Introduction to supergravity 2015: Exercise 3.
Institute for Theoretical Physics, Masaryk University,
611 37 Brno, Czech Republic
Here we work with the linearized new-minimal supergravity [1]. Our superspace conventions
are found in [2]. The component field definitions for the gravitational superfield in the linearized
new-minimal formulation are (in the Wess-Zumino gauge)
−
i
4
¯D2
Dαφm| = ψmα,
−
1
2
[Dα, ¯D˙α]φm| = hα ˙αm + Bα ˙αm,
−
1
8
Dα ¯D2
Dαφm| = Am,
(1)
where hmn = hnm and Bmn = −Bnm.
The definition of the global supersymmetry transformations is
δO| = ξα
DαO| + ¯ξ˙α
¯D ˙α
O|. (2)
For example for the gravitino
δψmα = δ −
i
4
¯D2
Dαφm| = ξβ
Dβ −
i
4
¯D2
Dαφm | + ¯ξ˙α
¯D ˙α
−
i
4
¯D2
Dαφm |. (3)
Similarly one finds the transformations for all the component fields in (1).
Do the following
1. Find the transformations for all the component fields in (1). Calculate in the WZ gauge.
2. Does the algebra of these supersymmetry transformations close off-shell (up to Poincar´e and
gauge transformations)? Calculate the commutator [δξ, δη]ψmα for example. What is the
difference here compared to the on-shell linearized supergravity?
References
[1] S. Cecotti, S. Ferrara, M. Porrati and S. Sabharwal, “New Minimal Higher Derivative Supergravity
Coupled To Matter,” Nucl. Phys. B 306, 160 (1988).
[2] J. Wess and J. Bagger, “Supersymmetry and supergravity,” Princeton, USA: Univ. Pr. (1992)
259 p
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