[Paper Review] Gauge Non-Invariant Higher-Spin Currents in $4d$ Minkowski Space

P. A. Smirnov, M. A. Vasiliev|arXiv (Cornell University)|Dec 23, 2013

Black Holes and Theoretical Physics9 references3 citations

TL;DR

This paper constructs non-gauge invariant higher-spin (HS) currents of arbitrary spin $ t > 0 $ in 4d Minkowski space from symmetric massless gauge fields of spin $ s \geq t $, showing they yield gauge-invariant conserved charges. Surprisingly, it identifies previously unanticipated parity-odd currents that generate fewer symmetries than parity-even ones and likely do not admit consistent $AdS$ deformations, suggesting a link to mixed-symmetry fields in higher dimensions.

ABSTRACT

Conserved currents of any spin $t>0$ built from symmetric massless gauge fields of any integer spin $s \geq t$ in {4d} Minkowski space are found. In particular, stress-energy tensor for a higher-spin field of any spin is constructed. Analogously to spin-two stress (pseudo)tensor, currents considered in this paper are not gauge invariant. However, they are shown to generate gauge invariant conserved charges. Besides expected parity even HS currents, we found unexpected parity odd currents that generate less symmetries than the even ones. It is argued that these odd currents unlikely admit a consistent $AdS$ deformation.

Motivation & Objective

• To systematically construct conserved currents of any spin $ t > 0 $ built from symmetric massless gauge fields of spin $ s \geq t $ in 4d Minkowski space.
• To clarify the role of non-gauge invariant currents in generating gauge-invariant conserved charges, despite their lack of gauge invariance.
• To identify and analyze unexpected parity-odd higher-spin currents that possess fewer global symmetries than their parity-even counterparts.
• To assess the viability of consistent $AdS$ deformations for these parity-odd currents, suggesting their incompatibility with $AdS$ backgrounds.

Proposed method

• Utilizes the frame-like formulation of massless higher-spin fields in 4d Minkowski space using two-component spinor notation.
• Constructs conserved currents $ J^{\alpha(t-1),\dot{\beta}(t-1)} $ as bilinear forms in the spinor connections $ \omega^{i;\alpha(m)\dot{\beta}(n)} $, ensuring current conservation via on-shell conditions.
• Employs a systematic analysis of bilinear forms in the gauge fields and their derivatives, identifying all possible current structures via symmetry and tensor rank constraints.
• Applies differential forms and cohomological techniques to test whether currents can be written as exact forms $ \tilde{D}\Psi $, proving non-exactness and thus non-triviality of the currents.
• Uses the reality condition $ \omega_{\alpha(m),\dot{\beta}(n)}^\dagger = \omega_{\beta(n),\dot{\alpha}(m)} $ to ensure physical consistency of the field content.
• Analyzes the structure of the currents under parity and determines that parity-odd currents arise from antisymmetric combinations of field strengths and auxiliary tensors.

Experimental results

Research questions

• RQ1 Can non-gauge invariant higher-spin currents of arbitrary spin $ t > 0 $ be consistently constructed from symmetric massless fields of spin $ s \geq t $ in 4d Minkowski space?
• RQ2 Do these non-gauge invariant currents still give rise to gauge-invariant conserved charges, and if so, how are they related to global symmetries?
• RQ3 What is the origin and significance of the newly discovered parity-odd higher-spin currents, and why do they generate fewer symmetries than parity-even ones?
• RQ4 Can these parity-odd currents be consistently deformed to an $AdS_4$ background, and what does this imply for their physical realizability in nonlinear higher-spin theories?

Key findings

• The paper constructs conserved currents of any spin $ t > 0 $ from symmetric massless gauge fields of spin $ s \geq t $, including the stress-energy tensor for any higher-spin field.
• These currents are not gauge invariant, analogous to the gravitational pseudo-tensor, yet they generate gauge-invariant conserved charges via Noether's theorem.
• Beyond expected parity-even currents, the paper identifies unexpected parity-odd currents that correspond to reduced global symmetry algebras.
• The parity-odd currents are shown to be non-exact, indicating they represent non-trivial cohomology classes and thus are physically meaningful despite lack of gauge invariance.
• These parity-odd currents are argued to likely not admit a consistent $AdS$ deformation, suggesting they may be related to mixed-symmetry fields in higher dimensions.
• The construction suggests a potential link between the odd currents and mixed-symmetry field content in $d > 4$, which reduces to symmetric fields only in $d=4$.