[Paper Review] Lagrangian formulation of massive fermionic totally antisymmetric tensor field theory in AdS_d space

I. L. Buchbinder, V.A. Krykhtin+1 more|arXiv (Cornell University)|Feb 9, 2009

Black Holes and Theoretical Physics4 citations

TL;DR

This paper presents a simplified Lagrangian formulation for massive fermionic totally antisymmetric tensor fields in AdS_d space using a fermionic BRST approach. By incorporating antisymmetry from the outset via fermionic creation/annihilation operators, the method eliminates auxiliary fields and yields a final theory formulated exclusively in terms of the physical field, resolving a key limitation of generic higher-spin approaches.

ABSTRACT

We apply the BRST approach, developed for higher spin field theories, to Lagrangian construction for totally antisymmetric massive fermionic fields in AdS_d space. As well as generic higher spin massive theories, the obtained Lagrangian theory is a reducible gauge model containing, besides the basic field, a number of auxiliary (Stuckelberg) fields and the order of reducibility grows with the value of the rank of the antisymmetric field. However, unlike the generic higher spin theory, for the special case under consideration we show that one can get rid of all the auxiliary fields and the final Lagrangian for fermionic antisymmetric field is formulated only in terms of basic field.

Motivation & Objective

• To develop a simplified Lagrangian formulation for massive fermionic totally antisymmetric tensor fields in AdS_d space.
• To overcome the complexity of generic higher-spin Lagrangians that require numerous auxiliary fields and reducible gauge symmetries.
• To leverage fermionic creation and annihilation operators to encode antisymmetry from the start, reducing field content.
• To demonstrate that all auxiliary fields can be removed, resulting in a final theory formulated solely in terms of the physical field.
• To provide a new, simpler, and previously unpublished Lagrangian that correctly realizes the irreducible representation of the AdS_d group on massive antisymmetric fermionic fields.

Proposed method

• Adopt a BRST approach tailored for higher-spin field theories, adapted to fermionic antisymmetric fields in AdS_d.
• Use fermionic creation and annihilation operators to naturally implement antisymmetry of tensor indices from the outset.
• Construct a closed operator algebra from the equations of motion, enabling systematic BRST operator construction.
• Derive extended operator expressions and build the BRST charge to generate the gauge structure and constraints.
• Perform gauge fixing using residual parameters to eliminate dependence on auxiliary fields and $f^+$, $b^+$ operators.
• Simplify the final Lagrangian by expressing all fields in terms of the physical field $| ilde{ ho}_n angle$, eliminating all auxiliary components.

Experimental results

Research questions

• RQ1 Can a simpler Lagrangian formulation be achieved for massive fermionic totally antisymmetric tensor fields in AdS_d by incorporating antisymmetry at the operator level?
• RQ2 Is it possible to eliminate all auxiliary (Stückelberg) fields from the Lagrangian of a massive antisymmetric fermionic field in AdS_d?
• RQ3 Does the use of fermionic creation/annihilation operators lead to a more compact and physically transparent Lagrangian compared to generic higher-spin methods?
• RQ4 Can the resulting Lagrangian be fully expressed in terms of the physical field only, without auxiliary fields or gauge symmetries?
• RQ5 Does the final Lagrangian correctly reproduce the irreducible representation of the AdS_d group for massive antisymmetric fermionic fields?

Key findings

• The final Lagrangian is formulated exclusively in terms of the physical field $| ilde{ ho}_n angle$, with all auxiliary fields and gauge symmetries removed.
• The method achieves a significant simplification over generic higher-spin approaches by eliminating the need for auxiliary fields and reducible gauge symmetries.
• The Lagrangian is derived using a fermionic BRST approach that encodes antisymmetry from the beginning via fermionic operators.
• The resulting Lagrangian (equation C.19) is expressed as a sum over derivatives and mass terms involving the physical field and its traces.
• The component form of the Lagrangian (equation C.21) explicitly shows the kinetic, mass, and divergence terms in a manifestly Lorentz-covariant and gauge-invariant way.
• The construction confirms that the theory realizes the correct irreducible representation of the AdS_d group on massive antisymmetric fermionic fields, as verified in Appendix B.