On Born-Infeld-type higher-spin interactions and supersymmetry

Technical Summary: On Born-Infeld-type higher-spin interactions and supersymmetry

Problem Statement
The construction of consistent cubic interactions for massless higher-spin fields in four-dimensional flat space is a fundamental step toward understanding higher-spin gravity. While the classification of such interactions (specifically parity-invariant vertices) was established by Metsaev, explicit Lorentz-covariant constructions for the "Born-Infeld" (BI) type vertices—those containing the maximal number of derivatives, $s + s_1 + s_2$—have been lacking. Previous attempts, such as the Berends–Burgers–van Dam (BBvD) currents, utilized Weyl tensors of different chiralities ($J \sim \partial C \partial \bar{C}$). However, the specific structure required for BI-type vertices involves currents constructed from Weyl tensors of the same chirality ($J \sim \partial C \partial C$). Furthermore, while these interactions are known to be necessary for constructing consistent higher-order deformations, their explicit off-shell supersymmetric extensions for $N=1$ and $N=2$ had not been systematically derived.

Methodology
The paper employs a manifestly Lorentz-covariant approach using two-component spinor notation. The methodology proceeds through the following steps:

1. Ansatz Construction: The author formulates a general ansatz for the higher-spin current $J$ and its trace $T$ in terms of the higher-spin Weyl tensors ($C$) of the interacting fields. The ansatz includes arbitrary complex coefficients and sums over possible derivative distributions.
2. Conservation Constraints: The coefficients are fixed by imposing the conservation equations on the mass shell of the lower-spin fields ($s_1$ and $s_2$). This leads to a system of recurrence relations for the coefficients.
3. Elimination of "Fake" Vertices: The author distinguishes between physical interactions and "fake" vertices. Fake vertices are those that can be removed by local field redefinitions and correspond to terms involving the gauge-invariant Fronsdal curvature tensors ($R$) rather than the Weyl tensors. By solving the recurrence relations and identifying the trivial solutions associated with these fake vertices, the author isolates the unique non-trivial conserved current.
4. Supersymmetric Generalization: The construction is extended to $N=1$ and $N=2$ supersymmetry using off-shell superfield formalisms:
   • For $N=1$, the author utilizes the standard superfield formalism, constructing supercurrents from Weyl supertensors ($W$) of the same chirality. They impose strengthened conservation equations ($D_\alpha J \approx 0$ and $\bar{D}_{\dot{\alpha}} J \approx 0$) to select the physical representative.
   • For $N=2$, the harmonic superspace formalism is employed. The author constructs a principal complex supercurrent using $N=2$ Weyl supertensors and harmonic derivatives ($D^{\pm}$), solving the corresponding conservation equations to uniquely fix the coefficients.

Key Contributions and Results

• Non-Supersymmetric Case ($N=0$):

  • The paper derives the unique conserved higher-spin current for arbitrary spins $s \ge s_1 \ge s_2$ with $s + s_1 + s_2$ derivatives.
  • It is demonstrated that for identical fields ($s_1 = s_2$), the traceless conserved current vanishes identically if the spin $s$ is odd. For even $s$, a non-trivial current exists.
  • In the range $s \le s_1 + s_2$, the interaction is shown to admit a manifestly gauge-invariant representation purely in terms of higher-spin Weyl tensors, establishing the equivalence between the current formulation and the Weyl tensor formulation.
  • The analysis confirms that non-trivial BI vertices require two distinct fields of the same chirality; interactions involving a single field or fields of opposite chirality yield only fake vertices.

• $N=1$ Supersymmetry:

  • A general conserved supercurrent for integer superspin $s$ is constructed, bilinear in the Weyl supertensors of superspins $Y_1$ and $Y_2$.
  • The result generalizes previous findings by Gates and Koutrolikos. Specifically, for identical superspins ($Y_1 = Y_2$), the supercurrent vanishes for odd integer superspin $s$, mirroring the bosonic case.
  • For $s \le Y_1 + Y_2$, the interaction is shown to possess a chiral superspace representation involving the product of three Weyl supertensors.

• $N=2$ Supersymmetry:

  • Using harmonic superspace, the author constructs the principal higher-spin supercurrent with coefficients uniquely fixed by conservation equations.
  • A significant distinction is found compared to the $N=0$ and $N=1$ cases: for identical $N=2$ superfields, the constructed vertex does not vanish. This is attributed to the structure of $N=2$ supermultiplets, which combine integer and half-integer $N=1$ superspins, allowing different bosonic components to contribute non-trivially even if one sector vanishes.

Significance
The paper provides the first systematic, explicit construction of Born-Infeld-type higher-spin interactions and their $N=1$ and $N=2$ supersymmetric extensions in a manifestly Lorentz-covariant form. The central observation is that these vertices are generated by conserved currents built from same-chirality Weyl tensors ($J \sim \partial C \partial C$), distinguishing them from the previously studied BBvD currents.

The author claims that these results provide a necessary framework for constructing consistent interactions at higher orders, as BI-type vertices are known to be required for higher-order deformations of the cubic action. By providing explicit forms for these currents, the work offers a starting point for analyzing higher-order deformations and extends the understanding of higher-spin dynamics into the supersymmetric domain. The paper does not propose experimental applications or extend the results to anti-de Sitter space or $N \ge 3$ supersymmetry, noting these as areas where formalisms are less developed.