N=2 Topological Yang-Mills Theory on Compact K\"{a}hler Surfaces

We study a topological Yang-Mills theory with $N=2$ fermionic symmetry. Our formalism is a field theoretical interpretation of the Donaldson polynomial invariants on compact K\"{a}hler surfaces. We also study an analogous theory on compact oriented Riemann surfaces and briefly discuss a possible application of the Witten's non-Abelian localization formula to the problems in the case of compact K\"{a}hler surfaces.Comment: ESENAT-93-01 & YUMS-93-10, 34pages: [Final Version] to appear in Comm. Math. Phy

Quantum Metrology: Towards an alternative definition for the meter

The motivation for this article came from an attempt to give an alternative definition for the meter, the SI unit for measuring length. As a starting point towards this goal, in this piece of work we present the underlying theory behind our approach which uses ideas from quantum field theory and noncommutative geometry, in particular the notion of an odd K-cycle which is based on the Dirac operator (and its inverse, the Dirac propagator). Using (the perhaps more familiar) physics terminology, the key point in our strategy is this: instead of measuring length directly in space-time we measure the "algebraic (spectral) length" in the space of the corresponding quantum states of some particle (fermion) acted upon by the Dirac propagator. This approach shares the spirit of the unanimus vote of the 24th General Conference of Standards and Measures (21st October 2011) in Serves, France for the redefinition of the fundamental units using Planck's constant.Comment: Extended version of an invited talk during the 4th Tactical Conference on Metrology, 3-4 February 2012, National Technical University of Athens, Athens Greec