As Mazur says in his 2004 paper in the Bulletin of the AMS: "one can learn a lot about a mathematical object by studying how it behaves under small perturbations." In this vein, it is natural to ask what happens to the Killing superalgebra under such perturbations. But first some prehistory...
One of my main interests during my graduate student days was homological algebra. Incidentally, this was sparked by feedback from Alan Guth's to an essay for his MIT "freshman seminar" on Cosmology and Particle Physics. My essay was about Kałuża–Klein reduction and contained a discussion of Maxwell's equations in the language of differential forms. With the naiveté of youth, I asserted that dF = 0 implied that F = dA and Alan correctly pointed out that whereas this was OK in Minkowski space, in general it depends on the cohomology of the spacetime. That was the first time I had come across the notion of cohomology and from that point on I was intrigued by it in all its manifestations. Along the way I came across the fact that the theory of deformations demands a cohomology theory, in the sense that deformations are always controlled by some cohomology theory, which if it doesn't exist has to be invented for the purpose. Clearly one had to understand deformations!
During the winter break of 1988-9, and as a desperate attempt to relieve the sheer boredom of spending Christmas in Florida, I studied deformation theory of Lie algebras. The motivation, apart from studying the deformation theory of objects I was familiar with, came from Physics. We postulate symmetry principles on the basis of empirical data which are subject to errors, whence in the absence of a compelling theoretical reason why the symmetry ought to be exact, one is forced to view it as only approximate. A case in point is the Galilean group: a kinematical symmetry known to be approximately valid for speeds small compared to that of light. I asked myself the question — which had surely been asked before — of which kinematical symmetries were "close" to the Galilean symmetry. Such a question can be answered systematically using the theory of Lie algebra deformations, which is controlled, for example, by the cohomology theory of Lie algebras in the sense of Chevalley and Eilenberg. My answer to this question was given in my paper on the deformations of the Galilean algebra.
More than 18 years later (¡que se dice pronto!) I was motivated again to think about deformations, this time of Killing superalgebras. Supergravity is after all an effective theory, be it of a string theory, M-theory,... Assuming that the Killing superalgebra persists under stringy/quantum corrections, one is led to wonder about its fate. This line of investigation is of necessity rather speculative because it has not been demonstrated — or even argued convincingly — that the notion of Killing superalgebra persists. Even if it persists, it is not clear that it persists as a Lie superalgebra — after all, a Lie (super)algebra has a number of alternative equivalent characterisation, e.g., as a Hopf (super)algebra or as a differential graded (super)algebra and there is little reason to choose one category over the other. But following popular wisdom (e.g., if it ain't broke, don't fix it!), I decided as a first step to consider deformations in the category of Lie superalgebras. Furthermore I made the simplifying assumption that the Killing superalgebra does not change dimension, which is itself perhaps a strong hypothesis.
The result of this investigation is contained in two papers. In Deformations of M-theory Killing superalgebras, I calculated the deformations of many M-theory backgrounds with large Killing superalgebras. This restriction is one of convenience, but also of relevance. The best known M-theory backgrounds are (perhaps not surprisingly) highly symmetric, and it is only for superalgebras with large semisimple factors that the deformation analysis is tractable, thanks to the factorisation theorem of Hochschild and Serre. Most backgrounds analysed turn out to have rigid Killing superalgebras, but I did find integrable one-parameter deformations of the Killing superalgebras of the M2 brane and of the Kałuża–Klein monopole. The M2 Killing superalgebra deformation suggests that the worldvolume of the brane becomes AdS₃, whereas that of the Kałuża–Klein monopole suggests a non-geometric deformation of the background!
Bert Vercnocke, a student of Toine Van Proeyen's, was visiting me in 2007, on an Early Stage Researcher contract from the EU RTN "ForcesUniverse" to which both Edinburgh and Leuven belong. I was fortunate to enlist his help in this project and together we calculated the Killing superalgebra deformations of a number of ten-dimensional supergravity backgrounds. Our results, contained in our joint paper, revealed an interesting property of the Killing superalgebra deformations, namely their good behaviour under Kałuża–Klein reduction. This suggests that these deformations are geometric in origin, and hence in some sense meaningful. The question remains whether this meaning is to be found within supergravity or, on the contrary, whether one has to find it in the full quantum theory.