Mathematically speaking, perturbative string theory is the study of certain natural functors on the category of Riemann surfaces or complex curves. This relationship is easily explained. A fundamental string is a loop in space. If we consider the history of such a string in space-time, it will sweep out a two-dimensional surface known as a world-sheet. The internal consistency of the theory demands that this world-sheet carries a complex structure and thus becomes a Riemann surface. For a free propagating string the topology of this worldsheet is a cylinder, but if we include interactions, under which the string can split and join, the surfaces can have arbitrary topology and represent the possible classical trajectories of a collection of interacting strings.
It is a very general principle that in any quantum theory we should associate an amplitude to any possible trajectory. Concretely this means that if we have a Riemann surface with, say, n incoming boundaries and m outgoing boundaries, as depicted in figure 1, there will be a linear map associated to the surface, where is the Hilbert space describing the posssible quantum states of a single string. These maps are not arbitrary, but satisfy all kinds of consistency relations, making it into a functor from the category of Riemann surfaces to the category of Hilbert spaces. Examples of such functors can be defined by maps of the surface into a complex manifold, explaining the relevance to the problem of counting curves in algebraic geometry. The simple existence of the `pair of pants' surface, the three-holed sphere, immediately tells us that there is some kind of natural algebraic structure indicating the relation with representation theory.
Since string theory is a quantum theory, Feynman's principle of sums over histories tells us to consider not only one particular surface but to sum over all possible topologies, i.e. all genera , and integrate over all possible complex structures. In this way we are led to consider formal expressions as
where we integrate the amplitudes over the moduli space of Riemann surfaces of genus g. Here the string coupling constant or Planck's constant controls the perturbative quantisation; the classical theory is recovered in the limit where only spherical topologies contribute.
Unfortunately, straightforward estimates show that the above expression for can be at most an asymptotic expansion. So the definition of string theory in terms of Riemann surfaces is at least incomplete. This is a well-known phenomenon in quantum field theories, where the Feynman diagrams rarely capture the full story. To add insult to injury, there furthermore seem to be five independent perturbative string theories, i.e. consistent functors , that differ rather dramatically in their basic properties, destroying the uniqueness of the theory.