[Paper Review] A Codicil To Massless Gauge Superfields of Higher Half-Odd Integer Superspins

S. James Gates, Konstantinos Koutrolikos|arXiv (Cornell University)|Mar 18, 2011

Black Holes and Theoretical Physics2 references8 citations

TL;DR

This paper presents a new formulation of 4D, $/mathcal{N}=1$ supersymmetric theories with massless higher half-odd-integer superspins, using a systematic algorithm based on gauge invariance and compensator superfields. It identifies a new class of theories for arbitrary superspin $Y = s + 1/2$, with the $s=1$ case recovering the original off-shell supergravity formulation by Breitenlohner, and introduces a new-minimal supergravity action via dual on-shell constraints.

ABSTRACT

We study theories of 4D, N=1 supersymmetric massless, arbitrary higher half odd-integer superspins. A new series of such theories is found to exist for arbitrary superspin Y (Y=s+1/2 for any integer s). The lowest member (s=1) of the series is the original off-shell formulation of 4D, N=1 supergravity first presented by Breitenlohner in 1977.

Motivation & Objective

• To re-express and extend the known formulations of massless higher half-odd-integer superspin multiplets in 4D, $/mathcal{N}=1$ supersymmetry.
• To identify new off-shell formulations beyond the two known ones established by Kuzenko, Postnikov, and Sibiryakov.
• To generalize the algorithm used for integer superspins to the half-odd-integer case, ensuring gauge invariance and consistency.
• To derive a new-minimal supergravity formulation by imposing a linear compensator constraint on the auxiliary superfield $U$.

Proposed method

• Construct a real, gauge-invariant action for a bosonic superfield $H_{{ar{ heta}}(s){ heta}(s)}$ with zero mass dimension and $2s+1$ symmetric indices.
• Use a general quadratic action with four free coefficients $c_1, c_2, c_3, c_4$, including terms with $D$, $ar{D}$, $ar{D}D$, and $ar{D}D$ derivatives.
• Impose gauge symmetry on $H_{{ar{ heta}}(s){ heta}(s)}$ via a complex parameter superfield $L_{{ar{ heta}}(s){ heta}(s-1)}$, ensuring invariance under $ar{D}L - Dar{L}$ transformations.
• Introduce auxiliary superfields $U$, $ar{ heta}$, and $ heta$ as compensators to enforce consistency and reduce degrees of freedom.
• Solve the Bianchi identities by demanding gauge invariance of the action, leading to constraints on coefficients: $c_3 = c_4$, $e=0$, $b=6c_4$, $g=4c_4$.
• Derive two equivalent formulations: one using the equation of motion for $U$ (yielding minimal supergravity), and another using the equation of motion for $ heta$ (yielding new-minimal supergravity).

Experimental results

Research questions

• RQ1 Can a new off-shell formulation be constructed for massless higher half-odd-integer superspin multiplets in 4D, $/mathcal{N}=1$ supersymmetry?
• RQ2 What are the necessary conditions on the action and gauge transformations to ensure consistency and gauge invariance for arbitrary superspin $Y = s + 1/2$?
• RQ3 Does the algorithm used for integer superspins generalize to the half-odd-integer case, and if so, what new theories emerge?
• RQ4 Can the new-minimal supergravity formulation be derived from a unified action with auxiliary compensators and consistent constraints?
• RQ5 What is the role of the auxiliary superfield $U$ in realizing the new-minimal formulation, and how does it differ from the minimal one?

Key findings

• A new series of off-shell formulations for massless 4D, $/mathcal{N}=1$ higher half-odd-integer superspins is constructed for arbitrary $Y = s + 1/2$, with $s$ an integer.
• The $s=1$ case of the new formulation reproduces the original off-shell supergravity action first presented by Breitenlohner in 1977.
• The action with $c_3 = c_4$ and $c_1 = 2c_4$, $c_2 = c_4$, $c_3 = c_4$, $c_4 = c_4$ is invariant under the gauge transformations $ar{D}_{{ar{ heta}}}L - D_{ heta}ar{L}$, ensuring consistency.
• The minimal supergravity formulation arises when the auxiliary superfield $U$ is eliminated via its equation of motion, yielding $S_1$ with a term proportional to $irac{4}{3}c_4 H^{ar{ heta} heta} ar{ heta} heta$.
• The new-minimal supergravity formulation is derived by imposing $ar{D}^2 U = 0$ as a constraint, leading to $S_2$ with a $6c_4 U^2$ term and a gauge-invariant action under $ar{D}^2 U = 0$.
• The new-minimal formulation is explicitly invariant under the gauge transformations $ar{D}_{{ar{ heta}}}L - D_{ heta}ar{L}$ and $ar{D}^2 D^{ar{ heta}}L_{ar{ heta}}$, confirming its consistency.