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# Supersymmetry and Supergravity

Lectures by Peter van Nieuwenhuizen (Stony Brook)

The lectures take place from January 9 – March 6, 2017 as follows:

during January every Monday 14:15-15:45 and Wednesday 12:15-13:45,

February: Wed 1, Fri 3, Mon 13, Wed 15, Fri 17, Mon 27 (always 14:15-15:45)

March: Wed 1, Fri 3, Mon 6

Because of an illness, the lectures scheduled for Feb 15 and 17 have been cancelled;
lectures resumed on Feb 27.

Location: Sem.R. DB gelb 10, Freihaus, TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien

Abstract: In these lectures we assume no knowledge of supersymmetry (susy) or supergravity (sugra). We begin with a very simple model in d=1 (quantum mechanics) where one does not need complicated Fierz-rearrangements, but later we go to d=4 and follow the same approach to construct the corresponding susy and sugra models.
The model is the “WZ” model in d=1 (one real scalar, one real one-component fermion). We establish its rigid supersymmetry (susy). We construct the charges Q and H for rigid susy and time translations. The (anti)commutators of these charges lead to the superalgebra of this model, and to evaluate the (anti)commutators, we introduce Dirac brackets. Then we use the Noether method to find the coupling to the supergravity (sugra) gauge fields (gravitational field h and gravitino field psi). There are no auxiliary fields and no pure gauge action (the Einstein action vanishes in d=1 and the gravitino action already in d=2), and the rigid and local gauge algebras close. Next we repeat this derivation for the first-order formulation of the theory with canonically conjugate momenta for the two matter fields. Then no Dirac brackets are needed but one still must handle the fields h and psi “by hand”.  For that reason we turn to a completely general Hamiltonian approach with canonical momenta for all fields (not only for h and psi, but also for example for the ghosts AND antighosts). The action in this approach follows from Hamiltonian-BRST quantization with a so-called “gauge fermion”. Integrating out redundant fields yields the Lagrangian counterpart of this action, but with a surprise: the gauge fixing term is not proportional to h and psi (as usual in the Lagrangian approach) but to their time derivatives.
Next we go over to superspace. We introduce the general coset formalism and construct supervielbeins, (super) “covariant derivatives” and (super) Lie-derivatives. First we construct the action in the prepotential approach, but then in the covariant approach; in the latter we impose constraints on the super-torsions and super-curvatures (there are only “conventional” constraints in this model) and a partial gauge choice. Instead of a partial gauge choice one can (more satisfactorily) make a field redefinition for which we use an operator formulation of diffeomorphisms which treats gravitational symmetries as Yang-Mills symmetries, with the derivatives (in x-space or in superspace) as generators.
The d=4 topics to be discussed will be determined in discussion with the participants.

Lecture notes/literature:

Exercises: (For more details, contact Abhiram Kidambi – mkabhiram[AT]gmail.com)

Start:
January 9, 2017
End:
March 6, 2017
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