Gravity and Strings

13.2 Supersymmetric vacua and residual (unbroken) supersymmetries 375
fields in supergravity theories are Lorentz tensors (with vector or spinor indices) and the
standard Lie derivative is not covariant under local Lorentz transformations and its action
on Lorentz tensors is frame-dependent.
This has annoying consequences: for instance the Lie derivatives of Vielbeins with respect
to a Killing vector will not be zero in general, even though the same Lie derivative of
the metric always will.
On the other hand, Lorentz tensors (and, in particular, spinors) in curved spaces are
treated as scalars under GCTs in (Weyl’s) standard formalism explained in Section 1.4.
Then, if we work in Minkowski spacetime in curvilinear coordinates using Weyl’s formalism
and perform a Lorentz transformation, all Lorentz tensors and spinors will be invariant.
This looks strange, but is not unphysical: in practice one always makes a choice of frame
based on some simplicity criterion. For instance, we could always set the Vielbein matrix
in an upper-triangular form using local Lorentz transformations. This choice can be seen as
agauge-fixing condition that uses up all the Lorentz gauge symmetry. If we now perform
a GCT (for instance, the Lorentz transformation we were discussing), it will be necessary
to implement a compensating local Lorentz transformation in order to keep the Vielbein
matrix upper-triangular. This local Lorentz transformation will act on all Lorentz tensors
and can be understood as the effect of the GCT on them.
It is necessary for our purposes to find an operator acting on Lorentz tensors that implements
the adequate compensating local Lorentz transformation for each GCT. This operator
is the Lie–Lorentz derivative [748], which was first introduced for spinors by Lichnerowicz
and Kosmann in [632, 633, 655] and used in supergravity by Figueroa-O’Farrill in [390]
(see also [586, 919, 920]). In simple terms, it is just a Lorentz-covariant Lie derivative.
Analogous problems arise whenever there are additional local symmetries. For instance,
in N = 2, d = 4supergravity there is a local U(1) symmetry. In the Poincaré case only the
gauge potential Aµ transforms under it, but in the AdS case (“gauged N = 2, d = 4 supergravity”)
the gravitinos and infinitesimal supersymmetry parameters transform as doublets
(they are charged). A U(1)-covariant derivative (Lie–Maxwell derivative)isneeded in order
to represent infinitesimal GCTs on these fields.
Covariant Lie derivatives can be found also in the context of the geometry of reductive
coset spaces G/H (see Appendix A.4) on which there is a well-defined action of H.Infact,
the Lie–Lorentz derivative coincides with it in coset spaces in which spinors can be defined
and H is a subgroup of the Lorentz group [25].
More generally, they can be defined in principal bundles with a reductive G-structure 7
[460], but here we will not make use of this formalism.
13.2.1 Covariant Lie derivatives
The Lie–Lorentz derivative The spinorial Lie–Lorentz derivative with respect to any vector
v of a Lorentz tensor T transforming in the representation r is given by
LvT ≡ v ρ ∇ρT + 1
2 ∇[avb] Ɣr(M ab )T, (13.9)
7 Recall that G/H is a principal bundle over G/H with structure group H, sothis is a special case.