Abstract:We define a classical $(1+1)$-dimensional conformal field theory whose fields on space-time (a conformal surface) are maps to a Riemannian string manifold $M$. Another way to view the construction is to say that it gives the spinor bundle over the loop space $LM$, together with a {\em conformal} connection. We explain why the precise analogue of our construction in dimension $(0+1)$ leads to the heat kernel of the Dirac operator on $M$ via the Feynman-Kac formula. Such a Lagrangian quantization unfortunately does not yet exist to give the desired operator on $LM$.
On the way, we give precise definitions and proofs for the facts that orientations of $LM$ are in canonical 1-1 correspondence to spin structures on $M$, and that spin structures on $LM$ are in canonical 1-1 correspondence to string structures on $M$. These facts have been previously known only on the existence side, and under the additional assumption that $M$ is $1$- respectively $2$-connected. Our new idea is to make systematic use of the {\em fusion} operation on $LM$, which already has been successfully applied to define a level preserving product of finite energy representations of loop groups.