We study index theory for some special infinite-dimensional manifolds
equipped with a “proper cocompact” action of the loop group
of the
circle
,
from the viewpoint of noncommutative geometry. We introduce
-equivariant
-theory and construct
three
-elements:
the index element, the Clifford symbol element and the Dirac element.
These elements satisfy a certain equality, which should be called the
(-theoretical) index
theorem, or the
-theoretical
Poincaré duality for infinite-dimensional manifolds. We also discuss the assembly
maps.
Keywords
index theory, noncommutative geometry, equivariant
$KK\mkern-2mu$-theory, loop groups, infinite-dimensional
Heisenberg group, $C^*$-algebra of Hilbert space
