Constructing stable Hilbert bundles via Diophantine approximation

Abstract

On any complex smooth projective curve with positive genus, we construct Hilbert bundles that admit Hermitian--Einstein metrics. Our main constructive step is by investigating the arithmetic property of the upper half plane in Bridgeland's definition of stability conditions and its homological countparts. The main analytic ingredient in our proof is a notion called a well-approximating sequence of stable bundles. This notion helps us to apply the Diophantine approximation to Donaldson's functional and bound the $L^\infty$ norm of Hermitian-Einstein metrics. We further study the continuous structures, smooth structures, and holomorphic structures on such Hilbert bundles. We hope that this construction can shed some new light on the geometric background of quantum field theory.