual ones ∂ [A F BC] = 0. At the same time, the expression of the physical bosonic field strength in terms of the bosonic components of the multiplet would be found.
• Thirdly, the ansatz for the action should be constructed by covariantizing the well-known bosonic action with respect to broken supersymmetry and by adding the Wess-Zumino term. Finally, using the standard techniques, the transformation laws of the components with respect to unbroken supersymmetry should be derived and the invariance of the action proven in the lowest nontrivial approximation in the fermions.
The commutation relations of the so(5) automorphism algebra in the basis with only one explicit su(2) can be written as The generators of so(5) commute with the supercharges as For the purposes of the latter construction, only the generators R ij and T ij , which form so(4), are relevant. The spontaneous breaking of half the supersymmetry can be achieved with the following coset element: Here, x αβ and θ α i are the coordinates of the superspace, and ψ α i (x, θ) are the Goldstone fermionic superfields. This is justified by their transformation laws. If the transformations in the coset space are induced by the left multiplication the variations of x, θ and ψ under unbroken and broken supersymmetry are As expected, x αβ and θ α i transform with respect to unbroken supersymmetry as the coordinates of the superspace, and the ψ α i remains inert. Conversely, θ α i are not touched by broken supersymmetry, while the variations of ψ α i and x αβ remind the transformation laws of the Goldstone fermion proposed by Volkov and Akulov [18] in four dimensions.
The Maurer-Cartan differential form Ω = g −1 dg is invariant with respect to the Q and S transformations: Expanding the differential of the arbitrary invariant function in terms of the forms △x αβ and dθ α i , one may construct derivatives covariant with respect to both supersymmetries: As D i α , D j β = 2iǫ ij ∂ αβ , their (anti)commutation relations are 2 The N = (1, 0), d = 6 vector multiplet Let us briefly recall the properties of the N = (1, 0), d = 6 vector multiplet. It was considered in the SU (2) non-covariant approach in [19] and [20]. The SU (2) covariant formulation can be found in [21]. The latter is most useful when the formalism of nonlinear realizations is used. In this case, the usual N = (1, 0), d = 6 vector multiplet is given by the spinorial superfield ψ α i , subjected to the following irreducibility conditions One can check that these conditions imply that only the following components of the multiplet are independent: Acting on the ψ α i field by two spinorial derivatives, one finds that the result always reduces to the spacetime derivatives of ψ α i . It should be noted that as a consequence of the constraints (9) the component F α β satisfies the differential identities known as the Bianchi identities. They indirectly imply that the antisymmetric tensor F α β = D i α ψ β i | θ→0 is the strength of some vector potential. The first identity can be obtained by acting by two derivatives on the condition D k γ ψ γ k = 0: The second one is a bit trickier. Analyzing the expression ǫ αµνλ D i µ D j ν D k λ ψ β k , one can note that its part, symmetric in α, β, is proportional to ǫ ij : Multiplying this by ǫ ij and using the fact that In the d = 6 vector notation, these two identities can be recognized as self-dual and anti self-dual parts of the identity ∂ [A F BC] = 0: To construct the N = (2, 0), d = 6 Born-Infeld action, it is required to find a proper covariant generalization of these constraints, which would be compatible with additional spontaneously broken supersymmetry. As the construction of the actions of the N = 2, d = 4 and N = 4, d = 3 Born-Infeld theories shows, in the case of the vector multiplets it is not sufficient to formally covariantize the constraints with respect to the broken supersymmetry only by replacing the spinor derivatives with fully covariant ones (8). It is also required to choose the constraints which are covariant with respect to the automorphism group of the considered superalgebra.
Actually, the irreducibility conditions should be covariantized with respect to only the SO(4) subgroup of the whole automorphism group SO (5). Moreover, the SU (2) part of the SO(4) is realized by the linear transformations which rotate the indices i, j, and to preserve this symmetry, it would be sufficient to keep the balance of these indices. The transformations of the coset SO(4)/SU (2) are realized on the variables Now one can immediately derive variations of the differential forms △x αβ , dθ α i , dψ α i with respect to these transformations and, finally, of the derivatives of ψ α i : It can be noted that δ∇ i α ψ β j experiences a shift by the transformation parameter under these transformations, though it affects only its trace part over the Lorentz indices symmetrized with respect to i, j, ∇ (i α ψ j)α . The first component of this combination is the auxiliary field of the multiplet.
