Harmonic expansions in solutions in d=11 supergravity with SU(3)×SU(2)×U(1) or SU(2)×SU(2)×SU(2)×(1) symmetry

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Abstract
The spectrum of the scalar Laplacian on the SU(3)*SU(2)*U(1) invariant spaces introduced by Witten is calculated. Contrary is previous results in the literature, it is shown that the harmonics can carry non-zero triality representations of SU(3). Similar results hold also for spinor harmonics, and indeed the quark triplet representation can occur on certain of the spaces. The scalar spectrum on the Qpqr spaces with SU(2)*SU(2)*SU(2)*U(1) symmetry is also calculated. The Q111 space, which yields on N=2 supersymmetric theory in four dimensions, is shown to give rise to additional massless Wess-Zumino supermultiplets over and above the expected graviton and Yang-Mills supermultiplets. There are also further massless scalars in massive supermultiplets.

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