Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds

We extend classical Euclidean stability theorems corresponding to the nonrelativistic Hamiltonians of ions with one electron to the setting of non parabolic Riemannian 3-manifolds.Comment: 20 pages; to appear in Annales Henri Poincar

Once more on the Witten index of 3d supersymmetric YM-CS theory

The problem of counting the vacuum states in the supersymmetric 3d Yang-Mills-Chern-Simons theory is reconsidered. We resolve the controversy between its original calculation by Witten at large volumes and the calculation based on the evaluation of the effective Lagrangian in the small volume limit. We show that the latter calculation suffers from uncertainties associated with the singularities in the moduli space of classical vacua where the Born-Oppenheimer approximation breaks down. We also show that these singularities can be accurately treated in the Hamiltonian Born-Oppenheimer method, where one has to match carefully the effective wave functions on the Abelian valley and the wave functions of reduced non-Abelian QM theory near the singularities. This gives the same result as original Witten's calculation.Comment: 27 page

Infinite dimensional degree theory and Ramer's finite co-dimensional differential forms

Infinite-dimensional degree theory, especially for Fredholm maps with positive index as developed with Tromba, is combined with Ramer’s unpublished thesis work on finite co-dimensional differential forms. As an illustrative example, the approach of Nicolaescu and Savale to the Gauss–Bonnet–Chern theorem for vector bundles is reworked in this framework. Other examples mentioned are Kokarev and Kuksin’s approach to periodic differential equations and to forced harmonic maps. A discussion about how such forms and their constructions and cohomology relate to constructions for diffusion measures on path and loop spaces is also included