Representation theory and projective geometry

We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.Comment: 37 pages with picture

Aspects of Space-Time Dualities

Duality groups of Abelian gauge theories on four manifolds and their reduction to two dimensions are considered. The duality groups include elements that relate different space-times in addition to relating different gauge-coupling matrices. We interpret (some of) such dualities as the geometrical symmetries of compactified theories in higher dimensions. In particular, we consider compactifications of a (self-dual) 2-form in 6-D, and compactifications of a self-dual 4-form in 10-D. Relations with a self-dual superstring in 6-D and with the type IIB superstring are discussed.Comment: 10 pages, plain Latex. Minor misprint correcte

Modular Symmetries of N=2 Black Holes

We discuss the transformation properties of classical extremal N=2 black hole solutions in S-T-U like models under S and T duality. Using invariants of (subgroups of) the triality group, which is the symmetry group of the classical BPS mass formula, the transformation properties of the moduli on the event horizon and of the entropy under these transformations become manifest. We also comment on quantum corrections and we make a conjecture for the one-loop corrected entropy.Comment: 12 pages, LaTe

Comments about quantum symmetries of SU(3) graphs

For the SU(3) system of graphs generalizing the ADE Dynkin digrams in the classification of modular invariant partition functions in CFT, we present a general collection of algebraic objects and relations that describe fusion properties and quantum symmetries associated with the corresponding Ocneanu quantum groupo\"{i}ds. We also summarize the properties of the individual members of this system.Comment: 36 page

Octonionic Cayley Spinors and E6

Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group E6, and of its subgroups. We are therefore led to a description of E6 in terms of 3x3 octonionic matrices, generalizing previous results in the 2x2 case. Our treatment naturally includes a description of several important subgroups of E6, notably G2, F4, and (the double cover of) SO(9,1), An interpretation of the actions of these groups on the squares of 3-component "Cayley spinors" is suggested.Comment: 14 pages, 1 figure, contributed talk at 2nd Mile High Conference (Denver 2009

Massive Neutrinos and (Heterotic) String Theory

String theories in principle address the origin and values of the quark and lepton masses. Perhaps the small values of neutrino masses could be explained generically in string theory even if it is more difficult to calculate individual values, or perhaps some string constructions could be favored by generating small neutrino masses. We examine this issue in the context of the well-known three-family standard-like Z_3 heterotic orbifolds, where the theory is well enough known to construct the corresponding operators allowed by string selection rules, and analyze the D- and F-flatness conditions. Surprisingly, we find that a simple see-saw mechanism does not arise. It is not clear whether this is a property of this construction, or of orbifolds more generally, or of string theory itself. Extended see-saw mechanisms may be allowed; more analysis will be needed to settle that issue. We briefly speculate on their form if allowed and on the possibility of alternatives, such as small Dirac masses and triplet see-saws. The smallness of neutrino masses may be a powerful probe of string constructions in general. We also find further evidence that there are only 20 inequivalent models in this class, which affects the counting of string vacua.Comment: 18 pages in RevTeX format. Single-column postscript version available at http://sage.hep.upenn.edu/~bnelson/singpre.p

Automorphic forms: a physicist's survey

Motivated by issues in string theory and M-theory, we provide a pedestrian introduction to automorphic forms and theta series, emphasizing examples rather than generality.Comment: 22 pages, to appear in the Proceedings of Les Houches Winter School ``Frontiers in Number Theory, Physics and Geometry'', March 9-21, 2003; v2: minor changes and clarifications, section 3.5 on pure spinors has been rewritte