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Unitarity in three-dimensional flat space higher spin theories

D. Grumiller, M. Riegler, J. Rosseel

TL;DR

We study unitarity constraints for 3D flat-space higher-spin theories and find a no-go: for generic non-linear $\mathcal{W}$-algebras arising from contractions of AdS higher-spin symmetries, insisting on unitarity forces $c_M\to0$ and decouples the higher-spin content. This no-go applies to principal embeddings and many non-principal contractions, though a linear counterexample exists via a flat-space contraction of the linear $\mathcal{W}_\infty$ (hs$(1)$) sector, which can be unitary with non-trivial HS states. A second line of results shows that certain non-linear contracted algebras (e.g., GB-style and PB-type) admit tight central-charge bounds that further constrain unitary representations, sometimes eliminating multi-graviton excitations. The Yes-go is realized by linear flat-space HS algebras built from contractions of $\mathcal{W}_\infty$ (or $\mathcal{W}_{1+\infty}$), where unitarity is compatible with non-trivial HS content, suggesting that flat-space chiral higher-spin gravity may exist in a linearized framework. Overall, the work clarifies when flat-space higher-spin theories can be unitary and points to linearized or chiral constructions as viable directions for consistent theories with HS content.

Abstract

We investigate generic flat-space higher spin theories in three dimensions and find a no-go result, given certain assumptions that we spell out. Namely, it is only possible to have at most two out of the following three properties: unitarity, flat space, non-trivial higher spin states. Interestingly, unitarity provides an (algebra-dependent) upper bound on the central charge, like c=42 for the Galilean $W_4^{(2-1-1)}$ algebra. We extend this no-go result to rule out unitary "multi-graviton" theories in flat space. We also provide an example circumventing the no-go result: Vasiliev-type flat space higher spin theory based on hs(1) can be unitary and simultaneously allow for non-trivial higher-spin states in the dual field theory.

Unitarity in three-dimensional flat space higher spin theories

TL;DR

We study unitarity constraints for 3D flat-space higher-spin theories and find a no-go: for generic non-linear -algebras arising from contractions of AdS higher-spin symmetries, insisting on unitarity forces and decouples the higher-spin content. This no-go applies to principal embeddings and many non-principal contractions, though a linear counterexample exists via a flat-space contraction of the linear (hs) sector, which can be unitary with non-trivial HS states. A second line of results shows that certain non-linear contracted algebras (e.g., GB-style and PB-type) admit tight central-charge bounds that further constrain unitary representations, sometimes eliminating multi-graviton excitations. The Yes-go is realized by linear flat-space HS algebras built from contractions of (or ), where unitarity is compatible with non-trivial HS content, suggesting that flat-space chiral higher-spin gravity may exist in a linearized framework. Overall, the work clarifies when flat-space higher-spin theories can be unitary and points to linearized or chiral constructions as viable directions for consistent theories with HS content.

Abstract

We investigate generic flat-space higher spin theories in three dimensions and find a no-go result, given certain assumptions that we spell out. Namely, it is only possible to have at most two out of the following three properties: unitarity, flat space, non-trivial higher spin states. Interestingly, unitarity provides an (algebra-dependent) upper bound on the central charge, like c=42 for the Galilean algebra. We extend this no-go result to rule out unitary "multi-graviton" theories in flat space. We also provide an example circumventing the no-go result: Vasiliev-type flat space higher spin theory based on hs(1) can be unitary and simultaneously allow for non-trivial higher-spin states in the dual field theory.

Paper Structure

This paper contains 19 sections, 90 equations.

Table of Contents

  1. Introduction
  2. Flat space contractions and unitarity
  3. Flat space holography
  4. Unitarity in Galilean conformal algebras
  5. Contraction for higher spin theories in principal embedding ($\mathcal{W}_N$)
  6. Unitarity in contracted Polyakov--Bershadsky ($\mathcal{W}_3^{(2)}$)
  7. Galilean Polyakov--Bershadsky
  8. Unitarity in Galilean Polyakov--Bershadsky
  9. No-go: Unitarity in contractions of general non-linear $\mathcal{W}$-algebras
  10. Upper bound on central charge ($\mathcal{W}_4^{(2-1-1)}$)
  11. Feigin--Semikhatov family of examples ($\mathcal{W}_N^{(2)}$)
  12. General no-go result
  13. Elimination of multi-graviton excitations ($\mathcal{W}_4^{(2-2)}$)
  14. Yes-go: Unitarity in flat space hs$(1)$
  15. The $\mathcal{W}_\infty$ algebra
  16. ...and 4 more sections