Topology of Singularities on Algebraic Loop Spaces

TL;DR

This paper investigates the topology of algebraic formal loop spaces, revealing conditions under which certain cohomologies vanish or produce simplified results, advancing understanding of infinite-dimensional singularities.

Contribution

It introduces new results on the cohomology of algebraic loop space singularities, including vanishing theorems and renormalized cohomology computations.

Findings

Nearby cycles cohomology vanishes in some cases.

Renormalized cohomology yields a non-zero, simplified result.

Provides insights into the topology of infinite-dimensional singularities.

Abstract

We study the topology of some simple infinite dimensional singularities arising from spaces of \emph{algebraic formal loops}. We prove that in some simple cases the natural analogue of nearby cycles cohomology for a function on the loop space vanishes, and show further that a suitably renormalized version of the above cohomology produces a simple (non-zero) result.