# An Analytic LT-equivariant Index and Noncommutative Geometry

## TL;DR

This paper develops an analytic $LT$-equivariant index theory for infinite-dimensional manifolds with loop group actions, connecting noncommutative geometry, spectral triples, and applications like Borel-Weil theory.

## Contribution

It constructs a novel $LT$-equivariant index framework using spectral triples and noncommutative geometry for infinite-dimensional manifolds with loop group actions.

## Key findings

- Defined an $LT$-equivariant index valued in the Verlinde ring.
- Constructed a twisted crossed product $LT$-C*-algebra.
- Applied the theory to Borel-Weil for loop groups.

## Abstract

Let $T$ be a circle and $LT$ be its loop group. Let $\mathcal{M}$ be an infinite dimensional manifold equipped with a nice $LT$-action. We construct an analytic $LT$-equivariant index for $\mathcal{M}$, and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space $\mathcal{H}$ consisting of "$L^2$-sections of a Clifford module bundle" and a "Dirac operator" $\mathcal{D}$ which acts on $\mathcal{H}$. Then, we define an analytic index of $\mathcal{D}$ valued in the representation group of $LT$, so called Verlinde ring. We also define a "twisted crossed product $LT\ltimes_\tau C_0(\mathcal{M})$," although we cannot define each concept "function algebra for $\mathcal{M}$ vanishing at infinity," "function from $LT$ to a $C^*$-algebra vanishing at infinity," and a Haar measure on $LT$. Moreover we combine all of them in terms of spectral triples and verify that the triple has an infinite spectral dimension. Lastly, we add some applications including Borel-Weil theory for $LT$.