Using the transformation laws (16), one can establish the covariant generalization of the constraints (9). The simplest task is to generalize the constraint D i α ψ α i = 0. One can observe that Therefore, in the following matrix power series variations of each term mutually cancel each other: As Tr arctanh ∇ i α ψ β j reduces to D i α ψ α i when all nonlinear terms are neglected, the condition is the suitable one. The second irreducibility condition should be generalized as Their transformation laws could be readily extracted from (16): Then collecting the coefficients of a ij , B ij in the variation of (20), one can find that Tr As we want to find the on-shell identity for the field strength, it is sufficient to know the irreducibility conditions in the first approximation in B ij , or Y α β in the limit B → 0. Then the second relation could be neglected, while the first one implies that It is convenient to write the approximate irreducibility condition as As these conditions are known only approximately, it is not possible to fully check their consistency. However, they are, at least partially, justified by the latter construction.
It should be noted that one can establish the covariance of the constraints with respect to R ij and T ij transformations but not others. For example, for any generator that mixes Q and S, like the generator R 0 , the transformation law for ∇ i α ψ β j will contain a shift by the transformation parameter. However, the irreducibility condition can be written as a relation that expresses the general superfield ∇ i α ψ β j in terms of the superfields , the first components of which are independent components of the multiplet: As the variation of the left-hand side contains the shift term, the variation of the right-hand side should contain such a term, too. Therefore, it is possible to covariantize the identity only with respect to the generators which can be associated with the auxiliary field of the multiplet. Let us also note that the first irreducibility condition (19) remains nonlinear even in the on-shell limit Therefore, Tr V is not equal to zero, unlike the linear case. It remains a nontrivial component, though it is expressed in terms of other components. Interestingly, this condition can be reduced to a much simpler cubic equation with the use of the formula det e A = e TrA : Tr arctanh Also, the derivative of this condition implies that

Bianchi identities
With the irreducibility conditions found, it is possible to derive differential identities that are satisfied by the components V α β = V α β | θ→0 . The derivation of the identities can be made simpler if one needs only the identities in the bosonic limit and with the auxiliary field eliminated by its equation of motion in the final result. To perform this task, one needs to take the irreducibility conditions in the lowest nontrivial approximation in B ij and perform differentiation neglecting B ij in all cases when less than two spinorial derivatives act on it. Much like the identities in the linear case, the first identity can be found by acting by two derivatives on one of the irreducibility conditions: The second identity can be found by the analysis of the expression ǫ αµνλ ∇ i µ ∇ j ν ∇ k λ ψ β k : These identities should be equivalent to the usual ones. This requires that, in particular, the matrices In principle, one may treat F α β as a polynomial of degree 3 in V α β , the matrices M (αβ) (µν) , N (αβ)(µν) -as double polynomials, and equate both sides of relations (29). This approach, however, is very tedious and does not shed light on the nature of the matrices M , N . Additionally, it requires to analyze two separate identities.
To avoid these difficulties, one should rewrite the identities in the vector notation. To additionally simplify these relations, one may note that in both of them the derivatives are found as part of the combination and G 0 could be canceled from the identities. The relation on the components of V α β (25) now implies With the help of (30), two identities (27), (28) can be written as follows: Here Taking into account the self-duality properties of γ ABC αβ , γ ABC αβ (see Appendix), two relations (32) are equivalent to the single one Using the identity ǫ [ABCMN P D K] = 0, (34) can also be presented as It is now clear that this identity should be multiplied by three matrices Σ −1 to be brought to the standard form because it is one and only way to make the indices of all derivatives ∂ A ′ free, as in the canonical identity. To prove exactly that after this multiplication (35) finally acquires the expected form, it is convenient to introduce the matrix In terms of this matrix, identity (35) reads After the multiplication by Σ −1 C ′ C and the integration by parts, the first term can be presented as Using the properties of Φ AB (36) and explicitly taking the derivative of Σ B L , one may find that the generated terms cancel all the extra terms in identity (37). Therefore, the right identity reads ∂ [A F BC] = 0, where For further considerations, it is useful to write it down in the spinor notation: Here relation (25) was used to express Tr V 3 in terms of Tr V and Tr V 2 . Let us also note that the numerator of (40) can be written as

Broken supersymmetry
The component approach to the actions with broken supersymmetry involves the construction of the ansatz for the action invariant with respect to broken supersymmetry by modifying the measure and the derivatives in the bosonic action and adding the Wess-Zumino terms and checking its invariance with respect to unbroken supersymmetry. As θ α i are invariant with respect to broken supersymmetry, the necessary transformation laws and invariant forms can be obtained from (7), (8) in the limit θ → 0. Therefore, the covariant derivative which acts on the components reads It is also useful to rewrite the derivatives and the matrices in the vector notation The active transformation laws of the fields and the vielbein in the vector notation are The invariant measure is, therefore, d 6 x det E.
