Supercurrents in N = 1 Minimal Supergravity in the Superconformal Formalism
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
Supercurrents in N = 1 Minimal Supergravity in the Superconformal Formalism Sergio Ferraraa, b, *, Marine Samsonyanc, **, Magnus Tournoyd, ***, and Antoine Van Proeyend, **** a
Theoretical Physics Department, CERN CH-1211 Geneva 23, Switzerland Nazionali di Frascati Via Enrico Fermi 40, Frascati, I-00044 Italy c Alikhanyan National Laboratory (Yerevan Physics Institute), Yerevan, 0036 Armenia dKU Leuven, Institute for Theoretical Physics, Celestijnenlaan 200D, Leuven, B-3001 Belgium *e-mails: [email protected] **e-mail: [email protected] ***e-mail: [email protected] ****e-mail: [email protected] bINFN—Laboratori
Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—We discuss the Einstein tensor, the supercurrent and their conservation laws of old and new minimal formulations of supergravity in the superconformal approach. The variation of the action with respect to the gauge field of the R -symmetry in the conformal approach (the auxiliary field in the super-Poincaré action) allows to find the Einstein tensor and supercurrent in any curved background. Hence generalized expressions for their Ward identities follow. This proceeding is based on [1, 2]. DOI: 10.1134/S1547477120050143
Gμν ≈ 0 . The canonical energy momentum-tensor is symmetric due to Lorentz rotation invariance: Tμν = Tνμ . It is also conserved once the field equations of the theory are used. Callan, Coleman and Jackiw (CCJ) have proved in [3] that the energy-momentum tensor can be made traceless once the theory possesses a scale and conformal invariance.
CONTENTS 1. Introduction 2. The Improved Energy-Momentum Tensor 2.1. From Conformal Action 3. Supercurrent and Einstein Tensor from Superconformal Approach 4. Non-linear Form of Ferrara–Zumino Equations of 1975 5. The Supercurrent 5.1. Aμ Field Equation 6. Old Minimal Supergravity with a FI Term 7. New Minimal Supergravity
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Θμν = Tμν + improvement term,
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μ
∂ Jμ ≈ 0
1.
Likewise, for pure gravity one finds
1 We will indicate equations valid modulo equations of motion by
(1.1)
This improved energy-momentum tensor found its place in the supercurrent multiplet J αα . In rigid supersymmetry the conservation law of a supercurrent J αα expressed through its supercurrent divergence is [4–9]
1. INTRODUCTION In general for every rigid symmetry there is a conserved current upon using equations of motion. Currents can be found from gauge couplings. For example, from the Lagrangian of electromagnetic field S = d 4 x − 1 2 Fμν F μν + Aμ J μ + ... , the variation 4g with respect to Aμ gives a current ∂ μ Fμν = J ν , which is conserved upon using the field equations of Aμ , μ
μ
∂ Θμν = 0, Θμν = Θνμ , Θμ = 0.
α D J αα ≈ DαY + ωα ,
(1.2)
where Y , ωα are chiral superfields and moreover ωα satisfies a reality condition Dαωα = D α ωα . In linearized supergravity [5, 8] the current couples to the Einstein multiplet Eαα :
E
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