Supergravity

Abstract

These notes are based on lectures presented at the 2001 Les Houches Summerschool ``Unity from Duality: Gravity, Gauge Theory and Strings''

Figures (5)

• Figure 1: States of the spinless representation in terms of the energy eigenvalues $E$ and the angular momentum $j$. Each point corresponds to the spherical harmonics of $S^{D-1}$: traceless, symmetric tensors $Y^{a_1\cdots a_l}$ of rank $l=j$.
• Figure 2: States of the spinor representation in terms of the energy eigenvalues $E$ and the angular momentum; the half-integer values for $j=l+{\frac{1}{2}}$ denote that we are dealing with a symmetric tensor-spinor of rank $l$. The small circles denote the original spinless multiplet from which the spinor multiplet has been constructed by a direct product with a spinor.
• Figure 3: The spin-0 and spin-${\frac{1}{2}}$ singleton representations. The solid dots indicate the states of the spin-0 singleton, the circles the states of the spin-${\frac{1}{2}}$ singleton. It is obvious that singletons contain much less degrees of freedom than a generic local field. The value of $E_0$, which denotes the spin-0 ground state energy, is equal to $E_0={\frac{1}{2}}(D-3)$. The spin-${\frac{1}{2}}$ singleton ground state has an energy which is one half unit higher, as is explained in the text.
• Figure 4: States of the spin-$1$ representation in terms of the energy eigenvalues $E$ and the angular momentum $j$. Observe that there are now points with double occupancy, indicated by the circle superimposed on the dots and states transforming as mixed tensors (with $l=j$) denoted by a $\star$. The double-occupancy points exhibit the structure of a spin-0 multiplet with ground state energy $E_0+1$. This multiplet becomes reducible and can be dropped when $E_0=D-2$, as is explained in the text. The remaining points then constitute a massless spin-$1$ multiplet, shown in fig. \ref{['massless-spin1-irrep']}.
• Figure 5: States of the massless $s=1$ representation in terms of the energy eigenvalues $E$ and the angular momentum $j$. Now $E_0$ is no longer arbitrary but it is fixed to $E_0=D-2$.