Abstract
We derive the characters of all unitary irreducible representations of the (d+1)-dimensional de Sitter spacetime isometry algebra mathfrak{so}left(1,kern0.5em d+1right) , and propose a dictionary between those representations and massive or (partially) massless fields on de Sitter spacetime. We propose a way of taking the flat limit of representations in (anti-) de Sitter spaces in terms of these characters, and conjecture the spectrum resulting from taking the flat limit of mixed-symmetry fields in de Sitter spacetime. We identify the equivalent of the scalar singleton for the de Sitter (dS) spacetime.
Highlights
The importance of mixed-symmetry fields is no longer to be emphasised: whether motivated by string theory — where they make up most of the spectrum — or, more generically, by quantum field theory in arbitrary dimensions — where they are the central objects of interest in the sense that they are the most general fields one may consider.1 At the free level, equations of motions for massless mixed-symmetry fields in flat spacetime were spelled out by Labastida [1, 2]2, and later given in the unfolded form [7]
On top of that, mixed-symmetry gauge fields in AdSd+1 were shown to have quite an interesting flat limit [23]: starting from a gauge field in AdSd+1 with symmetry encoded by an arbitrary so(d) Young diagram Y and sending the cosmological constant Λ to zero yields a spectrum of massless fields in flat spacetime composed of all possible fields labelled by so(d − 1) Young diagrams obtained from Y by removing boxes in each of the last rows of each block, leaving the first block untouched
We found for gauge fields of arbitrary shape, that unitary fields in dSd+1 are those whose gauge symmetry involves the lowest block of their Young diagram
Summary
The importance of mixed-symmetry fields (i.e. fields whose physical components carry representations of the little group described by Young diagrams of height greater than one) is no longer to be emphasised: whether motivated by string theory — where they make up most of the spectrum — or, more generically, by quantum field theory in arbitrary dimensions — where they are the central objects of interest in the sense that they are the most general fields one may consider. At the free level, equations of motions for massless mixed-symmetry fields in flat spacetime were spelled out by Labastida [1, 2]2 (see [4, 5] for a proof that his equations and trace constraints describe the right propagating degrees of freedom and [6] for the fermionic case), and later given in the unfolded form [7]. A nontrivial difference between the positive and negative cosmological constant cases is the question of unitarity of the fields and their corresponding (irreducible) representations, which is one of the main issues investigated in the present paper In deriving all these equations, the constraints imposed by gauge symmetry were crucial. On top of that, mixed-symmetry gauge fields in AdSd+1 were shown to have quite an interesting flat limit [23]: starting from a gauge field in AdSd+1 with symmetry encoded by an arbitrary so(d) Young diagram Y and sending the cosmological constant Λ to zero yields a spectrum of massless fields in flat spacetime composed of all possible fields labelled by so(d − 1) Young diagrams obtained from Y by removing boxes in each of the last rows of each block (until it reaches the length of the row just below), leaving the first (upper) block untouched For proofs of this spectrum, see [12, 13, 24]. The remaining generators D, Pi and Kj correspond respectively to infinitesimal dilations, translations and special conformal transformations of the Euclidean space Rd
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