On the Uniqueness of D = 11 Interactions Among a Graviton, a Massless Gravitino and a Three-Form Iii: Graviton and Gravitini

Abstract

Under the hypotheses of smoothness of the interactions in the coupling constant, locality, Poincaré invariance, Lorentz covariance and the preservation of the number of derivatives on each field in the Lagrangian of the interacting theory (the same number of derivatives like in the free Lagrangian), we prove that in D = 11 there are no cross-interactions between the graviton and the massless gravitino and also no self-interactions in the Rarita-Schwinger sector. A comparison with the case D = 4 is briefly discussed.

On the Uniqueness of D = 11 Interactions Among a Graviton, a Massless Gravitino and a Three-Form Ii: Three-Form and Gravitini

“…It is simple to see that (2) p 2 (which contributes to a A−ψ 0 ) produces field equations for the Rarita-Schwinger field with two spacetime derivatives, which disagrees with requirement (i) from the beginning of this section related to the derivative order assumption. Thus, we must set…”

“…Next, we approach equation (85). Due to the fact that each (2) M µ is a gaugeinvariant, 11 × 11 matrix with two spacetime derivatives, it contains precisely two three-form field strengths (since it cannot depend on ∂ [αψβ] , which is a spinor). As the elements (0) N αβ µνρ are derivative-free and gauge-invariant, they can only be constant.…”

“…In order to investigate the former condition, we represent so we can only have (2) p 1 = 0 in (94). Now, we investigate equation (95).…”

“…Assume that the second term from the right-hand side of (105) would give a γ-exact modulo d quantity. Comparing (105) to (85), we find that (2)…”

On the consistent interactions in D = 11 among a graviton, a massless gravitino and a three‐form

“…We have shown in [6][7][8][9] that the first-order deformation S 1 can be decomposed as a sum of six components…”