Gravity and Strings

370 Unbroken supersymmetry
theories with more supercharges for Part III because these theories can be derived from
ten-dimensional superstring effective theories, but we will say what can be expected from
general arguments based on the structures of the respective superalgebras.
In Section 13.5 we will study the properties of solutions with partially broken supersymmetry
that cannot be considered vacua but instead can be considered as excitations of some
vacuum to which they tend asymptotically. By associating states in a quantum theory with
these solutions and using the vacuum superalgebra, general supersymmetry bounds for the
mass can be derived. These bounds are saturated by (supersymmetric or “BPS”) states with
partially unbroken supersymmetry. The bounds can be extended to solutions of the theory,
even in the absence of supersymmetry, if certain conditions on the energy–momentum tensor
are imposed. These are very powerful techniques.
In Section 13.5.2 we will review important examples of solutions with unbroken supersymmetries
in N = 1, 2, 4, d = 4 Poincaré supergravity, including the general families
of supersymmetric solutions which are known only for these cases. In particular, we will
discuss the relations among BH thermodynamics, cosmic censorship, and unbroken supersymmetry
in these theories.
13.1 Vacuum and residual symmetries
The solutions of the equations of motion of a given theory usually break most (or all) of its
symmetries. Sometimes a solution has (preserves) some of them, which receive the name
of residual (or unbroken) symmetries, and, being symmetries, they form a symmetry group.
The solution is said to be symmetric. The symmetries of the theory which are broken by
the symmetric solution can be used to generate new solutions of the theory. Let us see two
examples.
Classical mechanics. The Lagrangian of a free relativistic particle moving in Minkowski
spacetime is invariant under the whole Poincaré group ISO(1,3). However, every
solution is a straight line, invariant only under translations parallel to it and rotations
with it as the axis. These are the residual symmetries of every solution and form a
two-dimensional group R × SO(2). The remaining Poincaré transformations move
the line and generate other solutions.
Field theory. Einstein’s equations are invariant under the infinite-dimensional group of
GCTs. However, a given solution (metric) is invariant only under a finite-dimensional
group of isometries. Bydefinition, an infinitesimal isometry is an infinitesimal GCT
that leaves the metric invariant, that is
δξ gµν =−Lξ gµν =−2∇(µξν) = 0, (13.1)
which is known as the Killing equation. The solutions ξ µ = ξk µ are each the product
of an infinitesimal constant ξ times a Killing vector k µ , the generator of the isometry.
The isometries of a metric form an isometry group. This is a finite-dimensional Lie
group, whose generators are Killing vectors. The finite-dimensional Lie algebra of
isometries coincides with the Lie algebra of the Killing vectors with the Killing