Semi-Infinite de Rham Theory

I shall start by explaining just what “semi-infinite” means and how it gives rise to interesting structures in geometry. These structures are studied as part of String Theory and are closely related to Floer Theory. Manifolds which carry a “semi-infinite” structure include loop spaces (C∞(S1, M) for M a closed manifold) and the theory seems to have particular simplicities when the original manifold is symplectic. In time, it is hoped to develop many semi-infinite variants of “ordinary” geometrical objects such as K-Theory and Index Theory. For the moment, the main area of research appears to be into semi-infinite cohomology. Floer theory can be thought of as a “semi-infinite” variant of Morse theory. There is also a “semi-infinite” Lie algebra theory put forward by Feigin and Frenkel amongst others. I shall outline how one can construct a semi-infinite de Rham theory. Along the way, I hope to illustrate how one deals in general with semiinfinite objects (by “taming” them). I shall also give some indications of where elements of the construction can be used as starting points for considering semi-infinite K-Theory and Index Theory, and how the natural circle action on the loop space can be factored in.