The Tangent Bundle of an Almost-complex Free Loopspace

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 infinite cyclic cover g LV , has an equivariant decomposition as a completion of TV ⊗ (⊕C(k)), where TV is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of TV along evaluation at the basepoint (and ⊕C(k) denotes an algebra of Laurent polynomials). On a flat 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 finitedimensional 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 infinite-cyclic cover of the loopspace. The first 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. 1. The free loopspace and its universal cover 1.1 If V is a connected compact almost-complex manifold of real dimension 2n, the space of smooth maps from the circle S = {x ∈ C | |z| = 1} to V is an (infinite-dimensional) manifold LV , with local charts defined by the vector bundle neighborhoods of [5 §13]; the tangent space at the loop σ is a vector space TσLV = ΓS1(σTV ) of sections of the pullback of the tangent bundle along σ. The circle group T acts on LV by rotating loops, and rotation through the angle α lifts to the complex-linear transformation α∗ : TσLV → Tσ◦αLV Date: 1 May 2001. 1991 Mathematics Subject Classification. 58Dxx; 53C29, 55P91. The author was supported in part by the NSF. 1