Introduction to supergravity 2015: Exercise 5.
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]. A Bianchi identity for this supergravity reads
Wα = −
1
2
¯D ˙α
Eα ˙α, (1)
which can be used as a definition for Wα. Here Wα is a superfield of the new-minimal supergravity
which also has the property
Wα ∼ θαR, (2)
where R is the linearized Ricci scalar.
The Lagrangian for linearized new-minimal R + R2
supergravity reads
LR+R2 = d4
θ φmEm
+ α d2
θ Wα
Wα + c.c. . (3)
Do the following
1. Explain why (3) gives R + R2
.
2. Write (3) in the linearized superconformal supergravity framework with the help of a real
linear compensator L.
3. Make the compensator unconstrained by introducing the term
ΦL + ¯ΦL, (4)
where Φ is a chiral superfield.
4. Show that we can not integrate out L. Verify that we could integrate it out for α = 0. What
does this mean?
5. Find the dual theory to (3). You will find standard linearized supergravity and a massive
vector multiplet [1,3].
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
[3] S. Cecotti, S. Ferrara and L. Girardello, “Massive Vector Multiplets From Superstrings,”
Nucl. Phys. B 294, 537 (1987).
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