The New Loop Space Index Theorems and String Theory

Abstract

These lectures are concerned with index theorems as seen from the point of view of field theory: not with the various uses of index theorems in field and string theory —like the study of anomalies—, but merely with the use of supersymmetric quantum field theory to prove (or maybe more accurately to derive) index theorems. Of course we are eventually more interested by the physical consequences of the theorems than by their proofs, but I would like to convince you that in the process of understanding the structure of physical theories —in this case field and string theories— one often recovers deep mathematical results and sometimes discovers exciting new ones. As we will see it is quite remarkable that one of the most profound result of the last twenty years in mathematics, the Atiyah-Singer index theorem [1][2], has an extremely simple rewriting in terms of quantum mechanics language. But perhaps the most surprising fact is that the field theoretical approach leads naturally to extensions of this theorem to infinite dimensional spaces, namely the space of loops of any given compact closed manifold. In fact the story I will try to present has three very different versions and I am going to concentrate mainly on one of them which stems from the study of the Dirac-Ramond operator introduced long time ago in the study of string theory (see for example [3]). This operator is the generalization to loop space of the Dirac operator and we are going to compute its index just as easily as one computes the index of the Dirac operator [4,5,6,7,8,9]. The second version of the story, due to Schellekens and Warner [10][11], came independently from the study of a generating function for all the field theory anomalies coming from string theory.