Minuscule representations, invariant polynomials, and spectral covers

Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy classes of the Lie algebra. There are partial results in the case of a quasiminuscule representation, and a conjecture in the case of a general irreducible finite-dimensional representation. The method of proof is to relate the question to a problem concerning holomorphic principal bundles over cuspidal cubic curves.Comment: LaTeX, 42 pages, final version, to appear in the proceedings of the University of Missouri conference on Hilbert schemes, vector bundles and representation theory, new material on extensions and the adjoint representation of a simply laced Lie algebra adde

Existence and uniqueness theorems for massless fields on a class of spacetimes with closed timelike curves

We study the massless scalar field on asymptotically flat spacetimes with closed timelike curves (CTC's), in which all future-directed CTC's traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries of this type, and for smooth initial data with all derivatives in $L_2$ on {\cI}^{-}, we prove existence of smooth solutions which are regular at null and spatial infinity (have finite energy and finite $L_2$-norm) and have the given initial data on \cI^-. A restricted uniqueness theorem is obtained, applying to solutions that fall off in time at any fixed spatial position. For a complementary class of spacetimes in which CTC's are confined to a compact region, we show that when solutions exist they are unique in regions exterior to the CTC's. (We believe that more stringent uniqueness theorems hold, and that the present limitations are our own.) An extension of these results to Maxwell fields and massless spinor fields is sketched. Finally, we discuss a conjecture that the Cauchy problem for free fields is well defined in the presence of CTC's whenever the problem is well-posed in the geometric-optics limit. We provide some evidence in support of this conjecture, and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular, both existence and uniqueness can fail in smooth, asymptotically flat spacetimes with a compact nonchronal region.Comment: 47 pages, Revtex, 7 figures (available upon request