The Quantum Structure of Space and Time (293 pages)

102 The Quantum Structure of Space and Time

4.1.3.3 Functorial description

In the functorial description of conformal field theory the maps are abstracted
away from the concrete sigma model definition. Starting point is now an arbitrary
(closed, oriented) Riemann surface C with boundary. This boundary consists of
a collections of oriented circles. One declares these circles in-coming or out-going
depending on whether their orientation matches that of the surface C or not. To
a surface C with m in-coming and n out-going boundaries one associates a linear

map

These maps are not independent but satisfy gluing axioms that generalize the simple
composition law (1)

where C is obtained by gluing C1 and C2 on their out-going and incoming boundaries

respectively.

In this way we obtain what is known as a modular functor. It has a rich algebraic

structure. For instance, the sphere with three holes

gives rise to a product

Using the fact that a sphere with four holes can be glued together from two copies

of the three-holed sphere one shows that this product is essentially commutative

and associative

Once translated in terms of transition amplitudes, these relation lead to non-trivial

differential equations and integrable hierarchies. For more details see e.g. [4, 201.