Duality invariance of all free bosonic and fermionic gauge fields

We give a simple general extension to all free bosonic and fermionic massless gauge fields of a recent proof that spin 2 is duality invariant in flat space. We also discuss its validity in (A)dS backgrounds and the relevance of supersymmetry. Recently [1], it was shown that the duality invariance originally established [2] for Maxwell theory in D=4, extends to free massless spin 2. It was also conjectured there that higher spin gauge bosons, and perhaps General Relativity (GR) might enjoy a similar property. In this Note, a simple method is used to exhibit the duality invariance of all free gauge fields, bosonic and fermionic; some implications of supersymmetry and the extent of flat space duality in (A)dS are also discussed. We begin with bosons. The key to our derivation is the remark that since free gauge field actions are (abelian) gauge invariant, they can be uniformly expressed–after elimination of constraints–in terms of the fundamental spatial gauge invariant symmetric transverse-traceless (TT) conjugate variables [3] (π ij··· T T , q T T ij···), ∂ i π ij··· T T ≡ ∂ i q T T i j··· ≡ 0 ; π i··· T T i ≡ q T T i i··· ≡ 0. (1) In this connection, we recall that only the above dynamical variables can be meaningfully varied; neither constraint, gauge nor Lagrange multipliers play any role here. Only spatial indices appear (s of them for spin s) and we drop the " TT " henceforth. The obvious extension of the Maxwell action is