Asymptotic symmetries of three dimensional Anti-de-Sitter space from a perspective of representation theory

Abstract

As Wigner showed in 1939 for the Poincaré algebra, fundamental particles can be classified using symmetry algebras. In a universe including gravity, the Poincaré algebra cannot be the correct symmetry algebra, as this is the symmetry algebra for flat space. Instead, one should consider a symmetry algebra of asymptotic symmetries, which preserve only the asymptotic structure of gravity. Many of these asymptotic symmetries are not physically useful and are therefore considered 'trivial'. In this thesis we give a new, quantum, definition of trivial symmetries, namely that a symmetry is trivial if it does not change which fundamental particles are found in a classification. We then specialize to three-dimensional asymptotically Anti-de-Sitter space. To calculate which symmetries are trivial we first determine the second cohomology group of the asymptotic symmetries.
Using the second cohomology group it is then found that the useful asymptotic symmetry algebra is given by $\mathfrak w\oplus \mathfrak w\oplus \mathbb R$, where $\mathfrak w$ is the Witt (centerless Virasoro) algebra, whereas the standard definition of trivial symmetry gives $\mathfrak w\oplus \mathfrak w$ as the useful symmetry algebra. The extra factor of $\mathbb R$ is interpreted to be a kind of center-of-mass momentum.