Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $σ_-$ Cohomology Analysis for Coxeter HS $B_2$ model

A. A. Tarusov, K. A. Ushakov

TL;DR

The paper develops a comprehensive $\sigma_-$-cohomology analysis for the $B_2$ Coxeter CHS model in $AdS_4$, focusing on one-forms in $(adj\otimes adj)$ and zero-forms in $(tw\otimes adj)$ and $(adj\otimes tw)$. It computes $H^0$, $H^1$, and $H^2$ for the $(adj\otimes adj)$ sector, establishing that the dynamics include symmetric massless and partially massless fields with a rich PM-depth structure, while unitary truncation yields standard Weyl tensors and on-shell Fronsdal dynamics; in the zero-form sectors, it reveals indecomposable $ ext{sl}(2) imes ext{sl}(2)$ structures and primary fields with diverse cocycles. The work also analyzes linear vertices gluing one- and zero-form sectors, finding many non-cohomological terms that impede a simple one-to-one gluing, and shows that suitable field redefinitions and unitary truncations can reorganize the theory so that dynamics are governed by 2-cocycles and on-shell Fronsdal-type equations. Appendices provide a detailed account of non-split extensions and momentum actions, supporting the understanding of indecomposable modules and their role in the CHS spectrum. Overall, the results confirm the presence of partially massless fields in $B_2$ CHS and clarify how the unfolded, cohomological framework encodes the spectrum and its (potential) unitary truncation.

Abstract

The dynamical content of equations resulting from rank-two covariant derivatives in $B_2$ Coxeter theory in $AdS_4$ are analyzed in terms of $σ_-$-complexes. Primary fields and gauge-invariant differential operators on primary fields are classified for $(adj \otimes adj)$ one-form fields $ω$ and $(tw\otimes adj)$ zero-form fields $C$. It is shown that one-forms $ω$ in the $(adj \otimes adj)$ sector encode symmetric massless fields and partially massless fields of all spins and depth of masslessness. Gluing of the one-form module to the zero-form modules at the linear vertices is studied.

The $σ_-$ Cohomology Analysis for Coxeter HS $B_2$ model

TL;DR

The paper develops a comprehensive -cohomology analysis for the Coxeter CHS model in , focusing on one-forms in and zero-forms in and . It computes , , and for the sector, establishing that the dynamics include symmetric massless and partially massless fields with a rich PM-depth structure, while unitary truncation yields standard Weyl tensors and on-shell Fronsdal dynamics; in the zero-form sectors, it reveals indecomposable structures and primary fields with diverse cocycles. The work also analyzes linear vertices gluing one- and zero-form sectors, finding many non-cohomological terms that impede a simple one-to-one gluing, and shows that suitable field redefinitions and unitary truncations can reorganize the theory so that dynamics are governed by 2-cocycles and on-shell Fronsdal-type equations. Appendices provide a detailed account of non-split extensions and momentum actions, supporting the understanding of indecomposable modules and their role in the CHS spectrum. Overall, the results confirm the presence of partially massless fields in CHS and clarify how the unfolded, cohomological framework encodes the spectrum and its (potential) unitary truncation.

Abstract

The dynamical content of equations resulting from rank-two covariant derivatives in Coxeter theory in are analyzed in terms of -complexes. Primary fields and gauge-invariant differential operators on primary fields are classified for one-form fields and zero-form fields . It is shown that one-forms in the sector encode symmetric massless fields and partially massless fields of all spins and depth of masslessness. Gluing of the one-form module to the zero-form modules at the linear vertices is studied.
Paper Structure (23 sections, 2 theorems, 204 equations)

This paper contains 23 sections, 2 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Coxeter HS model and $\sigma_-$ cohomology
  3. Coxeter groups and framed Cherednik algebra
  4. General CHS theory
  5. $B_2$ CHS
  6. Interpretation of unfolded systems through $\sigma_-$ cohomology
  7. One-form module $(adj \otimes adj)$
  8. Computation of $H^\bullet(\sigma_-)$
  9. $H^0(\sigma_-)$
  10. $H^1(\sigma_-)$
  11. $H^2(\sigma_-)$
  12. Summary for $H^{0,1,2}(\sigma_-)$
  13. Zero-form module $(tw\otimes adj)$ and $(adj\otimes tw)$
  14. Computation of $H^\bullet(\sigma_-)$
  15. $H^0(\sigma_-)$
  16. ...and 8 more sections

Key Result

Theorem 2.1

The following is true for a $p$-form $\mathcal{W}$: 1) Differential gauge symmetry parameters $\varepsilon$ span $H^{p-1}(\sigma_-)$ 2) Nontrivial dynamical fields $\mathcal{W}$ span $H^p(\sigma_-)$ 3) Gauge-invariant differential operators on the nontrivial dynamical fields, contained in $\mathcal{

Theorems & Definitions (3)

  • Theorem 2.1
  • Lemma 7.1
  • proof