  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 $LT$.
