Geometry and analysis on isospectral sets. II: Grassmannian, determinant bundle and the tau function, asymptotic case (Q1912264)

The author defines the infinite dimensional Grassmannian \(G\) over the Hilbert space \(L^2[0,1]\) polarized into the even and odd functions; this is a Hilbert manifold modelled on the space of Hilbert-Schmidt operators. The determinant bundle and tau function are constructed over \(G\) and the group action is derived; this is in contrast to the usual construction involving regularized determinants. For fixed elements of the isospectral set and the variation of the parameter \(x\), the second group which acts on \(G\) is derived. A unified construction for both group actions can be carried out by comparing it to the Fock bundle construction of 3+1 dimensional Dirac-Yang-Mills theories. [For part I see ibid., No. 2, 93-117 (1996).].