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**Example text**

In finite dimensions, determinants are exponentiated traces of logarithms; we extend weighted traces to logarithms of pseudo-differential operators in order to define determinants in infinite dimensions. p. tr((log A)Q−z ) |z=0 . (12) As before, Q is referred to as the weight and tr Q (log A) as the Q-weighted trace of log A. Underlying this definition, is a choice of a determination of the logarithm which we shall not make explicit in the notation unless it is strictly necessary. ∗adm (M, E) with orders q , q Theorem [O] (see also [Du]).

1 + 1 ⊗ ∇AW ), A ∈ C(W )}. (43) + Associated to the family {DA , A ∈ C(W )}, there is a determinant bundle LD + on X = C(W ). We set as before − + := DA DA ∗ , + A − + := DA DA , − A + − := DA DA , A := + A ⊕ − A. The gauge group action. The gauge group G := C ∞ (M, Aut (W )) is a Fr´echet Lie group with Lie algebra Lie (G) := C ∞ (M, H om(W )). If W = adP , where P → M is a trivial principal G bundle, G the structure group, then Lie (G) := C ∞ (M, Lie(G)), where Lie(G) is the Lie algebra of G.

Moreover, Q being self-adjoint the range of Q is given by R(Q) = (ker Q∗ )⊥ = (ker Q)⊥ so that Q := Q + PQ is onto. Q being injective and onto is invertible and ∗adm (M, E) (it has the same being self-adjoint, and therefore admissible, it lies in Ellord>0 Q Q order as Q) and we can define tr (A), resp. str (A). p. tr(Ae− Q ) | , =0 resp. p. str(Ae− Q ) | . (3) =0 We pay a price for having left out divergences when taking the finite part of otherwise diverging expressions, namely the occurrence of weighted trace anomalies.