The role of singletons in S7S^7 compactifications

We derive the isometry irrep content of squashed seven-sphere compactifications of eleven-dimensional supergravity, i.e., the left-squashed ($LS^7$) with ${\mathcal N}=1$ and right-squashed ($RS^7$) with ${\mathcal N}=0$ supersymmetry, in a manner completely independent of the round sphere. Then we compare this result with the spectrum obtained by Higgsing the round sphere spectrum. This way we discover features of the spectra which makes it possible to argue that the only way the round spectrum can be related by a Higgs mechanism to the one of $LS^7$ is if the singletons are included in the round sphere spectrum. For this to work also in the $RS^7$ case it seems that the gravitino of the $LS^7$ spectrum must be replaced by a fermionic singleton present in the $RS^7$ spectrum.Comment: 24 pages including appendix with 12 figure, v2 minor typos correcte

Permutation invariance, partition algebras and large N matrix models

Models with degrees of freedom that naturally arrange themselves into matrices have a long history in science. Statistical models of large Hermitian matrices are believed to capture universal features of chaotic quantum systems. The various connections between matrix theory and string theory have been so prolific that one might argue that matrix models capture generic features of string theories. The first sign of this connection (gauge-string duality) was discovered by 't Hooft, where string worldsheets emerge from the combinatorics of Feynman diagrams in U(N → ∞) Yang-Mills theory. Many aspects of this emergence can be understood from the mathematical duality known as Schur-Weyl duality. It relates two algebraic structures: the representation theory of U(N) and the representation theory of symmetric groups $S_k$. This has implications for U(N) matrix models where observables find an eloquent description in terms of the group algebras $\mathbb{C}(S_k)$. The duality underlies the geometric construction of gauge-string duality, where string worldsheets emerge from a connection between symmetric group elements, ribbon graphs and Riemann surfaces. In this thesis we will study matrix models with discrete gauge group $S_N$. We will put these matrix models into a generalized Schur-Weyl duality framework where dual algebras, known as partition algebras, emerge. These form generalizations of the symmetric group algebras -- they are semi-simple finite-dimensional associative algebras with a basis labelled by diagrams. We review the structure and representation theory of partition algebras. These algebras are then used to compute expectation values of $S_N$ invariant observables. This is a step towards studying the emergence of new geometric structures in their Feynman diagram expansion. Matrix models also appear in the form of quantum mechanical models of matrix oscillators. We explore the implications of the Schur-Weyl duality framework to matrix quantum mechanics with permutation symmetry