Semi-Infinite Theory

In this seminar, we shall introduce semi-infinite manifolds and show how one may define cohomology theories dependent on this semiinfinite structure. In particular, we shall define a de Rham theory. With certain alterations, this is calculable for Wiener manifolds. The goal, however, is to calculate the theory for loop spaces and this is, as yet, an unsolved problem. 1 Semi-Infinite Theory The term “semi-infinite theory” is a rather loose one. It describes a collection of work all of which have a common theme. This theme is the concept of being “half-way” between zero and infinity. The principle behind “semi-infinity”, or “infinity over two” as some prefer, is the following: we choose some well-defined point which represents half way between zero and infinity and consider finite differences to this point. To illustrate this, consider N sitting inside Z. This is a prime candidate for a semi-infinite subset of Z since it is infinite itself but also has an infinite complement. We consider subsets of Z which differ from N by a finite amount. Thus a semi-infinite subset of Z is a set which contains almost all the positive numbers and almost no negative ones. En route to manifolds, the next step is to look at vector spaces. The analogue of subsets is closed subspaces and so the semi-infinite concept when translated into vector space theory tells us that we should be looking at objects of the form X = X− ⊕ X+ where X± are closed, infinite dimensional subspaces. Choosing one particular such decomposition, we consider decompositions X = X ′ −⊕X ′ + which differ from the chosen one by a finite amount.