Index Theorems and Superstring Compactification.
This dissertation addresses some issues in the compactification of superstring theories, particularly the low-energy spectrum of the resulting four-dimensional theory, using techniques which are a generalization of the Atiyah-Singer index theorem. Any symmetries of the internal space K become symmetries of the low-energy lagrangian; continuous symmetries become gauge symmetries and discrete symmetries become global symmetries. The action of these symmetries on the low-energy fields can be partially determined using character-valued index theorems. A new derivation of these theorems using the techniques of supersymmetric quantum mechanics is presented here. The first result is that these symmetries act on the low-energy matter generations in a more general way than would be possible in conventional grand unified models in four dimensions. The particular possibility of family-dependent gauge transformations is discussed in some detail, and an example of such a symmetry is given. The second application to the potential appearance of massless E(,6) singlets, with the physical consequence of breaking E(,6) gauge symmetry down to an O(10) or SU(5) subgroup. Mathematically, these massless singlets are related to harmonic vector-bundle-valued differential forms whose ordinary index vanishes. The character-valued indices do not necessarily vanish, however, and when they do not they signal the presence of these massless singlets.
- Pub Date:
- Physics: Elementary Particles and High Energy