# Non-commutative analytic torsion form on the transformation groupoid convolution algebra

## TL;DR

This paper develops a non-commutative analytic torsion form for transformation groupoids, extending classical index theory to non-commutative spaces with applications to Riemann-Roch-Grothendieck formulas.

## Contribution

It introduces a new construction of analytic torsion forms in non-commutative geometry for groupoid convolution algebras, under weaker spectral assumptions.

## Key findings

- Constructed a well-defined non-commutative torsion form
- Derived a transgression formula involving the torsion form
- Established a non-commutative Riemann-Roch-Grothendieck index formula

## Abstract

Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our construction is well defined under the weaker assumption of positive Novikov-Shubin invariant. We prove that this torsion form appears in a transgression formula, from which a non-commutative Riamannian-Roch-Grothendieck index formula follows.