Determinant Bundles over Grassmannians
Denoting by H the Hilbert space of square-integrable Dirac spinor fields on a manifold M, transforming according to a unitary representation p of a gauge group G, we have a linear representation of the group g of gauge transformations in the space H. If ρ is faithful we can consider g as a subgroup of the general linear group GL(H). By constructing representations of GL(H) we automatically obtain representations of g. It turns out that in the case when the dimension d of M is odd, g is contained in a smaller group GL p ⊂ GL(H) which has the property that it perturbs the subspace H+ ⊂ H consisting of eigenvectors of a Dirac operator belonging to positive eigenvalues, by an operator A for which the trace tr|A|2p exists. The Schatten index depends on the dimension of M. The statement above is true when p ≥ (d + 1)/2.
KeywordsDirac Operator Central Extension Fredholm Operator Hermitian Form Spin Bundle
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