Quantum fields as derivatives

The basic concept in euclidean quantum field theory is the
partition function, which for given space dimension D can be defined
as a map from the space of compact D-dimensional real manifolds with
Riemannian metric to the positive real numbers. This map should be smooth
with respect to local changes of the metric, including those that change
the topology. Given a partition function, quantum fields can be defined
as the corresponding derivatives with respect to the metric.
For conformally covariant partition functions and D=2 this approach is
calculationally effective. The simplest example relates to well known
mathematical structures: Hypergeometric functions that are algebraic,
the Rogers-Ramanujan functions and Deligne's exceptional series.