The tangent bundle of an almost-complex free loopspace

Abstract

The space LV of free loops on a manifold V inherits an action of the circle group T. When V has an almost-complex structure, the tangent bundle of the free loopspace, pulled back to a certain inÞnite cyclic cover g , has an equivariant decomposition as a completion of TV a (aC(k)), where TV is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of TV along evaluation at the basepoint (and aC(k) denotes an algebra of Laurent polynomials). On a sat manifold, this analogue of Fourier analysis is classical. The purpose of this note is to show that the study of the equivariant tangent bundle of a free smooth loopspace can be reduced to the study of a certain Þnitedimensional vector bundle over that loopspace – at least, provided the underlying manifold has an almost-complex structure (e.g. it might be symplectic), and if we are willing to work over a certain interesting inÞnite-cyclic cover of the loopspace. The Þrst section below summarizes the basic facts we’ll need from equivariant differential topology and geometry, and the second is a quick account of the universal cover of a symmetric product of circles, which is used in the third section to construct the promised decomposition of the equivariant tangent bundle. It is interesting that the covering transformations and the circle act compatibly on the tangent bundle of the covering, while their action on the splitting commutes only up to a projective factor.