With these transformations at hand, one can restore the fermionic contributions to the field strength. As is easy to note, the unmodified Bianchi identity ∂ [A F BC] = 0 is not invariant (due to variation of F AB , when active transformations are considered, or due to nontrivial variation of x A if one considers usual transformations). Therefore, the true physical field strength F AB should have another transformation law with respect to broken supersymmetry. The comparison with the N = 2, d = 4 Born-Infeld theory suggests that the right field strength , one may find that it transforms proportionally to itself and its derivatives: Therefore, the identity ∂ [A F BC] = 0 is compatible with broken supersymmetry in all approximations in the fermions.
The simple transformation law of F AB , in comparison with F AB , suggests that the bosonic core of the action should be generalized as Comparing the lowest nontrivial limit of (46) with the free action, one may immediately determine C 1 = 1. It can be rewritten in terms of the variables V α β too, The Wess-Zumino term also should be constructed. As the main action (47) involves the terms of even power in the fermions and in the bosons, the Wess-Zumino term which could make a useful contribution to the action should also be quadratic in the field strengths and at least quadratic in the fermions. Also, its variation with respect to broken supersymmetry transformations (44) should reduce to the full derivative. Therefore, one can expect this term to approximately read Indeed, in the lowest order in the fermions, the only field which transforms is ψ α i without the derivative, δ S ψ α i ∼ ε α i . Then δ S L W Z can be integrated by parts, and the appearing terms with the derivatives of F AB will vanish due to the Bianchi identity.
By adding more terms with the fermions, one can make the Wess-Zumino action invariant with respect to broken supersymmetry in all approximations in the fermions: Indeed, varying this term with respect to transformations (44), one can find that which is full divergence due to the previously established Bianchi identities. In the last line we used two relations It would be useful to rewrite the Wess-Zumino term in the spinor notation. In the lowest approximation in the fermions it reads

Unbroken supersymmetry
The last point in constructing the action is checking its invariance with respect to unbroken supersymmetry. As one of the coefficients in the action (46) was already fixed by the invariance with respect to unbroken supersymmetry in lowest approximation, only one free constant C W Z remains: It should be determined by the invariance with respect to the complete unbroken supersymmetry transformations taken in the lowest approximation in ψ α i . The transformations of the components can be derived with the help of the formula As we plan to prove the invariance of the action in the first order in the fermions, the H µν terms are not relevant and all broken supersymmetry covariant derivatives can be replaced with the usual ones D αβ → ∂ αβ .
The transformations of the basic components in the lowest approximation in the fermions then read The variation of det E is relatively simple, while the variations of the basic bosonic invariants whose action depends on, Tr V , Tr V 2 and Tr V 4 , are too large to be written explicitly. Also, only the fermions have to be varied in the Wess-Zumino term (48), where, up to the full derivative, Then the whole variation of the action (52) can be written as a sum of terms with the structure Φ (k)(m) = ǫ α i V k Tr Though it is far from obvious, the terms with the third power of F α β actually cancel out. To prove this, let us write all pieces of (57) in the vector notation: It is now evident that all cubic terms cancel out, and the only linear term is full divergence due to the bosonic Bianchi identity. Therefore, the final action reads

Conclusion
In this article, the N = (2, 0), d = 6 Born-Infeld theory was considered in the component approach. It was shown that it is possible to construct its component action using the principles already successfully employed in the construction of the component N = 2, d = 4 Born-Infeld theory [7]. These include the use of the standard nonlinear realization formalism with the exponential parametrization of the coset space to find the transformation laws of the superfields with respect to both unbroken and spontaneously broken supersymmetries and automorphisms, as well as the differential forms and the derivatives covariant with respect to these transformations. Another important idea used in this paper, already employed in [7], is that the properly generalized irreducibility conditions of the vector multiplets should be invariant not only with respect to broken supersymmetry but also with respect to the subgroup of the external automorphisms of the supersymmetry algebra. With these ideas implemented, it becomes a difficult though technical problem to calculate the Bianchi identity, which is satisfied by the bosonic field strength, and prove its equivalence to the standard one. The fermionic contributions to the identity can be unambiguously restored by demanding its covariance with respect to broken supersymmetry. The rest of the procedure is common to all studied component actions with partial spontaneous breaking of supersymmetry. It involves modifying the bosonic action following the recipe of Volkov and Akulov [18], adding the Wess-Zumino term, and checking the invariance with respect to unbroken supersymmetry, fixing the remaining arbitrary constants in the process. Let us also mention the observation made during the analysis of the bosonic Bianchi identity for the field strength. This identity involves the matrix which, at the same time, relates the anticommutator of two spinor derivatives to the x A derivative, relates the physical bosonic field strength to the tensor component of the multiplet, and is used to multiply the original identity to bring it to the proper form. Therefore, the role of this matrix is likely fundamental for the component D-brane actions and requires further investigation.
Another important unsolved problem is to perform the reduction of this theory to four dimensions, which should result in the N = 4, d = 4 Born-Infeld theory.