Character Formulas from Matrix Factorisations

(With an Appendix by Constantin Teleman) In the spirit of Freed, Hopkins, and Teleman I establish an equivalence between the category of discrete series representations of a real semisimple Lie group G and a category of equivariant matrix factorisations on a subset of the dual of the Lie algebra, in analogy with the situation in [FT] which treated the case when G is compact or a loop group thereof. The equivalence is implemented by a version of the Dirac operator used in [FHT1-3], squaring to the superpotential W defining the matrix factorisations. Using the structure of the resulting matrix factorisation category as developed in [FT] I deduce the Kirillov character formula for compact Lie groups and the Rossman character formula for the discrete series of a real semi-simple Lie group. The proofs are a calculation of Chern characters and use the Dirac family constructed in [FHT1-3]. Indeed, the main theorems of [FHT3] and [FT] are a categorification of the Kirillov correspondence, and this paper establishes that this correspondence can be recovered at the level of characters.Comment: Appendix (by Constantin Teleman) to appear in final versio

Completed K-theory and Equivariant Elliptic Cohomology

Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of $S^1$-equivariant $K$-theory for spaces. I present a $G$-equivariant version of their construction, which is a completed version of the Freed-Hopkins-Teleman model of $K$-theory for local quotient groupoids and resolves the issues concerning twisting and degree that arise in a first attempt to relate their work to elliptic cohomology.Comment: 23 page

Strongly nonlinear convection in binary fluids: Minimal model for extended states using symmetry decomposed modes

Spatially extended stationary and traveling states in the strongly nonlinear regime of convection in layers of binary fluid mixtures heated from below are described by a few-mode-model. It is derived from the proper hydrodynamic balance equations including experimentally relevant boundary conditions with a non-standard Galerkin approximation that uses numerically obtained, symmetry decomposed modes. Properties of the model are elucidated and compared with full numerical solutions of the field equations.Comment: 16 pages, including 5 postscript figure