The Determinant Line Bundle over Moduli Spaces of Instantons on Abelian Surfaces

. We study the determinant line bundle over moduli space of stable bundles on abelian surfaces. We evaluate their analytic torsions. We extend Mukai's version of the Parseval Theorem to L 2 metrics on cohomology groups. We prove that the Mukai transform preserves the determinant line bundle as a hermitian line bundle. This is done by induction via the natural boundary of the moduli spaces. Introduction The determinant map in homological algebra which assigns to a vector space its top exterior power was extended to sheaves by Grothendieck and Ferrand in a paper which was intended to appear in [SGA6] but did not. The details were later published by Mumford and Knudsen in [KM]. Subsequently, Quillen showed that there is another way to define a line bundle over families of @ operators on Riemann surfaces ([Q]). In this case the determinant line bundle carries a natural hermitian structure. This work was extended by Bismut and Freed in [BF] to more general families of differential operat..