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

TL;DR

This paper establishes an index theorem for gauge-invariant families of elliptic operators under solvable Lie group actions, extending Atiyah-Singer formulas to parameter-dependent pseudodifferential operators and analyzing boundary conditions.

Contribution

It introduces the gauge-equivariant index for families invariant under solvable Lie group actions and computes its Chern character using an Atiyah-Singer type formula, including topological data.

Findings

Derived an index theorem for gauge-invariant elliptic families with solvable group actions.

Computed the Chern character of the gauge-equivariant index incorporating topological information.

Extended index theory to parameter-dependent pseudodifferential operators and boundary conditions.

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.