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An index theorem for gauge-invariant families: The case of solvable groups

Victor Nistor
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Abstract

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family $\GR \to B$ of Lie groups (these families are called ``gauge-invariant families'' in what follows). If the fibers of $\GR \to B$ are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah-Singer type formula that incorporates also topological information about the bundle $\GR \to B$. The algebras of invariant pseudodifferential operators that we study, $\Psm {\infty}Y$ and $\PsS {\infty}Y$, are generalizations of ``parameter dependent'' algebras of pseudodifferential operators (with parameter in $\mathbb R^q$), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex.

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Victor Nistor. 2002-01-21. An index theorem for gauge-invariant families: The case of solvable groups. https://arxiv.org/abs/math/0201196

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